Recent zbMATH articles in MSC 34https://zbmath.org/atom/cc/342021-01-08T12:24:00+00:00WerkzeugNumerical solution of fractional differential equation by wavelets and hybrid functions.https://zbmath.org/1449.651692021-01-08T12:24:00+00:00"Refahi Sheikhani, A. H."https://zbmath.org/authors/?q=ai:sheikhani.amirhossein-refahi"Mashoof, M."https://zbmath.org/authors/?q=ai:mashoof.mahamadSummary: In this paper, we introduce methods based on operational matrix of fractional order integration for solving a typical \(n\)-term non-homogeneous fractional differential equation (FDE). We use Block pulse wavelets matrix of fractional order integration where a fractional derivative is defined in the Caputo sense. Also we consider Hybrid of Block-pulse functions and shifted Legendre polynomials to approximate functions. By the use of these methods we translate an FDE to an algebraic linear equations which can be solved. Methods have been tested by some numerical examples.Some oscillation criteria for certain second-order differential equations.https://zbmath.org/1449.342282021-01-08T12:24:00+00:00"Qin, Guijiang"https://zbmath.org/authors/?q=ai:qin.guijiang"Ji, Zhanjiang"https://zbmath.org/authors/?q=ai:ji.zhanjiang"Lu, Zhenkun"https://zbmath.org/authors/?q=ai:lu.zhenkunSummary: We study the oscillatory behavior of second-order generalized Emden-Fowler-type differential equations with a nonlinear neutral term. By using the generalized Riccati transformation and integral averaging technique, we establish some new oscillation criteria for the equations. Some examples are provided to show that our results generalize and improve known results.Stability of two-dimensional complex oscillator networks with time delay.https://zbmath.org/1449.342462021-01-08T12:24:00+00:00"Li, Xue"https://zbmath.org/authors/?q=ai:li.xue"Xu, Xu"https://zbmath.org/authors/?q=ai:xu.xu.2|xu.xuSummary: We consider the stability of two-dimensional complex oscillator network systems with time delay. By analyzing the stability switching criteria, we discuss the difference of the stability switching between complex and real systems. Extending the ring structure to the two-dimensional case makes it possible to generate or the storage visual patterns.A modified Euler method for solving fuzzy differential equations under generalized differentiability.https://zbmath.org/1449.340042021-01-08T12:24:00+00:00"Ahmady, N."https://zbmath.org/authors/?q=ai:ahmady.n"Allahviranloo, T."https://zbmath.org/authors/?q=ai:allahviranloo.tofigh"Ahmady, E."https://zbmath.org/authors/?q=ai:ahmady.elhamSummary: In this paper, we intend to introduce a modified approach for solving fuzzy differential equations (FDEs) under generalized differentiability. Modified Euler method estimated FDEs by using a two-stage predictor-corrector algorithm with local truncation error of order two. The consistency, convergence, and stability of the proposed method are also investigated in detail. The acceptable accuracy of the modified Euler method is illustrated by some examples.Some coincidence and common fixed point theorems concerning \(F\)-contraction and applications.https://zbmath.org/1449.541012021-01-08T12:24:00+00:00"Tomar, Anita"https://zbmath.org/authors/?q=ai:tomar.anita"Sharma, Ritu"https://zbmath.org/authors/?q=ai:sharma.rituSummary: The aim of this paper is to establish coincidence and common fixed point theorems for a discontinuous noncompatible pair of self-maps in noncomplete metric space without containment requirement of range space of involved maps acknowledging the notion of \(F\)-contraction introduced by \textit{D. Wardowski} [Fixed Point Theory Appl. 2012, Paper No. 94, 6 p. (2012; Zbl 1310.54074)]. Our results generalize, extend and improve analogous results existing in the literature and are supported with the help of illustrative examples associated with pictographic validations to demonstrate the authenticity of the postulates. Solutions of two-point boundary value problem of a second order differential equation arising in electric circuit and a Volterra type integral equation using Ćirić type as well as Hardy-Rogers-type \(F\)-crontactions are also given to exhibit the usability of obtained results.Periodic solutions for seasonally forced SIRS model with pulse vaccination.https://zbmath.org/1449.341232021-01-08T12:24:00+00:00"Wang, Lin"https://zbmath.org/authors/?q=ai:wang.lin.2|wang.lin.1|wang.lin|wang.lin.3|wang.lin.4"Pang, Yanni"https://zbmath.org/authors/?q=ai:pang.yanni"Li, Wenjin"https://zbmath.org/authors/?q=ai:li.wenjinSummary: Using the coincidence degree theory of Gaines-Mawhin, we prove the existence of periodic solutions for seasonally forced SIRS models with pulse vaccination. The effects of different loss of immunity rates on the infectious disease models are compared with numerical simulations.On the solutions of a Caputo-Katugampola fractional integro-differential inclusion.https://zbmath.org/1449.450132021-01-08T12:24:00+00:00"Cernea, Aurelian"https://zbmath.org/authors/?q=ai:cernea.aurelianSummary: We consider a Cauchy problem associated to an integro-differential inclusion of fractional order defined by Caputo-Katugampola derivative and by a set-valued map with nonconvex values and we prove that the set of selections corresponding to the solutions of the problem considered is a retract of the space of integrable functions on unbounded interval.Relationship between number of zeros of nonlinear term and number of positive solutions of second-order periodic boundary value problems.https://zbmath.org/1449.340892021-01-08T12:24:00+00:00"Wang, Suyun"https://zbmath.org/authors/?q=ai:wang.suyun"Wei, Jinying"https://zbmath.org/authors/?q=ai:wei.jinying"Zhang, Yanhong"https://zbmath.org/authors/?q=ai:zhang.yanhongSummary: By using the fixed point index theorem, we give the existence of multiple positive solutions for periodic boundary value problems of second-order ordinary differential equations: \[\begin{cases}
u''-qu + \lambda f (u) = 0,\, t \in (0, T),\\
u (0) = u (T),\, u' (0) = u' (T),
\end{cases}\]
where \(0 < q < +\infty\), \(f \in C ([0;\infty), [0;\infty))\), \(f (a_i) = 0\), \(f (b_i) = 0\) and \(f (u) > 0\) in \((a_i,b_i)\) with \(a_i,b_i \in (0, \infty)\) and \(a_i < b_i \leq a_{i+1} < b_{i+1}\) for \(i =1,2,\cdots, n\). The results reveal the relationship between the number of zeros of the nonlinear term \(f\) and the number of positive solutions of periodic boundary value problems.Existence of solutions for ordinary differential equations with non-instantaneous impulses in Banach space.https://zbmath.org/1449.340512021-01-08T12:24:00+00:00"Xin, Zhen"https://zbmath.org/authors/?q=ai:xin.zhen"Chen, Pengyu"https://zbmath.org/authors/?q=ai:chen.pengyuSummary: By using the fixed point theorem of \(k\)-set contraction mapping and a new estimation of the measure of noncompactness, we prove the existence of the solution for the initial value problem of ordinary differential equations with non-instantaneous impulses, and then obtain the existence of the solution for the initial value problem of ordinary differential equations with non-instantaneous impulses under the assumption that the nonlinear term satisfies some weaker growth condition and noncompactness measure condition, and the non-instantaneous impulsive functions satisfies some Lipschitz conditions.Existence of positive solutions for nonhomogeneous boundary value problems of fractional differential equations with sign changing nonlinearities.https://zbmath.org/1449.340772021-01-08T12:24:00+00:00"Li, Lin"https://zbmath.org/authors/?q=ai:li.lin.1|li.lin.2|li.lin"Jia, Mei"https://zbmath.org/authors/?q=ai:jia.mei"Liu, Xiping"https://zbmath.org/authors/?q=ai:liu.xiping"Song, Junqiu"https://zbmath.org/authors/?q=ai:song.junqiuSummary: We consider the existence of positive solutions for a class of nonhomogeneous integral boundary value problems of fractional differential equations with sign changing nonlinearities. By using fixed point theorems of cone expansion and cone compression, we establish and prove the existence of positive solutions for the boundary value problem, and give some examples to illustrate the conclusions.Existence, uniqueness and global asymptotic stability for a class of complex-valued neutral-type neural networks with time delays.https://zbmath.org/1449.370222021-01-08T12:24:00+00:00"Tan, Manchun"https://zbmath.org/authors/?q=ai:tan.manchun"Xu, Desheng"https://zbmath.org/authors/?q=ai:xu.deshengSummary: This paper explores the problem of delay-independent and delay-dependent stability for a class of complex-valued neutral-type neural networks with time delays. Aiming at the neutral-type neural networks, an appropriate function is constructed to derive the existence of equilibrium point. On the basis of homeomorphism theory, Lyapunov functional method and linear matrix inequality techniques, several LMI-based sufficient conditions on the existence, uniqueness and global asymptotic stability of equilibrium point for complex-valued neutral-type neural networks are obtained. Finally, numerical examples are given to illustrate the feasibility and the effectiveness of the proposed theoretical results.The growth of meromorphic solutions of homogeneous and non-homogeneous complex linear equations for composite functions.https://zbmath.org/1449.343152021-01-08T12:24:00+00:00"Chen, Haiying"https://zbmath.org/authors/?q=ai:chen.haiying"Zheng, Xiumin"https://zbmath.org/authors/?q=ai:zheng.xiuminSummary: The growth of meromorphic solutions of a class of homogenous and non-homogeneous complex linear equations for composite functions with meromorphic coefficients is investigated by the Nevanlinna's value distribution of meromorphic function, which is generalized to the more general case of complex linear differential equations for composite functions. When more than one coefficient of the involved equations have the maximal order or the maximal lower order, some estimates on the lower bound of the order or the lower order of non-zero meromorphic solutions of involved equations are obtained under some conditions.A criterion for the existence and uniqueness of solution involving fractional order and impulsive boundary conditions.https://zbmath.org/1449.341022021-01-08T12:24:00+00:00"Zheng, Fengxia"https://zbmath.org/authors/?q=ai:zheng.fengxia"He, Cong"https://zbmath.org/authors/?q=ai:he.cong"Tang, Yuping"https://zbmath.org/authors/?q=ai:tang.yupingSummary: By using the fixed point theorem for the sum of operators, a criterion for the existence and uniqueness of solution involving fractional order and impulsive boundary conditions was obtained. Also, an iterative sequence was constructed to approximate the solution. To illustrate the main results, an example was given in the paper.Sufficient condition of Hilbertness of some eigenfunctions systems of the second order differential operator.https://zbmath.org/1449.343122021-01-08T12:24:00+00:00"Tsareva, A. S."https://zbmath.org/authors/?q=ai:tsareva.a-sSummary: We obtain a sufficient condition for the Hilbertness of the system of eigenfunctions of an ordinary linear non-selfadjoined differential operator of the second order under the condition that the imaginary parts of spectral parameters are large enough by absolute value.The dynamic behaviors of a new impulsive predator prey model with impulsive control at different fixed moments.https://zbmath.org/1449.341612021-01-08T12:24:00+00:00"Wang, Linjun"https://zbmath.org/authors/?q=ai:wang.linjun"Xie, Youxiang"https://zbmath.org/authors/?q=ai:xie.youxiang"Deng, Qicheng"https://zbmath.org/authors/?q=ai:deng.qichengSummary: In this paper, we propose a new impulsive predator prey model with impulsive control at different fixed moments and analyze its interesting dynamic behavior. Sufficient conditions for the globally asymptotical stability of the semi-trivial periodic solution and the permanence of the present model are obtained by Floquet theory of impulsive differential equation and small amplitude perturbation theory. Existences of the ``infection-free'' periodic solution and the ``predator-free'' solution are analyzed by bifurcation theory of impulsive differential equation. Finally, the analytical results presented in the work are illustrated by numerical simulation figures for this proposed model.Converse theorem for practical stability of nonlinear impulsive systems and applications.https://zbmath.org/1449.341942021-01-08T12:24:00+00:00"Ghanmi, Boulbaba"https://zbmath.org/authors/?q=ai:ghanmi.boulbaba"Dlala, Mohsen"https://zbmath.org/authors/?q=ai:dlala.mohsen"Hammami, Mohamed Ali"https://zbmath.org/authors/?q=ai:hammami.mohamed-aliSummary: The Lyapunov's second method is one of the most famous techniques for studying the stability properties of dynamic systems. This technique uses an auxiliary function, called Lyapunov function, which checks the stability properties of a specific system without the need to generate system solutions. An important question is about the reversibility or converse of Lyapunov's second method; i.e., given a specific stability property does there exist an appropriate Lyapunov function? The main result of this paper is a converse Lyapunov theorem for practical asymptotic stable impulsive systems. Applying our converse theorem, several criteria on practical asymptotic stability of the solution of perturbed impulsive systems and cascade impulsive systems are established. Finally, some examples are given to show the effectiveness of the derived results.Global attractivity of solutions for a class of multi-term fractional differential equations.https://zbmath.org/1449.341992021-01-08T12:24:00+00:00"Li, Yanfeng"https://zbmath.org/authors/?q=ai:li.yanfeng"Hao, Yanpeng"https://zbmath.org/authors/?q=ai:hao.yanpeng"Wang, Erjing"https://zbmath.org/authors/?q=ai:wang.erjing"Li, Qiaoluan"https://zbmath.org/authors/?q=ai:li.qiaoluanSummary: In this paper, we present results for the global attractivity of solutions of fractional differential equations involving Caputo-Katugampola fractional calculus. By transforming the differential equations into an integral equations, the existence of the solutions is obtained by using the Schauder's fixed point theorem.Anti-periodic solutions for time scale dynamic equations with exponential dichotomy.https://zbmath.org/1449.343222021-01-08T12:24:00+00:00"Meng, Xin"https://zbmath.org/authors/?q=ai:meng.xin"Lv, Xin"https://zbmath.org/authors/?q=ai:lv.xinSummary: We consider anti-periodic solutions for a class of time scale dynamic equations with exponential dichotomy. By applying the Banach fixed point theorem, we give sufficient conditions for the existence of anti-periodic solutions for nonhomogeneous linear time scale dynamic equations and semi-linear time scale dynamic equations, and give some examples to illustrate the applicability of the main results in practical problems.Lower approximation of systems of differential inclusions of one kind with slow and fast variables.https://zbmath.org/1449.340592021-01-08T12:24:00+00:00"Sokolovskaya, E. V."https://zbmath.org/authors/?q=ai:sokolovskaya.e-vSummary: We prove a theorem on lower approximation of systems of differential inclusions with slow and fast variables and one-sided Lipschitz condition.Stability analysis of a two-dose measles model.https://zbmath.org/1449.341582021-01-08T12:24:00+00:00"Sun, Dandan"https://zbmath.org/authors/?q=ai:sun.dandan"Zhang, Tailei"https://zbmath.org/authors/?q=ai:zhang.taileiSummary: In this paper, we establish an \(SV_1V_2EIR\) measles model with two vaccinations and incubation period. Firstly, we obtain the equilibrium point and the basic reproduction number of the model, and non-negativity and boundedness of the model solutions are determined by using comparison principle. Using the linearization, Hurwitz criterion and Lyapunov functions, we obtain the global asymptotic stability of the infection-free equilibrium when \({\mathcal{R}_0} < 1\). When \({\mathcal{R}_0} > 1\), the infection-free equilibrium is unstable, and there exists a unique globally asymptotically stable positive equilibrium. Finally, we simulate the model and study the key factors affecting measles transmission by sensitivity analysis and partial rank correlation coefficients. Combining with the real patient data from 2007 to 2017 in China, the model was simulated. The results of numerical simulations show the measles epidemic trends in China.Oscillation of impulsive differential equation with positive and negative coefficients.https://zbmath.org/1449.342212021-01-08T12:24:00+00:00"Chen, Jie"https://zbmath.org/authors/?q=ai:chen.jie.3|chen.jie|chen.jie.7|chen.jie.6|chen.jie.9|chen.jie.8|chen.jie.10|chen.jie.4|chen.jie.5|chen.jie.2|chen.jie.1"Shen, Jianhua"https://zbmath.org/authors/?q=ai:shen.jianhua|shen.jianhua.1Summary: This paper studies the oscillation properties of the solutions to the impulsive delay differential equation with positive and negative coefficients \[\begin{cases}[x (t)-R (t)x (t-r)]' + P(t)x (t-\tau) - Q (t)x (t-\sigma) = 0,\; t \ge {t_0}, \\ x (t_k^+) = {b_k}x (t_k),\; k = 1, 2, \cdots,\end{cases}\] where \(R (t), P(t), Q (t) \in PC ([{t_0},\infty), \text bf{R}^+)\), \(r > 0, \tau \ge 0, \sigma \ge 0\) are some constants, \(\{t_k\}\) and \(\{b_k\}\) are real sequences satisfying certain conditions.A mathematical basis for the graphene.https://zbmath.org/1449.343072021-01-08T12:24:00+00:00"Conca, Carlos"https://zbmath.org/authors/?q=ai:conca.carlos"Martín, Jorge San"https://zbmath.org/authors/?q=ai:san-martin.jorge-alonso"Solano, Viviana"https://zbmath.org/authors/?q=ai:solano.vivianaSummary: We present a new basis of representation for the graphene honeycomb structure that facilitates the solution of the eigenvalue problem by reducing it to one dimension. We define spaces in these geometrical basis that allow us to solve the Hamiltonian in the edges of the lattice. We conclude that it is enough to analyze a one-dimensional problem in a set of coupled ordinary second-order differential equations to obtain the behavior of the solutions in the whole graphene structure.Asymptotical stability of numerical methods for semi-linear impulsive differential equations.https://zbmath.org/1449.651562021-01-08T12:24:00+00:00"Zhang, Gui-Lai"https://zbmath.org/authors/?q=ai:zhang.guilaiSummary: This paper is concerned with asymptotical stability of a class of semi-linear impulsive ordinary differential equations. First of all, sufficient conditions for asymptotical stability of the exact solutions of semi-linear impulsive differential equations are provided. Under the sufficient conditions, some explicit exponential Runge-Kutta methods can preserve asymptotically stability without additional restriction on stepsizes. Moreover, it is proved that some explicit Runge-Kutta methods can preserve asymptotical stability without additional restriction on stepsizes under stronger conditions.On some fractional integro-differential inclusions with nonlocal multi-point boundary conditions.https://zbmath.org/1449.450122021-01-08T12:24:00+00:00"Cernea, Aurelian"https://zbmath.org/authors/?q=ai:cernea.aurelianSummary: Existence of solutions for two classes of fractional integro-differential inclusions with nonlocal multi-point boundary conditions is investigated in the case when the values of the set-valued map are not convex.Existence of mild solutions of second order evolution integro-differential equations in the Fréchet spaces.https://zbmath.org/1449.342662021-01-08T12:24:00+00:00"Jawahdou, Adel"https://zbmath.org/authors/?q=ai:jawahdou.adelSummary: In this article, we shall establish sufficient conditions for the existence of mild solutions for second order semilinear integro-differential evolution equations in Fréchet spaces \(C(\mathbb{R}_+ , E)\), where \(E\) is a Banach space. Our approach is based on the concept of a measure of noncompactness and Tychonoff fixed point theorem. For illustration we give an example.Limit cycle flutter and chaotic motion of two-dimensional airfoil system.https://zbmath.org/1449.341392021-01-08T12:24:00+00:00"He, Dongping"https://zbmath.org/authors/?q=ai:he.dongping"Huang, Wentao"https://zbmath.org/authors/?q=ai:huang.wentao"Wang, Qinlong"https://zbmath.org/authors/?q=ai:wang.qinlongSummary: Limit cycle flutter and the motion of chaos of two-dimensional airfoil with quadratic nonlinear pitching stiffness in incompressible flow on nonzero equilibrium points are investigated. The center manifold theory is used to reduce a four-dimensional system to a two-dimensional system, and the bifurcation points of the system are determined by bifurcation theory. The type and stability of bifurcation points are determined by computing focal values of system. The type of Hopf bifurcation is determined by the second Lyapunov method of bifurcation problem. The theoretical analysis presented here provides a good agreement with numerical simulations obtained by using a fourth-order Runge-Kutta method. Furthermore, the way leads to chaos in the airfoil system is found and there exits large field of the period-five motion. The results indicate that the bifurcation point is a stable weak focus, when the supercritical Hopf occurs, there exists a stable limit cycle. The motion of chaos occurs due to period-doubling bifurcation.Horseshoe dynamics in Duffing oscillator with fractional damping and multi-frequency excitation.https://zbmath.org/1449.370272021-01-08T12:24:00+00:00"Priyatharsini, S. Valli"https://zbmath.org/authors/?q=ai:priyatharsini.s-valli"Meenakshi, M. V. Sethu"https://zbmath.org/authors/?q=ai:sethu-meenakshi.m-v"Chinnathambi, V."https://zbmath.org/authors/?q=ai:chinnathambi.v"Rajasekar, S."https://zbmath.org/authors/?q=ai:rajasekar.shanmuganathanSummary: The occurrence of horseshoe chaos in Duffing oscillator with fractional damping and multi-frequency excitation is analyzed by using analytical and numerical techniques. Applying Melnikov method, analytical threshold condition for the onset of horseshoe chaos is obtained. The effect of damping exponent and the number of periodic forces on the dynamics of the Duffing oscillator is also analyzed. Due to fractional damping and multi-frequency excitation, suppression of chaos and various nonlinear phenomena are predicted. Analytical predictions are demonstrated through numerical simulations.On the graphene Hamiltonian operator.https://zbmath.org/1449.343082021-01-08T12:24:00+00:00"Conca, C."https://zbmath.org/authors/?q=ai:conca.carlos"Orive, R."https://zbmath.org/authors/?q=ai:orive.rafael"Martín, J. San"https://zbmath.org/authors/?q=ai:martin.jo-san|san-martin.jorge-alonso"Solano, V."https://zbmath.org/authors/?q=ai:solano.vivianaSummary: We solve a second-order elliptic equation with quasi-periodic boundary conditions defined on a honeycomb lattice that represents the arrangement of carbon atoms in graphene. Our results generalize those found by \textit{P. Kuchment} and \textit{O. Post} [Commun. Math. Phys. 275, No. 3, 805--826 (2007; Zbl 1145.81032)] to characterize not only the stability but also the instability intervals of the solutions. This characterization is obtained from the solutions of the energy eigenvalue problem given by the lattice Hamiltonian. We employ tools of the one-dimensional Floquet theory and specify under which conditions the one-dimensional theory is applicable to the structure of graphene. The systematic study of such stability and instability regions provides a tool to understand the propagation properties and behavior of the electrons wavefunction in a hexagonal lattice, a key problem in graphene-based technologies.On the inverse problem for Sturm-Liouville-type operators with frozen argument: rational case.https://zbmath.org/1449.342652021-01-08T12:24:00+00:00"Buterin, Sergey"https://zbmath.org/authors/?q=ai:buterin.sergey-alexandrovich"Kuznetsova, Maria"https://zbmath.org/authors/?q=ai:kuznetsova.mariya-andreevnaSummary: We study the inverse problem of recovering the potential \(q(x)\) from the spectrum of the operator
\[-y''(x)+q(x)y(a),y^{(\alpha )}(0)=y^{(\beta )}(1)=0,\]
where \(\alpha ,\beta \in \{0,1\}\) and \(a\in [0,1]\) is an arbitrary fixed rational number. We completely describe the cases when the solution of the inverse problem is unique and non-unique. In the last case, we describe sets of iso-spectral potentials and provide various restrictions on the potential under which the uniqueness holds. Moreover, we obtain an algorithm for solving the inverse problem along with necessary and sufficient conditions for its solvability in terms of characterization of the spectrum.Solvability analysis of a special type fractional differential system.https://zbmath.org/1449.340282021-01-08T12:24:00+00:00"Marynets, Kateryna"https://zbmath.org/authors/?q=ai:marynets.katerynaSummary: Some new results are obtained in the investigation of solutions of boundary-value problems (BVPs) for fractional differential systems (FDS), subject to anti-periodic boundary conditions. The approximate solution of the given BVP is built in the form of successive sequences of functions by using main ideas of the numerical-analytic technique [the author, Electron. J. Qual. Theory Differ. Equ. 2016, Paper No. 6, 14 p. (2016; Zbl 1363.34056); \textit{M. I. Ronto} and the author, Nonlinear Oscil., N.Y. 14, 379--413 (2012; Zbl 1334.34053); \textit{M. Rontó} et al., Tatra Mt. Math. Publ. 63, 247--267 (2015; Zbl 06545436)].A class of delayed HIV-1 infection models with latently infected cells.https://zbmath.org/1449.920502021-01-08T12:24:00+00:00"Yang, Junxian"https://zbmath.org/authors/?q=ai:yang.junxian"Xie, Baoying"https://zbmath.org/authors/?q=ai:xie.baoyingSummary: A class of delayed HIV-1 infection models with latently infected cells was proposed. The basic reproductive number \(R_0 \) was defined, and the existence conditions of disease-free equilibrium \(P_0 (x_0, 0, 0, 0)\) and chronic-infection equilibrium \(P^* (x^*, \omega^*, y^*, v^*)\) were given. First, the local asymptotic stability of infection-free equilibrium and chronic-infection equilibrium was obtained by the linearization method. Further, by constructing the corresponding Lyapunov functions and using LaSalle's invariant principle, it was proved that when the basic reproductive number \(R_0 \) was less than or equal to unity, the infection-free equilibrium \(P_0 (x_0, 0, 0, 0)\) was globally asymptotically stable. Moreover, when the basic reproductive number \(R_0 \) was greater than unity, the chronic-infective equilibrium \(P^* (x^* \omega^*, y^*, v^*)\) was globally asymptotically stable, but the infection-free equilibrium \(P_0 (x_0, 0, 0, 0)\) was unstable. The results showed that a latently infected delay and an intracellular delay did not affect the global stability of the model, and numerical simulations were carried out to illustrate the theoretical results.Oscillation of second-order generalized Emden-Fowler-type differential equations.https://zbmath.org/1449.342222021-01-08T12:24:00+00:00"Li, Jimeng"https://zbmath.org/authors/?q=ai:li.jimengSummary: The oscillatory behavior of a class of second-order generalized Emden-Fowler-type nonlinear variable delay neutral functional differential equations is studied in this article. By using the generalized Riccati transformation and some analytic techniques, we establish two new oscillation criteria for the equations under the condition \(\int_{t_0}^{+\infty} a^{-1/\beta} (t){\mathrm{d}}t < +\infty\). Illustrative examples are provided to show that our results extend and improve those previously reported in the literature, and the results are both practical and implementable.Stability of the stochastic \(\theta\)-method for super-linear stochastic differential equations with unbounded delay.https://zbmath.org/1449.650042021-01-08T12:24:00+00:00"Chen, Lin"https://zbmath.org/authors/?q=ai:chen.lin.2|chen.lin|chen.lin.6|chen.lin.4|chen.lin.1|chen.lin.5|chen.lin.3Summary: This paper deals with numerical stability properties of super-linear stochastic differential equations with unbounded delay. Sufficient conditions for mean square and almost sure decay stability of the above system and its stochastic \(\theta\)-method approximation are investigated in this paper. The author establishes numerical stability under a monotone-type condition in unbounded delay setting. An example is presented to illustrate the result.The Green's function of fourth-order difference equation with periodic boundary value problem.https://zbmath.org/1449.390172021-01-08T12:24:00+00:00"Jiang, Lingfang"https://zbmath.org/authors/?q=ai:jiang.lingfang"Liu, Aihua"https://zbmath.org/authors/?q=ai:liu.aihuaSummary: In this paper, we study the Green's function of fourth-order difference equation with periodic boundary value problem. We obtain some new results and generalize some results in a literature.Numerical solutions of nonautonomous stochastic delay differential equations by discontinuous Galerkin methods.https://zbmath.org/1449.650052021-01-08T12:24:00+00:00"Dai, Xinjie"https://zbmath.org/authors/?q=ai:dai.xinjie"Xiao, Aiguo"https://zbmath.org/authors/?q=ai:xiao.aiguoSummary: This paper considers a class of discontinuous Galerkin method, which is constructed by Wong-Zakai approximation with the orthonormal Fourier basis, for numerically solving nonautonomous Stratonovich stochastic delay differential equations. We prove that the discontinuous Galerkin scheme is strongly convergent, globally stable and analogously asymptotically stable in mean square sense. In addition, this method can be easily extended to solve nonautonomous Stratonovich stochastic pantograph differential equations. Numerical tests indicate that the method has first-order and half-order strong mean square convergence, when the diffusion term is without delay and with delay, respectively.Non-linear complex differential-difference equations admit meromorphic solutions.https://zbmath.org/1449.300702021-01-08T12:24:00+00:00"Liu, K."https://zbmath.org/authors/?q=ai:liu.kangping|liu.kuikui|liu.keliang|liu.keqing|liu.kangsheng|liu.kejing|liu.kaiqing|liu.kexuan|liu.kai.4|liu.keying|liu.kaiyu|liu.kejian|liu.kaizhen|liu.kefu|liu.kaifeng|liu.kaidi|liu.kehfei|liu.kexiu|liu.kaiyuan|liu.kexin|liu.keqin|liu.kaixin|liu.kaituo|liu.kanglin|liu.keji|liu.kan|liu.kenneth|liu.kaihua|liu.keping|liu.kexiao|liu.kepan|liu.kongjie|liu.kefeng|liu.ketao|liu.kaiying|liu.kai.3|liu.kai.5|liu.kuang|liu.kai|liu.kexi|liu.kunlun|liu.kefei|liu.kaisheng|liu.kebin|liu.kairan|liu.kehui|liu.kunlin|liu.keguang|liu.kui|liu.kuangyu|liu.kairong|liu.kuan|liu.kai.1|liu.kang|liu.kaifu|liu.kai.2|liu.kimfung|liu.kunkun|liu.kaijun|liu.keqiang|liu.kunlong|liu.kunqi|liu.kecheng|liu.kaizhou|liu.kejia|liu.kaifang|liu.kangyan|liu.kaihui|liu.kean|liu.kaien|liu.kening|liu.kangqi|liu.kaihe|liu.ke|liu.kangjie|liu.kun|liu.kunhong|liu.kaishin|liu.kunhui|liu.kewei|liu.kelan|liu.kuo|liu.kangze|liu.kangling|liu.kangman|liu.kesheng|liu.kaiming|liu.kaile|liu.keyu|liu.kunming|liu.kangni|liu.kiang|liu.keke"Song, C. J."https://zbmath.org/authors/?q=ai:song.chong-jae|song.changjiang|song.chuanjing|song.changjin|song.chengjun|song.chengjuSummary: We obtain necessary conditions for the non-linear complex differential-difference equations \[w(z + 1)w(z - 1) + a(z)\frac{w'(z)}{w(z)}= R(z,w(z))\] to admit transcendental meromorphic solutions \(w(z)\) such that \(\rho_2(w) < 1\), where \(R(z,w(z))\) is rational in \(w(z)\) with rational coefficients, \(a(z)\) is a rational function and \(\rho_2(w)\) is the hyper-order of \(w(z)\). Our results can be seen as the product versions on an equation of another type investigated by \textit{R. Halburd} and \textit{R. Korhonen} [Proc. Am. Math. Soc. 145, No. 6, 2513--2526 (2017; Zbl 1361.30049)]. We also provide an idea which implies that the case of \(\deg_w(R(z,w)) = 4\) in the original proof of Theorem 1.1 of the above mentioned paper can be organized in a short way.Power means of the Hurwitz zeta function over large intervals.https://zbmath.org/1449.110902021-01-08T12:24:00+00:00"Ashton, A. C. L."https://zbmath.org/authors/?q=ai:ashton.anthony-c-lSummary: In this note, we derive asymptotic formulas for power means of the Hurwitz zeta function over large intervals.The asymptotic behavior of a stochastic chemostat model with Michaelis-Menten food chain.https://zbmath.org/1449.341812021-01-08T12:24:00+00:00"Zhao, Yihan"https://zbmath.org/authors/?q=ai:zhao.yihan"Yang, Zhichun"https://zbmath.org/authors/?q=ai:yang.zhichunSummary: This paper investigates the asymptotic behavior of a stochastic chemostat model with Michaelis-Menten food chain in which the dilution rate is disturbed by white noise. First, the global existence and uniqueness of the positive solution of the model is proved. Then, by constructing Lyapunov function and using Itô's formula, sufficient condition for the stochastic global asymptotic stability of the washout equilibrium of the model is obtained. Finally, the long-time asymptotic behavior of the solution of the model are studied, which mainly reveals the oscillatory behavior of the solution around the predator-free equilibrium and positive equilibrium of the corresponding deterministic model under different conditions. The results improve and extend the relevant work of the existing literature.Random attractors for a class of stochastic lattice systems with time delay in \({X_\rho}\) space.https://zbmath.org/1449.342682021-01-08T12:24:00+00:00"Zhang, Yijin"https://zbmath.org/authors/?q=ai:zhang.yijin(no abstract)Existence criteria of positive solutions for a system of Riemann-Liouville type \(p\)-Laplacian fractional order boundary value problems.https://zbmath.org/1449.340212021-01-08T12:24:00+00:00"Krushna, Boddu Muralee Bala"https://zbmath.org/authors/?q=ai:krushna.boddu-muralee-balaSummary: This paper is concerned with determining the eigenvalue intervals of \(\lambda_1\) and \(\lambda_2\) for which there exist positive solutions to a coupled system of Riemann-Liouville type \(p\)-Laplacian fractional order boundary value problems by utilizing a fixed point theorem on a cone under suitable conditions.Dual combination function projective synchronization of chaotic systems with disturbances.https://zbmath.org/1449.370642021-01-08T12:24:00+00:00"Fang, Jie"https://zbmath.org/authors/?q=ai:fang.jie"Zhu, Fei"https://zbmath.org/authors/?q=ai:zhu.fei"Lou, Xinjie"https://zbmath.org/authors/?q=ai:lou.xinjie"Liu, Hua"https://zbmath.org/authors/?q=ai:liu.hua"Deng, Wei"https://zbmath.org/authors/?q=ai:deng.weiSummary: Based on dual synchronization and combined synchronization, the dual combination function projective synchronization of four drive chaotic systems and two response chaotic systems is researched in this paper. An adaptive feedback controller is theoretically designed based on Lyapunov stability theory, tracking control thought and adaptive control method, by which the state variables of the response systems are able to track the chaotic trajectory of the drive systems according to the desired function matrices and overcome the effects of the unknown disturbances. When synchronization is achieved, the drive and response systems of the two sets of synchronization systems are able to be combined arbitrarily, and thereby enhancing the flexibility of the synchronization system. MATLAB numerical simulation verifies the correctness and effectiveness of the theoretical analysis.Lie symmetry and conserved quantity for singular nonholonomic systems of Chetaev's type on time scales.https://zbmath.org/1449.700202021-01-08T12:24:00+00:00"Chen, Zhiwei"https://zbmath.org/authors/?q=ai:chen.zhiwei"Zhu, Jianqing"https://zbmath.org/authors/?q=ai:zhu.jianqingSummary: The Lie symmetry and conserved quantities of singular nonholonomic systems of Chetaev's type on time scales were studied. Firstly, the motion differential equations were established. Secondly, based on the invariance of differential equations under infinitesimal transformation on time scales, we gave the determining equation and the limiting equation of Lie symmetry for singular nonholonomic systems of Chetaev's type on time scales. Finally, a structural equation was established, and the conservation of Lie symmetry was given. At the end of the paper, an example was given to illustrate the application of the theorem.Growth and fixed point of solutions for second-order differential equations in unit disc.https://zbmath.org/1449.343162021-01-08T12:24:00+00:00"Chen, Yu"https://zbmath.org/authors/?q=ai:chen.yu.2|chen.yu.1|chen.yu.6|chen.yu.8|chen.yu.4|chen.yu.3|chen.yu.5|chen.yu.7"Deng, Guantie"https://zbmath.org/authors/?q=ai:deng.guantie(no abstract)Birkhoffian formulations of Bessel equation.https://zbmath.org/1449.700212021-01-08T12:24:00+00:00"Jiang, Wen'an"https://zbmath.org/authors/?q=ai:jiang.wenan"Xia, Lili"https://zbmath.org/authors/?q=ai:xia.lili"Xu, Yanli"https://zbmath.org/authors/?q=ai:xu.yanliSummary: The Birkhoffian mechanics is more general than the Hamilton mechanics, but only some dynamical systems can be realized as a Birkhoffian formulation. This paper proposes a novel Birkhoffian formulation for the classical Bessel equation. Based on the first method of Santilli, the Birkhoffian formulation of Bessel equation is established under the assumption that the Birkhoffian describes the total physical energy of the corresponding conservative systems. Zero and \(n\)-th order classical Bessel equations are studied to verify the effectiveness of the proposed formulation.A class of singular perturbed problems with nonlinear mixed boundary value conditions for forth-order differential equation.https://zbmath.org/1449.342022021-01-08T12:24:00+00:00"Liu, Yan"https://zbmath.org/authors/?q=ai:liu.yan.5|liu.yan.8|liu.yan|liu.yan.7|liu.yan.2|liu.yan.3|liu.yan.4|liu.yan.1|liu.yan.6Summary: A class of nonlinear mixed boundary value problems with singular perturbation for fourth-order differential equation is studied. The formal asymptotic solutions are constructed by the composite expansion method. The existence of solutions for the original problem and the uniform validity of the formal asymptotic solutions are proved by using the theory of differential inequalities. Finally, an example is given to illustrate the significance of the results.Global attractors of a Duffing system.https://zbmath.org/1449.341132021-01-08T12:24:00+00:00"Feng, Jinqian"https://zbmath.org/authors/?q=ai:feng.jinqian"Liu, Ya'ni"https://zbmath.org/authors/?q=ai:liu.yani"Wang, Yingxiao"https://zbmath.org/authors/?q=ai:wang.yingxiao"Li, Yuting"https://zbmath.org/authors/?q=ai:li.yuting(no abstract)On the nonlinear \(varPsi\)-Hilfer fractional differential equations.https://zbmath.org/1449.340232021-01-08T12:24:00+00:00"Kucche, Kishor D."https://zbmath.org/authors/?q=ai:kucche.kishor-d"Mali, Ashwini D."https://zbmath.org/authors/?q=ai:mali.ashwini-d"Sousa, J. Vanterler da C."https://zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.joseSummary: We consider the nonlinear Cauchy problem for \(\varPsi\)-Hilfer fractional differential equations and investigate the existence, interval of existence and uniqueness of solution in the weighted space of functions. The continuous dependence of solutions on initial conditions is proved via Weissinger fixed point theorem. Picard's successive approximation method has been developed to solve the nonlinear Cauchy problem for differential equations with \(\varPsi\)-Hilfer fractional derivative and an estimation has been obtained for the error bound. Further, by Picard's successive approximation, we derive the representation formulae for the solution of linear Cauchy problem for \(\varPsi\)-Hilfer fractional differential equation with constant coefficient and variable coefficient in terms of Mittag-Leffler function and generalized (Kilbas-Saigo) Mittag-Leffler function respectively.Numerical approximation to Prabhakar fractional Sturm-Liouville problem.https://zbmath.org/1449.651632021-01-08T12:24:00+00:00"Derakhshan, Mohammad Hossein"https://zbmath.org/authors/?q=ai:derakhshan.mohammad-hossein"Ansari, Alireza"https://zbmath.org/authors/?q=ai:ansari.alirezaSummary: In this paper, we treat a numerical scheme for the regular fractional Sturm-Liouville problem containing the Prabhakar fractional derivatives with the mixed boundary conditions. We show that the eigenfunctions corresponding to distinct numerical eigenvalues are orthogonal in the Hilbert spaces. The numerical errors and convergence rates are also investigated. Further, we consider a space-fractional diffusion equation and study the associated fractional Sturm-Liouville problem along with the convergence analysis.Analysis of an SEIRS model with nonlinear incidence rate and treatment rate.https://zbmath.org/1449.341352021-01-08T12:24:00+00:00"Gao, Xueli"https://zbmath.org/authors/?q=ai:gao.xueli"Wang, Hui"https://zbmath.org/authors/?q=ai:wang.hui.5"Hu, Zhixing"https://zbmath.org/authors/?q=ai:hu.zhixingSummary: In this paper, an SEIRS model with saturated incidence rate and nonlinear treatment rate is studied. The existence of an equilibrium point of the model is studied, and then the basic reproductive number \({R_0}\) is discussed and a new threshold for disease control \(R_0^*\) is obtained. Then, we find that there exist the forward bifurcation and the backward bifurcation, and sufficient conditions for a bifurcation. By using the Routh-Hurwitz criterion, the disease-free equilibrium \({P_0}\) and the endemic equilibrium \({P^*}\) are studied. \({P_2}\) is locally asymptotically stable and the endemic equilibrium \({P_1}\) is unstable. The global stability of the local disease equilibrium point is predicted. Finally, the conclusion and guess are illustrated by numerical simulation.Global residue harmonic balance method to periodic solutions of a class of strongly nonlinear oscillators.https://zbmath.org/1449.340532021-01-08T12:24:00+00:00"Ju, Peijun"https://zbmath.org/authors/?q=ai:ju.peijun"Xue, Xin"https://zbmath.org/authors/?q=ai:xue.xinSummary: A new approach, namely the global residue harmonic balance method, is presented to determine analytical approximate periodic solution of a class of strongly nonlinear oscillators. A class of nonlinear jerk equation containing velocity-cubed and velocity times displacements-squared is taken as a typical example. Unlike other harmonic balance methods, all the former residual errors are introduced in the present approximation to improve the accuracy. A comparison of the result obtained using this approach with the exact one and simplicity and efficiency of the proposed procedure is performed. The method can be easily extended to other strongly nonlinear oscillators.Modified numerical approaches for a class of Volterra integral equations with proportional delays.https://zbmath.org/1449.653682021-01-08T12:24:00+00:00"Taghizadeh, Elham"https://zbmath.org/authors/?q=ai:taghizadeh.elham"Matinfar, Mashallah"https://zbmath.org/authors/?q=ai:matinfar.mashallahSummary: This paper addresses modified-meshless numerical schemes for dynamical systems with proportional delays. The proposed mesh reduction techniques are based on a redial-point interpolation and moving least-squares methods. An optimal influence domain radius is constructed utilizing nodal connectivity and node-depending integration background mesh. Optimal shape parameters are obtained by the use of properties of the delta Kronecker and the compactly supported weight function. Numerical results are provided to justify the accuracy and efficiency of the proposed schemes.Adaptive finite-time generalized outer synchronization between two different dimensional chaotic systems with noise perturbation.https://zbmath.org/1449.342102021-01-08T12:24:00+00:00"Ma, Zhi-cai"https://zbmath.org/authors/?q=ai:ma.zhicai"Wu, Jie"https://zbmath.org/authors/?q=ai:wu.jie.6"Sun, Yong-zheng"https://zbmath.org/authors/?q=ai:sun.yongzhengSummary: This paper is further concerned with the finite-time generalized outer synchronization between two different dimensional chaotic systems with noise perturbation via an adaptive controller. First of all, we introduce the definition of finite-time generalized outer synchronization between two different dimensional chaotic systems. Then, employing the finite-time stability theory, we design an adaptive feedback controller to realize the generalized outer synchronization between two different dimensional chaotic systems within a finite time. Moreover, we analyze the influence of control parameter on the synchronous speed. Finally, two typical examples are examined to illustrate the effectiveness and feasibility of the theoretical result.Duplex selections, equilibrium points, and viability tubes.https://zbmath.org/1449.340572021-01-08T12:24:00+00:00"Kánnai, Zoltán"https://zbmath.org/authors/?q=ai:kannai.zoltanSummary: Existence of viable trajectories to nonautonomous differential inclusions are proven for time-dependent viability tubes. In the convex case we prove a double-selection theorem and a new Scorza-Dragoni type lemma. Our result also provides a new and palpable proof for the equilibrium form of Kakutani's fixed point theorem.Synchronization of time-delayed systems with discontinuous coupling.https://zbmath.org/1449.342582021-01-08T12:24:00+00:00"Shi, Hong-jun"https://zbmath.org/authors/?q=ai:shi.hongjun"Miao, Lian-ying"https://zbmath.org/authors/?q=ai:miao.lianying"Sun, Yong-zheng"https://zbmath.org/authors/?q=ai:sun.yongzhengSummary: This paper concerns the synchronization of time-delayed systems with periodic on-off coupling. Based on the stability theory and the comparison theorem of time-delayed differential equations, sufficient conditions for complete synchronization of systems with constant delay and time-varying delay are established. Compared with the results based on the Krasovskii-Lyapunov method, the sufficient conditions established in this paper are less restrictive. The theoretical results show that two time-delayed systems can achieve complete synchronization when the average coupling strength is sufficiently large. Numeric evidence shows that the synchronization speed depends on the coupling strength, on-off rate and time delay.Numerical solution of mixed-type fractional functional differential equations using modified Lucas polynomials.https://zbmath.org/1449.651432021-01-08T12:24:00+00:00"Moghaddam, B. P."https://zbmath.org/authors/?q=ai:moghaddam.behrouz-parsa"Dabiri, A."https://zbmath.org/authors/?q=ai:dabiri.azita|dabiri.arman"Lopes, António M."https://zbmath.org/authors/?q=ai:lopes.antonio-m"Machado, J. A. Tenreiro"https://zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiroSummary: There is an increasing interest in the field of functional and fractional differential equations. The lack of closed-form analytical solutions motivates the development of numerical methods for solving mixed-type fractional-order functional differential equations (MFFDEs) with retarded and neutral terms. This paper studies the solution of MFFDEs by a collocation technique with modified Lucas polynomials. The proposed method uses operational matrices to obtain an approximate solution by means of a system of linear algebraic equations. The accuracy of the proposed algorithm is verified with three illustrative examples.Epidemic model of leptospirosis containing fractional order.https://zbmath.org/1449.920442021-01-08T12:24:00+00:00"Khan, Muhammad Altaf"https://zbmath.org/authors/?q=ai:khan.muhammad-altaf"Saddiq, S. F."https://zbmath.org/authors/?q=ai:saddiq.syed-farasat"Islam, Saeed"https://zbmath.org/authors/?q=ai:islam.saeed"Khan, Ilyas"https://zbmath.org/authors/?q=ai:khan.ilyas"Ching, Dennis Ling Chuan"https://zbmath.org/authors/?q=ai:ching.dennis-ling-chuanSummary: We study an epidemic model of leptospirosis in fractional order numerically. The multistep generalized differential transform method is applied to find the accurate approximate solution of the epidemic model of leptospirosis disease in fractional order. A unique positive solution for the epidemic model in fractional order is presented. For the integer case derivative, the approximate solution of MGDTM is compared with the Runge-Kutta order four scheme. The numerical results are presented for the justification purpose.An inventive numerical method for solving the most general form of integro-differential equations with functional delays and characteristic behavior of orthoexponential residual function.https://zbmath.org/1449.653642021-01-08T12:24:00+00:00"Kürkçü, Ömür Kıvanç"https://zbmath.org/authors/?q=ai:kurkcu.omur-kivanc"Aslan, Ersin"https://zbmath.org/authors/?q=ai:aslan.ersin"Sezer, Mehmet"https://zbmath.org/authors/?q=ai:sezer.mehmetSummary: In this study, we constitute the most general form of functional integro-differential equations with functional delays. An inventive method based on Dickson polynomials with the parameter-\( \alpha \) along with collocation points is employed to solve them. The stability of the solutions is simulated according to an interval of the parameter-\( \alpha \). A useful computer program is developed to obtain the precise values from the method. The residual error analysis is used to improve the obtained solutions. The characteristic behavior of the residual function is established with the aid of the orthoexponential polynomials. We compare the present numerical results of the method with those obtained by the existing methods in tables.On the decay of solutions to a class of Hartree equations.https://zbmath.org/1449.353982021-01-08T12:24:00+00:00"Tarulli, Mirko"https://zbmath.org/authors/?q=ai:tarulli.mirko"Venkov, George"https://zbmath.org/authors/?q=ai:venkov.georgeIn the paper under consideration nonlinear defocusing Schrödinger equations with Hartree-type nonlinearity is studied. The authors prove that global solution of the Cauchy problem for such equation has peculiar decay property. To do this a combination of a localization trick, the nonlinear interaction Morawetz estimate and interpolation is applied. In this way the long-time behavior of the solutions of the problem under consideration in the space \( L^{q}(\mathbb{R}^{d})\) is obtained which leads to the scattering in the energy space.
Reviewer: Angela Slavova (Sofia)Numerical method of value boundary problem decision for 2D equation of heat conductivity with fractional derivatives.https://zbmath.org/1449.651722021-01-08T12:24:00+00:00"Beĭbalaev, Vetlugin Dzhabrilovich"https://zbmath.org/authors/?q=ai:beibalaev.vetlugin-dzhabrilovich"Shabanova, Mumina Ruslanovna"https://zbmath.org/authors/?q=ai:shabanova.mumina-ruslanovnaSummary: In this work a solution is obtained for the boundary problem for two-dimensional thermal conductivity equation with derivatives of fractional order on time and space variables by grid method. Explicit and implicit difference schemes are developed. Stability criteria of these difference schemes are proven. It is shown that the approximation order by time is equal one but by space variables it is equal two. A solution method is suggested using fractional steps. It is proved that the transition module, corresponding to two half-steps, approximates the transition module for the given equation.Global asymptotic stabilization control with variable periods for underactuated surface vessels.https://zbmath.org/1449.932262021-01-08T12:24:00+00:00"Zhang, Pengfei"https://zbmath.org/authors/?q=ai:zhang.pengfei"Guo, Ge"https://zbmath.org/authors/?q=ai:guo.geSummary: This paper investigates the problem of global asymptotic stabilization of the underactuated surface vessel. By introducing a diffeomorphism transformation, the stabilization of the underactuated surface vessel is converted into the stabilization of the second order underactuated system consisting of two cascade connected subsystems. For the transformed system, a smooth control method with variable periods is presented to ensure the global asymptotic convergence. Compared with the methods using constant periods in previous works, this method improves the convergence rate of the system near the origin by adjusting the periods based on the system states in real time. Then, on this basis, a control algorithm with variable periods for underactuated surface vessels is presented to globally asymptotically stabilize the original system. At last, simulations are given to demonstrate the effectiveness of the presented method.A new numerical algorithm for two-point boundary value problems.https://zbmath.org/1449.651592021-01-08T12:24:00+00:00"Guo, Lihua"https://zbmath.org/authors/?q=ai:guo.lihua"Wu, Boying"https://zbmath.org/authors/?q=ai:wu.boying"Zhang, Dazhi"https://zbmath.org/authors/?q=ai:zhang.dazhiSummary: We present a new numerical algorithm for two-point boundary value problems. We first present the exact solution in the form of series and then prove that the \(n\)-term numerical solution converges uniformly to the exact solution. Furthermore, we establish the numerical stability and error analysis. The numerical results show the effectiveness of the proposed algorithm.On the stability of hybrid homogeneous systems.https://zbmath.org/1449.340472021-01-08T12:24:00+00:00"Aleksandrov, Aleksandr Yur'evich"https://zbmath.org/authors/?q=ai:aleksandrov.aleksandr-yurevich"Platonov, Alekseĭ Viktorovich"https://zbmath.org/authors/?q=ai:platonov.aleksei-viktorovichSummary: The hybrid system consisting of the family of subsystems with homogeneous right-hand sides and a switching law is considered. It is assumed that the zero solution of each subsystem is asymptotically stable. By the use of the Lyapunov functions method, the classes of admissible switching laws are determined under which the corresponding hybrid system is also asymptotically stable. The region of asymptotic stability of the zero solution is investigated.Some aspects of initial value problems theory for differential equations with Riemann-Liouville derivatives.https://zbmath.org/1449.340302021-01-08T12:24:00+00:00"Ogorodnikov, Evgeniĭ Nikolaevich"https://zbmath.org/authors/?q=ai:ogorodnikov.evgenii-nikolaevichSummary: Some subjects of the well-formed initial value problem for ordinary differential equations with Riemann-Liouville derivatives are discussed. As an example the simplest linear homogeneous differential equation with two fractional derivatives is considered.Periodic solutions and asymptotic analysis of ordinary differential equations.https://zbmath.org/1449.000032021-01-08T12:24:00+00:00"Han, Maoan (ed.)"https://zbmath.org/authors/?q=ai:han.maoan"Yu, Pei (ed.)"https://zbmath.org/authors/?q=ai:yu.pei"Romanovski, Valery G. (ed.)"https://zbmath.org/authors/?q=ai:romanovski.valery-g"Zhang, Tonghua (ed.)"https://zbmath.org/authors/?q=ai:zhang.tonghuaFrom the text: In a single special issue, of course, there is no way to cover all recent advances in the area of Ordinary Differential Equations. But we do believe that the results published in this issue at least can reflect some of the most current trends in the area of Ordinary Differential Equations.Stability of a mathematical model of malaria transmission with relapse.https://zbmath.org/1449.920282021-01-08T12:24:00+00:00"Huo, Hai-Feng"https://zbmath.org/authors/?q=ai:huo.hai-feng"Qiu, Guang-Ming"https://zbmath.org/authors/?q=ai:qiu.guang-mingSummary: A more realistic mathematical model of malaria is introduced, in which we not only consider the recovered humans return to the susceptible class, but also consider the recovered humans return to the infectious class. The basic reproduction number \(R_0\) is calculated by next generation matrix method. It is shown that the disease-free equilibrium is globally asymptotically stable if \(R_0 \leq 1\), and the system is uniformly persistence if \(R_0 > 1\). Some numerical simulations are also given to explain our analytical results. Our results show that to control and eradicate the malaria, it is very necessary for the government to decrease the relapse rate and increase the recovery rate.Stability analysis of uncertain complex-variable delayed nonlinear systems via intermittent control with multiple switched periods.https://zbmath.org/1449.342512021-01-08T12:24:00+00:00"Zheng, Song"https://zbmath.org/authors/?q=ai:zheng.songSummary: In this paper, an intermittent control approach with multiple switched periods is proposed for the robust exponential stabilization of uncertain complex-variable delayed nonlinear systems with parameters perturbation, in which the considered complex systems have bounded parametric uncertainties. Based on the Lyapunov stability theory and comparison theorem of differential equations, some stability criteria are established for a class of uncertain complex delayed nonlinear systems with parameters perturbation. Finally, some numerical simulations are given to show the effectiveness and the benefits of the theoretical results.Identification of 4D Lü hyper-chaotic system using identical systems synchronization and fractional adaptation law.https://zbmath.org/1449.370612021-01-08T12:24:00+00:00"Abedini, Mohammad"https://zbmath.org/authors/?q=ai:abedini.mohammad-javad"Gomroki, Mehdi"https://zbmath.org/authors/?q=ai:gomroki.mehdi"Salarieh, Hassan"https://zbmath.org/authors/?q=ai:salarieh.hassan"Meghdari, Ali"https://zbmath.org/authors/?q=ai:meghdari.aliSummary: In this paper, the parameters of a 4D Lü hyper-chaotic system are identified via synchronization of two identical systems. Unknown parameters of the drive system are identified by an adaptive method. Stability of the closed-loop system with one state feedback controller is studied by using the Lyapunov theorem. Also the convergence of the parameters to their true values is proved. Then a fractional adaptation law is applied to reduce the time of parameter convergence. Finally the results of both integer and fractional methods are compared.Method of Duhamel integral for ordinary differential equations with constant coefficients in respect to the theory of distributions.https://zbmath.org/1449.340432021-01-08T12:24:00+00:00"Kogan, Iosif Leonidovich"https://zbmath.org/authors/?q=ai:kogan.iosif-leonidovichSummary: A new proof for the method of Duhamel integral is produced. This proof is based on the convolution algebra of distributions and allows to extend this method for the region \(x<0\). Universal formulas for solving equations with discontinuos right-hand side are obtained.Characterization of self-adjoint domains for regular even order \(C\)-symmetric differential operators.https://zbmath.org/1449.470822021-01-08T12:24:00+00:00"Sun, Jiong"https://zbmath.org/authors/?q=ai:sun.jiong"Bao, Qinglan"https://zbmath.org/authors/?q=ai:bao.qinglan"Hao, Xiaoling"https://zbmath.org/authors/?q=ai:hao.xiaoling"Zettl, Anton"https://zbmath.org/authors/?q=ai:zettl.antonSummary: Let \(C\) be a skew-diagonal constant matrix satisfying \(C^{-1}=-C=C^{\ast}\). We characterize the self-adjoint domains for regular even order \(C\)-symmetric differential operators with two-point boundary conditions. The previously known characterizations are a special case of this one.Finite-time synchronization of nonidentical neural networks with time-varying delay based on integral sliding mode control.https://zbmath.org/1449.342612021-01-08T12:24:00+00:00"Xiong, Jingjing"https://zbmath.org/authors/?q=ai:xiong.jingjing"Zhang, Guobao"https://zbmath.org/authors/?q=ai:zhang.guobao.1Summary: The finite-time synchronization problem of a class of nonidentical neural networks with time-varying delay is studied. Firstly, by using the drive-response concept to derive an error system, a suitable integral sliding mode manifold is constructed by applying the synchronization error. If the state trajectories of the error system are driven onto the sliding mode surface, the synchronization error will thereafter converge to zero in finite time. Then, by combining the bounded conditions on neuron activation functions, a proper sliding mode controller is designed. Based on the designed controller and the Lyapunov stability theory, the state trajectories of the error system can be driven onto the sliding mode surface, such that the finite-time synchronization of nonidentical neural networks with time-varying delay can be performed. Finally, numerical simulation results illustrate the effectiveness of the proposed method.Existence of positive solutions for regular fractional Sturm-Liouville problems.https://zbmath.org/1449.340182021-01-08T12:24:00+00:00"Haghi, Tahereh"https://zbmath.org/authors/?q=ai:haghi.tahereh"Ghanbari, Kazem"https://zbmath.org/authors/?q=ai:ghanbari.kazem"Mingarelli, Angelo B."https://zbmath.org/authors/?q=ai:mingarelli.angelo-b|mingarelli.angelo-bernadoSummary: In this article, we investigate existence and nonexistence results for some regular fractional Sturm-Liouville problems. We find the eigenvalues intervals of this problem may or may not have a positive solution. Some sufficient conditions for existence and nonexistence of positive solutions are given. Further, we study some special properties of positive solutions. We give some examples at the end.Alternative iterative technique.https://zbmath.org/1449.340522021-01-08T12:24:00+00:00"Avery, Richard"https://zbmath.org/authors/?q=ai:avery.richard-i"Anderson, Douglas"https://zbmath.org/authors/?q=ai:anderson.douglas-robert"Henderson, Johnny"https://zbmath.org/authors/?q=ai:henderson.johnnySummary: The standard methods of applying iterative techniques do not apply when the nonlinear term is neither monotonic (corresponding to an increasing or decreasing operator) nor Lipschitz (corresponding to a condensing operator). However, by applying the Layered Compression-Expansion Theorem in conjunction with an alternative inversion technique, we show how one can apply monotonicity techniques to a right focal boundary value problem.Analytical study of fractional-order multiple chaotic FitzHugh-Nagumo neurons model using multistep generalized differential transform method.https://zbmath.org/1449.920072021-01-08T12:24:00+00:00"Momani, Shaher"https://zbmath.org/authors/?q=ai:momani.shaher-m"Freihat, Asad"https://zbmath.org/authors/?q=ai:freihat.asad-a"AL-Smadi, Mohammed"https://zbmath.org/authors/?q=ai:al-smadi.mohammed-hSummary: The multistep generalized differential transform method is applied to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model. The algorithm is illustrated by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation. The fractional derivatives are described in the Caputo sense. Furthermore, we present figurative comparisons between the proposed scheme and the classical fourth-order Runge-Kutta method to demonstrate the accuracy and applicability of this method. The graphical results reveal that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient.Existence and stability results for nonlocal initial value problems for differential equations with Hilfer fractional derivative.https://zbmath.org/1449.340122021-01-08T12:24:00+00:00"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffak"Bouriah, Soufyane"https://zbmath.org/authors/?q=ai:bouriah.soufyane"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseSummary: In this paper, we establish sufficient conditions for the existence and stability of solutions for a class of nonlocal initial value problems for differential equations with Hilfer's fractional derivative. The arguments are based upon the Banach contraction principle. Two examples are included to show the applicability of our results.Existence of solutions to discrete and continuous second-order boundary value problems via Lyapunov functions and a priori bounds.https://zbmath.org/1449.390192021-01-08T12:24:00+00:00"Tisdell, Christopher"https://zbmath.org/authors/?q=ai:tisdell.christopher-c"Liu, Yongjian"https://zbmath.org/authors/?q=ai:liu.yongjian"Liu, Zhenhai"https://zbmath.org/authors/?q=ai:liu.zhenhaiSummary: This article analyzes nonlinear, second-order difference equations subject to ``left-focal'' two-point boundary conditions. Our research questions are: RQ1: What are new, sufficient conditions under which solutions to our ``discrete'' problem will exist?; RQ2: What, if any, is the relationship between solutions to the discrete problem and solutions of the ``continuous'', left-focal analogue involving second-order ordinary differential equations? Our approach involves obtaining new a priori bounds on solutions to the discrete problem, with the bounds being independent of the step size. We then apply these bounds, through the use of topological degree theory, to yield the existence of at least one solution to the discrete problem. Lastly, we show that solutions to the discrete problem will converge to solutions of the continuous problem.An SIRS epidemic model with pulse vaccination, birth pulse and Logistic death rate.https://zbmath.org/1449.341322021-01-08T12:24:00+00:00"Gao, Jianzhong"https://zbmath.org/authors/?q=ai:gao.jianzhong"Zhang, Tailei"https://zbmath.org/authors/?q=ai:zhang.taileiSummary: In this paper, we propose an SIRS epidemic model with pulse vaccination, birth pulse and logistic death rate. By using the stroboscopic map of a discrete dynamical system, the disease-free periodic solution (DFPS for short) of the model under pulse vaccination and birth pulse is obtained. Based on the Floquet theory and comparison theorem of impulsive differential equations, the global asymptotic stability of the DFPS is given, and sufficient conditions for the permanence are obtained. In addition, numerical simulations are done to illlustrate our theoretical results.Multiple positive solutions to singular fractional differential system with Riemann-Stieltjes integral boundary condition.https://zbmath.org/1449.340932021-01-08T12:24:00+00:00"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Li, Yaohong"https://zbmath.org/authors/?q=ai:li.yaohongSummary: In this paper, we study a class of singular fractional differential systems with Riemann-Stieltjes integral boundary condition by constructing a new cone and using the Leggett-Williams fixed point theorem. The existence of multiple positive solutions is obtained. An example is presented to illustrate our main results.A novel hybrid method for solving combined functional neutral differential equations with several delays and investigation of convergence rate via residual function.https://zbmath.org/1449.651662021-01-08T12:24:00+00:00"Kurkcu, Omur Kıvanc"https://zbmath.org/authors/?q=ai:kurkcu.omur-kivanc"Aslan, Ersin"https://zbmath.org/authors/?q=ai:aslan.ersin"Sezer, Mehmet"https://zbmath.org/authors/?q=ai:sezer.mehmetSummary: In this study, we introduce a novel hybrid method based on a simple graph along with operational matrix to solve the combined functional neutral differential equations with several delays. The matrix relations of the matching polynomials of complete and path graphs are employed in the matrix-collocation method. We improve the obtained solutions via an error analysis technique. The oscillation of them on time interval is also estimated by coupling the method with Laplace-Pad'{e} technique. We develop a general computer program, and so we can efficiently monitor the precision of the method. We investigate a convergence rate of the method by constructing a formula based on the residual function. Eventually, an algorithm is described to show the easiness of the method.Existence and stability of Langevin equations with two Hilfer-Katugampola fractional derivatives.https://zbmath.org/1449.340202021-01-08T12:24:00+00:00"Ibrahim, Rabha W."https://zbmath.org/authors/?q=ai:ibrahim.rabha-waell"Harikrishnan, Sugumaran"https://zbmath.org/authors/?q=ai:harikrishnan.sugumaran"Kanagarajan, Kuppusamy"https://zbmath.org/authors/?q=ai:kanagarajan.kuppusamySummary: In this note, we discuss the existence, uniqueness and stability results for a general class of Langevin equations. We suggest the generalization via the Hilfer-Katugampola fractional derivative. We introduce some conditions for existence and uniqueness of solutions. We utilize the concept of fixed point theorems (Krasnoselskii fixed point theorem (KFPT), Banach contraction principle (BCP)). Moreover, we illustrate definitions of the Ulam type stability. These definitions generalize the fractional Ulam stability.Global analysis for an epidemic model with the Beverton-Holt birth function and stage structure.https://zbmath.org/1449.341662021-01-08T12:24:00+00:00"Wang, Yuping"https://zbmath.org/authors/?q=ai:wang.yuping.1"Lin, Xiaolin"https://zbmath.org/authors/?q=ai:lin.xiaolin"Li, Jianquan"https://zbmath.org/authors/?q=ai:li.jianquanSummary: Based on the facts that some diseases spread only among adults, and that the growth of adult individuals is density-dependent, an adult epidemic model with stage structure is proposed in this paper, where it is assumed that individuals in the population consist of juveniles and adults, and that the birth function of juveniles is of the Beverton-Holt type. The global stability of the model is completely investigated by constructing the appropriate Lyapunov functions and qualitative analysis, and the basic reproduction numbers of the population growth and the disease transmission, determining the dynamics of the model, are found. The obtained results suggest that, when the basic reproduction number of the population growth is not greater than unity, the population eventually dies out; when the basic reproduction number of the population growth is greater than unity, and the number of the disease transmission is less than or equal to unity, the population persists and the disease dies out; when the basic reproduction number of the disease transmission is greater than unity, the disease persists and becomes endemic as the population survives.Stability of an eco-epidemiological model with stage structure and saturation incidence.https://zbmath.org/1449.342982021-01-08T12:24:00+00:00"Wang, Lingshu"https://zbmath.org/authors/?q=ai:wang.lingshu"Zhang, Ya'nan"https://zbmath.org/authors/?q=ai:zhang.yanan"Su, Huan"https://zbmath.org/authors/?q=ai:su.huanSummary: In this paper, an eco-epidemiological predator-prey model with saturation incidence and stage structure for the prey is investigated. A time delay describing the latent period of the disease in this model is discussed. By analyzing the characteristic equations, the local stability of the boundary equilibria and the positive equilibrium is established, respectively. Moreover, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium. By using Lyapunov functions and the LaSalle's invariance principle, the global stability of the boundary equilibria and the positive equilibrium is addressed, respectively. Therefore, the sufficient conditions are given for the disease extinction and permanence of the model.Accurate splitting approach to characterize the solution set of boundary layer problems.https://zbmath.org/1449.340312021-01-08T12:24:00+00:00"Sayevand, Khosro"https://zbmath.org/authors/?q=ai:sayevand.khosro"Machado, Jose Antonio Tenreiro"https://zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiroSummary: The boundary layer (BL) is an important concept and refers to the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. This paper studies singularly perturbed fractional differential equations where the fractional derivatives are defined in the Caputo sense. The solution of such equations, with appropriate boundary conditions, displays BL behavior. The solution out of the BL is estimated by the solution of the reduced problem and the layer solution is approximated by means of a modified truncated Chebyshev series. The coefficients of the truncated series are evaluated using a novel operational matrix technique. Moreover, the stability and the error analysis of the proposed method are analyzed. Several examples illustrate the validity and applicability of the method.Three positive solutions of \(N\)-dimensional \(p\)-Laplacian with indefinite weight.https://zbmath.org/1449.352382021-01-08T12:24:00+00:00"Chen, Tianlan"https://zbmath.org/authors/?q=ai:chen.tianlan"Ma, Ruyun"https://zbmath.org/authors/?q=ai:ma.ruyunSummary: This paper is concerned with the global behavior of components of positive radial solutions for the quasilinear elliptic problem with indefinite weight \[\text{div}(\varphi_p(\nabla u))+\lambda h(x)f(u)=0, \; \text{in}\ \; B,\] \[u=0, \; \text{on}\; \partial B,\] where \(\varphi_p(s)=\vert s\vert ^{p-2}s\), \(B\) is the unit open ball of \(\mathbb{R}^N\) with \(N\geq1\), \(1<p<\infty\), \(\lambda>0\) is a parameter, \(f\in C([0, \infty), [0, \infty))\) and \(h\in C(\bar{B})\) is a sign-changing function. We manage to determine the intervals of \(\lambda\) in which the above problem has one, two or three positive radial solutions by using the directions of a bifurcation.Time-delay Rucklidge system Hopf bifurcation analysis and circuit simulation.https://zbmath.org/1449.342862021-01-08T12:24:00+00:00"He, Hongjun"https://zbmath.org/authors/?q=ai:he.hongjun"Cui, Yan"https://zbmath.org/authors/?q=ai:cui.yan"Sun, Guan"https://zbmath.org/authors/?q=ai:sun.guanSummary: We consider a new type of single-delay Rucklidge system and analyze the stability of the system and the existence of Hopf bifurcation. Matlab numerical simulation illustrate the theoretical analysis. A chaotic circuit with switchable time-delay and without time-delay is designed and simulated by Multisim14.0.The radial distribution of Julia sets of some entire functions with infinite lower order.https://zbmath.org/1449.370322021-01-08T12:24:00+00:00"Qiu, Ling"https://zbmath.org/authors/?q=ai:qiu.ling"Xuan, Zuxing"https://zbmath.org/authors/?q=ai:xuan.zuxing"Zhao, Yan"https://zbmath.org/authors/?q=ai:zhao.yanSummary: This article investigates the radial distribution of Julia sets of some entire functions with infinite lower order which are solutions, the polynomial or differential polynomial of solutions of the equation \(f'' (z) + A (z)f' (z) + B (z)f (z) = 0\).Chaos synchronization of TSUCS unified chaotic system, a modified function projective control method.https://zbmath.org/1449.931212021-01-08T12:24:00+00:00"Tirandaz, Hamed"https://zbmath.org/authors/?q=ai:tirandaz.hamedSummary: The synchronization problem of the three-scroll unified chaotic system (TSUCS) is studied in this paper. A modified function projective synchronization (MFPS) method is developed to achieve this goal. Furthermore, the only parameter of the TSUCS unified chaotic system is considered unknown and estimated with an appropriate parameter estimation law. MFPS method is investigated for both identical and non-identical chaotic systems. Lyapunov stability theorem is utilized to verify the proposed feedback control laws and validate the proposed synchronization scheme. Finally, some numerical simulations are presented to assess the effectiveness of the theoretical discussions.Global analysis of a class of tumor-immune system dynamics.https://zbmath.org/1449.341402021-01-08T12:24:00+00:00"Huang, Pei"https://zbmath.org/authors/?q=ai:huang.pei"Lin, Xiaolin"https://zbmath.org/authors/?q=ai:lin.xiaolin"Li, Jianquan"https://zbmath.org/authors/?q=ai:li.jianquan"Song, Xiuchao"https://zbmath.org/authors/?q=ai:song.xiuchaoSummary: Based on the fact that tumor cells not only stimulate the proliferation of immune effector cells but also have the inhibiting effect on the growth of the cells, a tumor-immune dynamical model is described by expressing the comprehensive effect of tumor cells on immune system with a positive or negative action rate coefficient. By investigating the global dynamics of the model, it is found that the saddle-node bifurcation and the bistable phenomenon may occur, which implies that the final state of tumor development depends on the initial state, and the corresponding threshold conditions are obtained. The effect of the intrinsic input of effector cells and the action rate coefficient of tumor cells on effector cells on the dynamics of the model is analyzed. The obtained results show that the model may have complex dynamical behaviors when the inhibition effect of tumor cells on effector cells is strong enough.Analysis of local bifurcations of an enzyme catalyzed reaction system.https://zbmath.org/1449.341572021-01-08T12:24:00+00:00"Su, Juan"https://zbmath.org/authors/?q=ai:su.juanSummary: In this paper, the local bifurcations of an enzyme catalyzed system are studied. Firstly, it is shown that this system has either 1 or 2 isolated equilibria, or a singular line. The dynamical properties of each equilibrium are given. Then, when the isolated equilibria are non-hyperbolic, it is exhibited that this system may undergo transcritical bifurcation and Hopf bifurcation. By Lyapunov quantity, the order of weak focus is proved to be 1. Finally, numerical simulations are employed to illustrate the results obtained.Multiplicity of positive solutions to a class of multi-point boundary value problem.https://zbmath.org/1449.340962021-01-08T12:24:00+00:00"Zhao, Bao"https://zbmath.org/authors/?q=ai:zhao.bao"Yang, Yunrui"https://zbmath.org/authors/?q=ai:yang.yunrui"Zhou, Yonghui"https://zbmath.org/authors/?q=ai:zhou.yonghuiSummary: In this paper, the existence of multiple positive solutions to a class of nonlinear multi-point boundary value problems is established by using Guo-Krasnoselskii's fixed-point theorem. At the same time, an example is given to illustrate our conclusion.Existence of periodic solution for a kind of \( (m,n)\)-order generalized neutral differential equation.https://zbmath.org/1449.342362021-01-08T12:24:00+00:00"Yao, Shaowen"https://zbmath.org/authors/?q=ai:yao.shaowenSummary: In this paper, we consider the following higher-order \(p\)-Laplacian generalized neutral differential equation with variable parameter \[ (\varphi_p(x (t)-c (t)x (t-\sigma))^{(n)})^{(m)}+g (t,x (t),x (t-\tau (t)), x' (t),\cdots, x^{(m)} (t)) = e (t).\] By the coincidence degree theory, sufficient conditions for the existence of periodic solutions are established.Note on the stability property of the vanishing equilibrium point of an ecological system consisting of a predator and stage structure prey.https://zbmath.org/1449.341412021-01-08T12:24:00+00:00"Huang, Xiaoyan"https://zbmath.org/authors/?q=ai:huang.xiaoyuan"Chen, Fengde"https://zbmath.org/authors/?q=ai:chen.fengdeSummary: We revisit the stability property of an ecological system consisting of a predator and stage structure prey which was proposed by previous researchers. By constructing some suitable Lyapunov function and applying the differential inequality theory, we show that the conditions which ensure the local stability of the vanishing point are enough to ensure its global stability. Our result supplements and complements some known results.The qualitative analysis of a two species amensalism model with non-monotonic functional response and Allee effect on second species.https://zbmath.org/1449.341372021-01-08T12:24:00+00:00"Guan, Xinyu"https://zbmath.org/authors/?q=ai:guan.xinyu"Deng, Hang"https://zbmath.org/authors/?q=ai:deng.hangSummary: In this paper, we present a two species amensalism model with non-monotonic functional response and Allee effect on the second species. Local and global stability of the boundary and interior equilibrium are investigated. By introducing the Allee effect, we show that the boundary equilibria change from unstable node and saddle into saddle-node. Our results are supported by numeric simulations.Oscillation for second-order half-linear delay damped dynamic equations on time scales.https://zbmath.org/1449.343192021-01-08T12:24:00+00:00"Li, Jimeng"https://zbmath.org/authors/?q=ai:li.jimeng"Yang, Jiashan"https://zbmath.org/authors/?q=ai:yang.jiashanSummary: The oscillation of second-order half-linear variable delay damped dynamic equation
\[[a (t)|x^\Delta (t)|^{\lambda-1} x^\Delta (t)]^\Delta + b (t)|x^\Delta (t)|^{\lambda-1} x^\Delta (t) + p(t)|x (\delta (t))|^{\lambda-1} x (\delta (t)) = 0\]
is investigated on a time scale \(\mathbb{T}\), in which the equation is noncanonical, i.e., \(\int_{t_0}^\infty [a^{-1} (s) e_{-b/a} (s, {t_0})]^{1/\lambda} \Delta s <\infty\). By using the generalized Riccati transformation and the calculus theory on the time scales, and in combination with some inequality technique, some new oscillation criteria for the equation are established. The paper generalizes, improves and enriches known results.Stability analysis for a class of complex value fractional-order SIR model.https://zbmath.org/1449.341502021-01-08T12:24:00+00:00"Liu, Na"https://zbmath.org/authors/?q=ai:liu.na"Fang, Jie"https://zbmath.org/authors/?q=ai:fang.jie"Deng, Wei"https://zbmath.org/authors/?q=ai:deng.wei"Fang, Na"https://zbmath.org/authors/?q=ai:fang.naSummary: In this paper, the stability of a fractional-order SIR model with complex value is studied. Based on the Jacobian matrix and the FV\(^{-1}\) method, local stability and global stability of the equilibrium points are analyzed. Finally, the effectiveness of the theoretical results is illustrated by simulation results.Complex dynamics of an intraguild predation model.https://zbmath.org/1449.341692021-01-08T12:24:00+00:00"Yang, Xiaomin"https://zbmath.org/authors/?q=ai:yang.xiaomin"Qiu, Zhipeng"https://zbmath.org/authors/?q=ai:qiu.zhipeng"Ding, Ling"https://zbmath.org/authors/?q=ai:ding.lingSummary: The complex dynamics of an intraguild predation (IGP) model is investigated in this paper, and the model incorporates the Holling-II functional response functions. Sufficient conditions are obtained for the existence and local stability of boundary equilibria. Then, numerical simulations are applied to the model under given values of parameters. The numerical results show that the system may have an attracting invariant torus but no positive equilibrium. Furthermore, the Poincaré map and Fourier transform spectrum analysis are performed to study the complex dynamics of the system on the invariant torus. The results suggest that the dynamics on the invariant torus is almost periodic.Adaptive synchronization of dynamical networks via states of several nodes as target orbit.https://zbmath.org/1449.370752021-01-08T12:24:00+00:00"Xiao, Yuzhu"https://zbmath.org/authors/?q=ai:xiao.yuzhu"Tang, Sufang"https://zbmath.org/authors/?q=ai:tang.sufang"Yang, Xiaoli"https://zbmath.org/authors/?q=ai:yang.xiaoliSummary: In this paper, based on the invariance principle of differential equation, a simple adaptive control method is proposed to synchronize the dynamical networks with the general coupling functions. Comparing with other adaptive control methods, the weighted average of a few nodes' states is used as target orbit to design controller. To show the effectiveness of proposed method, some numerical simulations are performed.Some criteria for the global finite-time synchronization of two Lorenz-stenflo systems coupled by a new controller.https://zbmath.org/1449.370622021-01-08T12:24:00+00:00"Chen, Yun"https://zbmath.org/authors/?q=ai:chen.yun"Shi, Zhangsong"https://zbmath.org/authors/?q=ai:shi.zhangsong"Lin, Chunsheng"https://zbmath.org/authors/?q=ai:lin.chunshengSummary: This paper investigates the global finite-time synchronization of two chaotic Lorenz-Stenflo systems coupled by a new controller called the generalized variable substitution controller. First of all, the generalized variable substitution controller is designed to establish the master-slave finite-time synchronization scheme for the Lorenz-Stenflo systems. And then, based on the finite-time stability theory, a sufficient criterion on the finite-time synchronization of this scheme is rigorously verified in the form of matrix and the corresponding estimation for the synchronization time is analytically given. Applying this criterion, some sufficient finite-time synchronization criteria under various generalized variable substitution controllers are further derived in the algebraic form. Finally, some numerical examples are introduced to compare the results proposed in this paper with those proposed in the existing literature, verifying the effectiveness of the criteria obtained.Analysis of a predator-mutualism coupling system with a stage structure.https://zbmath.org/1449.341752021-01-08T12:24:00+00:00"Zhang, Yuxuan"https://zbmath.org/authors/?q=ai:zhang.yuxuan"Sun, Xue"https://zbmath.org/authors/?q=ai:sun.xue"Zhao, Zhitao"https://zbmath.org/authors/?q=ai:zhao.zhitaoSummary: Predation and mutualism are two important relationships in the population, which form the basis of the biological community. The dynamic properties of a predator-mutualism coupling system with a stage-structure are investigated. Based on the actual ecological problems, a predator-mutualism coupling system with a stage-structure is established, and the existence, uniqueness and stability of boundary equilibrium points and positive equilibrium points of the system are explored. Finally, the numerical simulation is used to illustrate the theoretical results obtained.Dynamics of a ratio-dependent Lotka-Volterra cooperative system with delays.https://zbmath.org/1449.342832021-01-08T12:24:00+00:00"Ahmadjan, Muhammadhaji"https://zbmath.org/authors/?q=ai:ahmadjan.muhammadhajiSummary: This paper studies a class of nonautonomous two-species ratio-dependent Lotka-Volterra cooperative system with delays. Some sufficient conditions on the boundedness, permanence, existence of periodic solution and global attractivity of the system are established by means of the comparison method and Lyapunov function method.Permanence for delayed Ayala-Gilpin cooperative systems with impulsive.https://zbmath.org/1449.342962021-01-08T12:24:00+00:00"Wang, Libo"https://zbmath.org/authors/?q=ai:wang.libo"Xu, Guigui"https://zbmath.org/authors/?q=ai:xu.guiguiSummary: This paper considers a class of delayed Ayala-Gilpin cooperative systems with impulses, and establishes sufficient conditions which ensure the system to be permanent. We improve and extend previously known results.Dynamic analysis of the predator-prey system in a polluted environment with impulsive.https://zbmath.org/1449.341482021-01-08T12:24:00+00:00"Liu, Juan"https://zbmath.org/authors/?q=ai:liu.juan"Hu, Jie"https://zbmath.org/authors/?q=ai:hu.jie"Zhao, Qing"https://zbmath.org/authors/?q=ai:zhao.qing"Li, Fuzhong"https://zbmath.org/authors/?q=ai:li.fuzhongSummary: We establish a predator-prey system with Holling II functional response and impulsive effect in a polluted environment. The comparison techniques and integral mean method are employed to obtain the conditions for population extinction and weak average persistent of system to save the species. This theory can be used to protect species, especially endangered species, from extinction.Dynamic analysis of a predator-prey system with the Allee effect.https://zbmath.org/1449.341702021-01-08T12:24:00+00:00"Yue, Zongmin"https://zbmath.org/authors/?q=ai:yue.zongmin"Wang, Jiao"https://zbmath.org/authors/?q=ai:wang.jiao"Lu, Kun"https://zbmath.org/authors/?q=ai:lu.kunSummary: In this paper, we establish a predator-prey model with a protection area and a open habitat for prey. The Allee effect in the prey of protection area and the environment carrying capacity of prey are considered. According to the biology of predator-prey system, fast and slow time scales are considered in some of the parameters. Based on two different time scales, the system is divided into a fast and a slow system. We analyze the dynamics of the slow system. Existence and global stability of equilibria, existence of Hopf bifurcation and the limit cycle are performed. Combining mathematical analysis and numerical simulation, we verify the results above. The results show that the Allee effect changes the conditions for coexistence of the two species. It makes the system dynamic behavior more complicated.Existence of mild solutions for a class of fractional semilinear integro-differential equation of mixed type.https://zbmath.org/1449.342772021-01-08T12:24:00+00:00"Zhu, Bo"https://zbmath.org/authors/?q=ai:zhu.bo"Han, Baoyan"https://zbmath.org/authors/?q=ai:han.baoyan"Liu, Lishan"https://zbmath.org/authors/?q=ai:liu.lishanSummary: In this paper, the authors studied the existence results of the mild solutions for a class of fractional semilinear integro-differential equation of mixed type by using the measure of noncompactness, \(k\)-set contraction and \(\beta\)-resolvent family. It is well known that the \(k\)-set contraction requires additional condition to ensure the contraction coefficient \(0 < k < 1\). We don't require additional condition to ensure the contraction coefficient \(0 < k < 1\). An example is introduced to illustrate the main results of this paper.Stability analysis of a class of fractional-order SIS models with pulse vaccination.https://zbmath.org/1449.341492021-01-08T12:24:00+00:00"Liu, Na"https://zbmath.org/authors/?q=ai:liu.na"Fang, Jie"https://zbmath.org/authors/?q=ai:fang.jie"Deng, Wei"https://zbmath.org/authors/?q=ai:deng.weiSummary: The stability of a class of fractional-order SIS models with pulse vaccination is analyzed in this paper. Based on the fractional-order comparison theorem, the trivial solutions of the impulsive fractional-order SIS systems are uniformly asymptotically stable. That is, the disease will eventually die out. Finally, the theoretical results are verified by simulation examples, and the effects of fractional-order parameter and vaccination ratio on the rate of disease attenuation are also simulated. This study has certain theoretical guidance to prevent and control the spread of infectious diseases.Positive solutions for the boundary value problem of Laplacian-like equation with parameter and delay.https://zbmath.org/1449.342192021-01-08T12:24:00+00:00"Yan, Shulin"https://zbmath.org/authors/?q=ai:yan.shulin"Li, Zhiyan"https://zbmath.org/authors/?q=ai:li.zhiyanSummary: Using the fixed point theorem on cone, we study a class of Laplacian-like equation with a parameter and a delay, and obtain the existence of positive solutions for the boundary value problem.On Poincaré compactification and the projectivization of polynomial vector fields.https://zbmath.org/1449.341142021-01-08T12:24:00+00:00"Li, Fengbai"https://zbmath.org/authors/?q=ai:li.fengbaiThe author proves that the Poincaré compactification defined through Pfaffian forms and the projectivization of polynomial vector fields in \(\mathbb{C}^n\) defined through homogeneous vector fields are equivalent. He firstly establishes the equivalence in dimension \(2\), and then in dimension \(n\).
Reviewer: Rodica Luca (Iaşi)Non-fragile Mittag-Leffler synchronization of fractional-order neural networks with time delay.https://zbmath.org/1449.342592021-01-08T12:24:00+00:00"Wang, Yougang"https://zbmath.org/authors/?q=ai:wang.yougang"Wu, Huaiqin"https://zbmath.org/authors/?q=ai:wu.huaiqinSummary: The paper is concerned with the global non-fragile Mittag-Leffler synchronization of discontinuous fractional-order neural networks with time delay. Under the designed non-fragile controller with two types of fluctuation, a sufficient criterion for synchronization with linear matrix inequality (LMI) is established by applying Lyapunov function method, non-smooth theory, matrix inequality techniques, etc. Finally, a numerical example is given to illustrate the feasibility of the designed controller.The existence of triple classical solutions to impulsive problems with small non-autonomous perturbations.https://zbmath.org/1449.341002021-01-08T12:24:00+00:00"Liu, Jian"https://zbmath.org/authors/?q=ai:liu.jian.1"Zhao, Zengqin"https://zbmath.org/authors/?q=ai:zhao.zengqin"Yu, Wenguang"https://zbmath.org/authors/?q=ai:yu.wenguangSummary: We study the existence of solutions to nonlinear impulsive boundary value problems with small non-autonomous perturbations on the half-line. We show the existence of at least three distinct classical solutions by using variational methods and a three critical points theorem.Stability of partial profit model with double time delays.https://zbmath.org/1449.342882021-01-08T12:24:00+00:00"Jiang, Yongsheng"https://zbmath.org/authors/?q=ai:jiang.yongsheng"Ji, Guilin"https://zbmath.org/authors/?q=ai:ji.guilin"Xu, Jiabo"https://zbmath.org/authors/?q=ai:xu.jiabo"Ge, Qing"https://zbmath.org/authors/?q=ai:ge.qingSummary: In this paper, we mainly study the partial profit model with time delay. The local stability of the positive equilibrium point and the boundary equilibrium point is obtained by the Hurwitz criterion and the sufficient conditions for the global stability are given by constructing a suitable Lyapunov function. It can be seen that the sufficient conditions of the local stability are also the sufficient conditions for the global stability. This results improves some of the existing results.Existence for nonoscillatory solutions of fractional differential equations.https://zbmath.org/1449.342322021-01-08T12:24:00+00:00"Zhao, Huanhuan"https://zbmath.org/authors/?q=ai:zhao.huanhuan"Liu, Youjun"https://zbmath.org/authors/?q=ai:liu.youjun"Yan, Jurang"https://zbmath.org/authors/?q=ai:yan.jurangSummary: In this paper, fractional neutral differential equations with forced terms are investigated. Krasnoselskii's fixed point theorem is used to obtain sufficient conditions for the existence of nonoscillatory solutions.Global Mittag-Leffler stability of fractional-order Cohen-Grossberg BAM neural networks with time-varying delays.https://zbmath.org/1449.342762021-01-08T12:24:00+00:00"Liu, Wei"https://zbmath.org/authors/?q=ai:liu.wei.6|liu.wei|liu.wei.5|liu.wei.3|liu.wei.2|liu.wei.1|liu.wei.7|liu.wei.9|liu.wei.8"Jiang, Wangdong"https://zbmath.org/authors/?q=ai:jiang.wangdong"Zhang, Yuehong"https://zbmath.org/authors/?q=ai:zhang.yuehongSummary: This paper mainly studies fractional-order Cohen-Grossberg BAM neural networks with time-varying delays. Using properties of fractional-order calculus, definition of the Mittag-leffler function, effective division of time intervals, differential mean value theorem and some analytical techniques, sufficient conditions are given to ensure global Mittag-Leffler stability of such fractional-order neural networks. Finally, numerical examples are given to illustrate the theoretical results.The existence of solutions for a class of second-order \(m\)-point boundary value problems.https://zbmath.org/1449.340612021-01-08T12:24:00+00:00"Wei, Xiaofei"https://zbmath.org/authors/?q=ai:wei.xiaofei"Cao, Wenjuan"https://zbmath.org/authors/?q=ai:cao.wenjuanSummary: In this paper, we mainly study the three-point boundary value problem \[\begin{cases}y'' (t) = f (t,y (t), y' (t)),\; t \in (a,b)\\ y (a) = A, y (b) = \alpha y (\eta), \end{cases}\] where \(\eta \in (a, b)\) and \(\alpha (\eta - a) \ne b - a\). Under certain conditions on the nonlinearity \(f\), we obtain the existence of the solutions by using the shooting method. Furthermore, we generalize this result to the \(m\)-point boundary value problem. Finally, we illustrate the result by MATLAB numerical simulation.Global stability of a measles epidemic model with partial immunity and environmental transmission.https://zbmath.org/1449.341452021-01-08T12:24:00+00:00"Jing, Xiaojie"https://zbmath.org/authors/?q=ai:jing.xiaojie"Zhao, Aimin"https://zbmath.org/authors/?q=ai:zhao.aimin"Liu, Guirong"https://zbmath.org/authors/?q=ai:liu.gui-rong.1Summary: In this paper, a measles epidemic model with partial immunity and environmental transmission is considered, and the basic reproduction number \({R_0}\) is obtained. By constructing Lyapunov functions, we prove the global asymptotic stability of the infection-free equilibrium and the endemic equilibrium. When \({R_0} < 1\), the infection-free equilibrium is globally asymptotically stable, which implies that measles dies out eventually; when \({R_0} > 1\), the model has a unique endemic equilibrium, which is globally asymptotically stable, and it means that the transmission of measles keeps a steady state. Finally, the simulations are carried out to illustrate the rationality of the results. This work has practical significance for guiding us to prevent and control the measles spread.Chaos control of fractional order Rabinovich-Fabrikant system and synchronization between chaotic and chaos controlled fractional order Rabinovich-Fabrikant system.https://zbmath.org/1449.370722021-01-08T12:24:00+00:00"Srivastava, M."https://zbmath.org/authors/?q=ai:srivastava.manjari-k|srivastava.muni-shanker|srivastava.manoj-kumar|srivastava.meera|srivastava.mani-b|srivastava.madhurima|srivastava.m-m|srivastava.milan|srivastava.mukesh|srivastava.mahesh-c|srivastava.manindra-kumar|srivastava.meenakshi|srivastava.mayank|srivastava.marindra-kumar|srivastava.mohit-kumar|srivastava.m-p|srivastava.madhu"Agrawal, S. K."https://zbmath.org/authors/?q=ai:agrawal.suresh-kumar|agrawal.sunil-kumar|agrawal.saurabh-k"Vishal, K."https://zbmath.org/authors/?q=ai:vishal.k"Das, S."https://zbmath.org/authors/?q=ai:das.subir-kSummary: In this article the local stability of the Rabinovich-Fabrikant (R-F) chaotic system with fractional order time derivative is analyzed using fractional Routh-Hurwitz stability criterion. Feedback control method is used to control chaos in the considered fractional order system and after controlling the chaos the authors have introduced the synchronization between fractional order non-chaotic R-F system and the chaotic R-F system at various equilibrium points. The fractional derivative is described in the Caputo sense. Numerical simulation results which are carried out using Adams-Boshforth-Moulton method show that the method is effective and reliable for synchronizing the systems.The contrast structure for the singularly perturbed problem with slow-fast layers and discontinuous righthand side.https://zbmath.org/1449.342012021-01-08T12:24:00+00:00"Chen, Huaxiong"https://zbmath.org/authors/?q=ai:chen.huaxiong"Wang, Yanyan"https://zbmath.org/authors/?q=ai:wang.yanyan"Ni, Mingkang"https://zbmath.org/authors/?q=ai:ni.mingkangSummary: This paper discusses the contrast structure solution for the singularly perturbed problem with slow-fast layers and discontinuous righthand side. By applying the boundary function method, the asymptotic solution of this problem is constructed. Then using the sewing connection method, the existence of the solution is shown and the asymptotic solution is proved to be uniformly valid. Finally, an example is given to illustrate the main results.Mean square exponential stability of the split-step \(\theta\) method for a class of neutral stochastic delay integro-differential equations.https://zbmath.org/1449.650072021-01-08T12:24:00+00:00"Peng, Wei"https://zbmath.org/authors/?q=ai:peng.wei"Zhu, Mengjiao"https://zbmath.org/authors/?q=ai:zhu.mengjiao"Wang, Wenqiang"https://zbmath.org/authors/?q=ai:wang.wenqiangSummary: In this paper, we are concerned with the mean square exponential stability of a class of neutral stochastic delay integro-differential equations with a split-step theta (SST) method. It is shown that the mean square exponential stability of the exact solution with the drift coefficient satisfies linear growth conditions. It is also proved that the SST method with \(\theta \in [0, 1/2]\) can recover the exponential mean square stability of the exact solution with some restrictive conditions on step-size \(h < {h^*}\) and the drift coefficient, but for \(\theta \in (1/2, 1]\), the SST method can reproduce the exponential mean square stability with the step-size \(h = \tau /m\), where \(m\) is a positive integer. Finally, the numerical test verifies the correctness of the theoretical results.Global attracting set for neutral type Hopfield neural networks with time-varying delays.https://zbmath.org/1449.342642021-01-08T12:24:00+00:00"Zhou, Qinghua"https://zbmath.org/authors/?q=ai:zhou.qinghua"Wan, Li"https://zbmath.org/authors/?q=ai:wan.li"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie.1|liu.jie.7|liu.jie.2|liu.jie.3|liu.jie|liu.jie.5|liu.jie.4Summary: This paper deals with the asymptotic properties of a class of nonlinear and non-autonomous neutral type Hopfield neural networks with time-varying delays. By applying the property of nonnegative matrix and an integral inequality, some sufficient conditions are derived to ensure the existence of the global attracting set and the stability in a Lagrange sense for the considered system. Finally, an example is given to demonstrate the effectiveness of our theoretical result.Pseudo almost automorphic solutions of hematopoiesis model with mixed delays.https://zbmath.org/1449.342842021-01-08T12:24:00+00:00"Aouiti, Chaouki"https://zbmath.org/authors/?q=ai:aouiti.chaouki"Dridi, Farah"https://zbmath.org/authors/?q=ai:dridi.farah"Kong, Fanchao"https://zbmath.org/authors/?q=ai:kong.fanchaoSummary: This paper is concerned with a hematopoiesis model with mixed delays. Under new conditions, we study the existence, uniqueness and global exponential stability of pseudo almost automorphic solutions for the suggested model. Our approach is mainly based on the exponential dichotomy of linear differential equation, Banach's fixed-point principle and suitable Lyapunov functional. At the end, some numerical examples are presented to demonstrate the effectiveness of our findings.Oscillation of second order nonlinear differential equations with neutral delay.https://zbmath.org/1449.342312021-01-08T12:24:00+00:00"Zhang, Zhiyu"https://zbmath.org/authors/?q=ai:zhang.zhiyu"Yu, Yuanhong"https://zbmath.org/authors/?q=ai:yu.yuanhong"Li, Shuping"https://zbmath.org/authors/?q=ai:li.shuping"Qiao, Shizhu"https://zbmath.org/authors/?q=ai:qiao.shizhuSummary: In this paper, the oscillatory behavior of solutions to a nonlinear second-order neutral differential equation is studied. Using double Riccati transformation and the technique of inequations, some new sufficient conditions are obtained for the solutions of all oscillations. The results generalize, improve and unify oscillation theorems of half linear functional differential equations, nonlinear equations and generalized Emden-Fowler type equations in the literature recently. At last, some examples are given to illustrate the effectiveness of our results.Existence and stability of periodic solution for a Lasota-Wazewska model with discontinuous harvesting.https://zbmath.org/1449.342352021-01-08T12:24:00+00:00"Yang, Chao"https://zbmath.org/authors/?q=ai:yang.chao.3|yang.chao.2|yang.chao.1"Li, Runjie"https://zbmath.org/authors/?q=ai:li.runjieSummary: In this paper, we study a class of mixed time-varying delayed Lasota-Wazewska model with discontinuous harvesting, which is described by a periodic nonsmooth dynamical system. Based on nonsmooth analysis, Kakutani's fixed point method and the generalized Lyapunov method, easily verifiable delay-independent criteria are established to ensure the existence and exponential stability of positive periodic solutions. Finally, we give an example to further illustrate the effectiveness of our main results.The discrete Poisson equation and the heat equation with the exponential nonlinear term.https://zbmath.org/1449.352442021-01-08T12:24:00+00:00"Li, Yafeng"https://zbmath.org/authors/?q=ai:li.yafeng"Xin, Qiao"https://zbmath.org/authors/?q=ai:xin.qiao"Mu, Chunlai"https://zbmath.org/authors/?q=ai:mu.chunlaiSummary: This paper mainly studies the relations between the solution of the discrete Poisson equation and the solution of the discrete heat equation with exponential nonlinear term by monotone iterative method and comparison principle. When the solutions of the discrete Poisson equation exist, we discuss the asymptotic stability of the solutions to the discrete heat equation with exponential nonlinear term.Periodic solutions of a class of nonlinear Hill's type equations with bounded restoring force.https://zbmath.org/1449.341222021-01-08T12:24:00+00:00"Wang, Chao"https://zbmath.org/authors/?q=ai:wang.chao.1|wang.chao.2|wang.chao.3Summary: In this paper, we study the existence and multiplicity of periodic solutions of a class of Hill's type equations with bounded restoring force. We prove the existence of infinite subharmonic solutions when the weight is positive. We also consider the existence, multiplicity and dense distribution of symmetric periodic solutions in case of even and periodic weight functions.Continuous dependence theorems on solutions of uncertain differential equations.https://zbmath.org/1449.340052021-01-08T12:24:00+00:00"Gao, Yuan"https://zbmath.org/authors/?q=ai:gao.yuan"Yao, Kai"https://zbmath.org/authors/?q=ai:yao.kaiSummary: In ordinary differential equation (ODE) and stochastic differential equation (SDE), the solution continuously depends on initial value and parameter under some conditions. This paper investigates the analogous continuous dependence theorems in uncertain differential equation (UDE). It proves two continuous dependence theorems, a basic one and a general one.Design of a soft variable structure controller for synchronization of fractional-order chaotic systems with different structures.https://zbmath.org/1449.930212021-01-08T12:24:00+00:00"Shao, Keyong"https://zbmath.org/authors/?q=ai:shao.keyong"Guo, Haoxuan"https://zbmath.org/authors/?q=ai:guo.haoxuan"Han, Feng"https://zbmath.org/authors/?q=ai:han.feng"Wang, Tingting"https://zbmath.org/authors/?q=ai:wang.tingtingSummary: Based on the fractional calculus and the Mittag-Leffler stability theory, the synchronization of fractional order chaotic systems with different structures is investigated. A soft variable structure controller is proposed for synchronization of different fractional order chaotic systems. The different structures synchronization of Chen chaotic system and Liu chaotic system are realized based on the designed soft variable structure controller. The Matlab simulation results verify the effectiveness of the designed controller.Fixed-time control synchronization for stochastic competitive neural networks with time-varying delays.https://zbmath.org/1449.342572021-01-08T12:24:00+00:00"Pu, Hao"https://zbmath.org/authors/?q=ai:pu.hao"Ran, Jie"https://zbmath.org/authors/?q=ai:ran.jie"Pan, Yonghui"https://zbmath.org/authors/?q=ai:pan.yonghui"Zhang, Zhuanzhou"https://zbmath.org/authors/?q=ai:zhang.zhuanzhou"Huang, Jianwen"https://zbmath.org/authors/?q=ai:huang.jianwenSummary: In this paper, the fixed-time control synchronization for stochastic competitive neural networks with time-varying delays is investigated. According to Lyapunov stability theory, fixed-time stability theory, the theory of stochastic differential equation, the Itô's formula and some inequality methods, some new and useful sufficient conditions for the fixed-time synchronization of competitive neural networks are obtained through \(p\)-norm.Asymptotic estimation of the trapezoidal method for a class of neutral differential equation with variable delay.https://zbmath.org/1449.651452021-01-08T12:24:00+00:00"Zhang, Gengen"https://zbmath.org/authors/?q=ai:zhang.gengen"Wang, Wansheng"https://zbmath.org/authors/?q=ai:wang.wansheng"Xiao, Aiguo"https://zbmath.org/authors/?q=ai:xiao.aiguoSummary: In this paper, we investigate the stability of the trapezoidal method for a class of neutral differential equation with variable delay and obtain the asymptotic estimation of numerical solution with the aid of a functional inequality. The asymptotic estimation is more accurate than asymptotic stability in describing the behaviours of the numerical solution, and gives the upper bound estimates of the numerical solution for the nonstable case.Existence of solutions for a new class of fuzzy differential inclusions with resolvent operators in Banach spaces.https://zbmath.org/1449.340072021-01-08T12:24:00+00:00"Nguyen Van Hung"https://zbmath.org/authors/?q=ai:nguyen-van-hung."Vo Minh Tam"https://zbmath.org/authors/?q=ai:vo-minh-tam."O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper, a new class of fuzzy differential inclusions with resolvent operators in Banach spaces using \((H(\cdot ,\cdot ),\eta)\)-monotone operators is introduced and studied. A continuous selection theorem and fixed point theory are used to establish the existence of solutions. Finally, as applications, we consider special cases of fuzzy differential inclusions with general \(A\)-monotone operators. Some examples are given to illustrate our results.Modelling intra-host competition between malaria parasites strains.https://zbmath.org/1449.920472021-01-08T12:24:00+00:00"Song, Tianqi"https://zbmath.org/authors/?q=ai:song.tianqi"Wang, Chuncheng"https://zbmath.org/authors/?q=ai:wang.chuncheng"Tian, Boping"https://zbmath.org/authors/?q=ai:tian.bopingSummary: An intra-host epidemiological model is formulated for the co-infection of drug-sensitive and drug-resistant malaria parasites to examine the impact of aggressive treatment on the effect of competitive release in an infected host. The analysis of the existence of equilibrium and their stability of the model is conducted, and the results reveal that the intra-host competition and treatment play a key role in the prevalence of drug-resistant strains. The mathematical outcomes qualitatively match the experimental fact, that the rapid elimination of drug-resistant strains could promote the very evolution it is intended to retard.On stability of the inverted pendulum motion with a vibrating suspension point.https://zbmath.org/1449.700262021-01-08T12:24:00+00:00"Demidenko, G. V."https://zbmath.org/authors/?q=ai:demidenko.gennadii-v"Dulepova, A. V."https://zbmath.org/authors/?q=ai:dulepova.a-vSummary: Under study is the stability of the inverted pendulum motion whose suspension point vibrates according to a sinusoidal law along a straight line having a small angle with the vertical. Formulating and using the contracting mapping principle and the criterion of asymptotic stability in terms of solvability of a special boundary value problem for the Lyapunov differential equation, we prove that the pendulum performs stable periodic movements under sufficiently small amplitude of oscillations of the suspension point and sufficiently high frequency of oscillations.Unsteady state fugacity model by a dynamic control system.https://zbmath.org/1449.800232021-01-08T12:24:00+00:00"Bru, Rafael"https://zbmath.org/authors/?q=ai:bru.rafael"Maria Carrasco, José"https://zbmath.org/authors/?q=ai:maria-carrasco.jose"Paraíba, Lourival C."https://zbmath.org/authors/?q=ai:paraiba.lourival-cSummary: A continuous time dynamic system of an unsteady state fugacity model is presented. Properties of this model as stability are studied. In order to evaluate numerical results a discretization preserving the stability and yielding the positivity property of the model is used. Finally, algorithms to determine the values of the fugacities, the concentrations and the dissipation time are given. The above study is illustrated with numerical results in a three compartmental environmental system.Adiabatic invariant for fractional generalized Birkhoffian system with variable order.https://zbmath.org/1449.700172021-01-08T12:24:00+00:00"Xie, Hanxing"https://zbmath.org/authors/?q=ai:xie.hanxing"Song, Chuanjing"https://zbmath.org/authors/?q=ai:song.chuanjing"Zhang, Jia'ning"https://zbmath.org/authors/?q=ai:zhang.jianing"Wu, Xueyan"https://zbmath.org/authors/?q=ai:wu.xueyan"Shen, Jingrong"https://zbmath.org/authors/?q=ai:shen.jingrongSummary: Based on the Caputo fractional order derivative of variable order, we studied the perturbation to symmetry and the adiabatic invariant for the fractional generalized Birkhoffian system. As special cases, we also discussed the adiabatic invariants for the fractional Birkhoffian system with variable order, the fractional generalized Birkhoffian system and the classical generalized Birkhoffian system. Finally, an example was given to illustrate the application of the methods and results.Eight positive almost periodic solutions to an delay predator-prey system with impulsive and harvesting terms.https://zbmath.org/1449.342912021-01-08T12:24:00+00:00"Lv, Xiaojun"https://zbmath.org/authors/?q=ai:lv.xiaojun"Xie, Haiping"https://zbmath.org/authors/?q=ai:xie.haiping"Zhao, Kaihong"https://zbmath.org/authors/?q=ai:zhao.kaihongSummary: By using Mawhin's continuation theorem of coincidence degree theory and differential inequalities, we study an delay predator-prey system with impulsive and harvesting terms. Finally, we find some sufficient conditions for the existence of eight positive almost periodic solutions for the system under consideration.An equilibrium point analysis of a class of planar cubic polynomial systems.https://zbmath.org/1449.341052021-01-08T12:24:00+00:00"Long, Neng"https://zbmath.org/authors/?q=ai:long.neng"Liang, Haihua"https://zbmath.org/authors/?q=ai:liang.haihuaSummary: The equilibrium points of a class of planar cubic polynomial systems
\[\dot{x} = -y+\alpha {x^2}-\alpha {y^2}+\beta {x^3}-3\beta x{y^2}, \, \dot{y} = x-2\alpha xy+3\beta {x^2}y-\beta {y^3}\]
are discussed. It is proved that when \(|\alpha-1| \ll 0, |\beta-1| \ll 0\), there are four infinite equilibrium points and all of them are saddle points, and there are three finite equilibrium points and all of them are focal points. The position, order and stability of the three focal points are given.Ulam-Hyers stability of fractional impulsive differential equations.https://zbmath.org/1449.342742021-01-08T12:24:00+00:00"Ding, Yali"https://zbmath.org/authors/?q=ai:ding.yaliSummary: In this paper, we first prove the existence and uniqueness for a fractional differential equation with time delay and finite impulses on a compact interval. Secondly, Ulam-Hyers stability of the equation is established by Picard operator and abstract Gronwall's inequality.Synchronization of fractional chaotic complex networks with delays.https://zbmath.org/1449.342542021-01-08T12:24:00+00:00"Hu, Jian-Bing"https://zbmath.org/authors/?q=ai:hu.jianbing"Wei, Hua"https://zbmath.org/authors/?q=ai:wei.hua"Feng, Ye-Feng"https://zbmath.org/authors/?q=ai:feng.ye-feng"Yang, Xiao-Bo"https://zbmath.org/authors/?q=ai:yang.xiaoboSummary: The synchronization of fractional-order complex networks with delay is investigated in this paper. By constructing a novel Lyapunov-Krasovskii function \(V\) and taking integer derivative instead of fractional derivative of the function, a sufficient criterion is obtained in the form of linear matrix inequalities to realize synchronizing complex dynamical networks. Finally, a numerical example is shown to illustrate the feasibility and effectiveness of the proposed method.Monotone positive solutions of fourth-order boundary value problems with integral boundary conditions.https://zbmath.org/1449.340742021-01-08T12:24:00+00:00"He, Yanqin"https://zbmath.org/authors/?q=ai:he.yanqin"Han, Xiaoling"https://zbmath.org/authors/?q=ai:han.xiaolingSummary: By using the monotone iterative technique, this article studies the existence of monotone positive solutions for fourth order boundary value problems with integral boundary conditions
\[\begin{cases}
u^{ (4)} (t) = f (t,u (t), u' (t)),\; t \in (0,1),\\
u (0) = u' (1) = u''' (1) = 0,\\
u'' (0) = \int_0^1 g (t)u'' (t){\mathrm{d}}t,
\end{cases}\]
where \(f:[0,1] \times [0,+\infty)^2 \to [0,+\infty)\) is continuous.Existence of positive solutions for a class of nonlinear fractional differential equations with boundary values.https://zbmath.org/1449.340712021-01-08T12:24:00+00:00"Cai, Huize"https://zbmath.org/authors/?q=ai:cai.huize"Han, Xiaoling"https://zbmath.org/authors/?q=ai:han.xiaolingSummary: In this paper, by using the Schauder fixed point theorem and Krasnoselskii's fixed point theorem, the existence of positive solutions for the boundary value problem of the nonlinear fractional differential equation
\[{}^{\mathrm{C}}D_{0^+}^\alpha u(t) = f(t,u(t),u'(t), u''(t)), t \in (0,1),\]
\[u'(0) + u''(0) = 0, u'(1) + u''(1) = 0, u(0) = 0\]
is obtained, where \(2 < \alpha \leq 3,{}^{\mathrm{C}} D_{0^+}^\alpha\) is the Caputo fractional derivative.The uniqueness of solution for initial value problems for fractional differential equation involving the Caputo-Fabrizio derivative.https://zbmath.org/1449.340402021-01-08T12:24:00+00:00"Zhang, Shuqin"https://zbmath.org/authors/?q=ai:zhang.shuqin"Hu, Lei"https://zbmath.org/authors/?q=ai:hu.lei"Sun, Sujing"https://zbmath.org/authors/?q=ai:sun.sujingSummary: In this paper, we study some results about the expression of solutions to some linear differential equations for the Caputo-Fabrizio fractional derivative. Furthermore, by the Banach contraction principle, the unique existence of the solution to an initial value problem for nonlinear differential equation involving the Caputo-Fabrizio fractional derivative is obtained.Oscillation analysis of second-order generalized Emden-Fowler-type delay differential equations.https://zbmath.org/1449.342232021-01-08T12:24:00+00:00"Li, Jimeng"https://zbmath.org/authors/?q=ai:li.jimengSummary: We study the oscillatory behavior of a class of second-order generalized Emden-Fowler-type nonlinear delay functional differential equations in this paper. By using the generalized Riccati transformation and some necessary analytic techniques, we establish some new oscillation criteria for the equations under both the cases canonical form and noncanonical form, which deal with some cases not covered by existing results in the literature. Three examples are given to illustrate the main results of this article.Using differentiation matrices for pseudospectral method solve Duffing oscillator.https://zbmath.org/1449.651682021-01-08T12:24:00+00:00"Nhat, L. A."https://zbmath.org/authors/?q=ai:nhat.le-anhSummary: This article presents an approximate numerical solution for nonlinear Duffing Oscillators by pseudospectral (PS) method to compare boundary conditions on the interval \([-1, 1]\). In the PS method, we have been used differentiation matrix for Chebyshev points to calculate numerical results for nonlinear Duffing Oscillators. The results of the comparison show that this solution had the high degree of accuracy and very small errors. The software used for the calculations in this study was Mathematica V.10.4.Fractional-order Legendre functions for solving fractional-order differential equations.https://zbmath.org/1449.330122021-01-08T12:24:00+00:00"Kazem, S."https://zbmath.org/authors/?q=ai:kazem.saeed"Abbasbandy, S."https://zbmath.org/authors/?q=ai:abbasbandy.saeid"Kumar, Sunil"https://zbmath.org/authors/?q=ai:kumar.sunilSummary: In this article, a general formulation for the fractional-order Legendre functions (FLFs) is constructed to obtain the solution of the fractional-order differential equations. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, an efficient and reliable technique for the solution of them is important, too. For the concept of fractional derivative we will adopt Caputo's definition by using Riemann-Liouville fractional integral operator. Our main aim is to generalize the new orthogonal functions based on Legendre polynomials to the fractional calculus. Also a general formulation for FLFs fractional derivatives and product operational matrices is driven. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.Asymptotic behavior of stochastic predator-prey model with epidemic in the predator.https://zbmath.org/1449.341742021-01-08T12:24:00+00:00"Zhang, Qiumei"https://zbmath.org/authors/?q=ai:zhang.qiumei"Wu, Jiajie"https://zbmath.org/authors/?q=ai:wu.jiajieSummary: In this paper, a stochastic predator-prey model with epidemic in the predator under the influence of white noise is formulated. The existence and uniqueness of the positive solution of the model are discussed, the asymptotic behavior of this model is studied.Oscillation for second-order Emden-Fowler-type differential equations with quasilinear neutral term.https://zbmath.org/1449.342292021-01-08T12:24:00+00:00"Qin, Guijiang"https://zbmath.org/authors/?q=ai:qin.guijiang"Yang, Jiashan"https://zbmath.org/authors/?q=ai:yang.jiashanSummary: The oscillatory behavior of certain second-order generalized Emden-Fowler-type differential equations with a quasilinear neutral term is analyzed, where the equation is in a noncanonical form, i.e., \(\int_{t_0}^{+\infty} {a^{-1/\beta}} (t){\mathrm{d}}t < +\infty\). By using the generalized Riccati transformation and inequality technique, some new oscillation criteria for the equations are established. Examples are given to illustrate the applicability of the obtained results.Bounds for the spectrum of a two-parameter eigenvalue problem in a Hilbert space.https://zbmath.org/1449.343112021-01-08T12:24:00+00:00"Gil', Michael"https://zbmath.org/authors/?q=ai:gil.michael-iosifThis paper is devoted to the following two-parameter eigenvalue problem (\(\Lambda\)):\[ T_1v_1-\mu_1v_1-\mu_2A_1v_1=0,\] \[T_2v_2-\mu_1A_2v_2-\mu_2v_2=0,\] where \(\mu_1, \mu_2\) are complex numbers, \(v_m\in{\mathcal{E}}_m, m=1,2,\) and \(T_m, A_m, m=1,2,\) are compact operators in \({\mathcal{E}},\) where \({\mathcal{E}}_m, m=1,2,\) and \({\mathcal{E}}\) are complex separable Hilbert spaces.
Assuming that \[\inf_{j,k=1,2,\dots}|1-\lambda_j(A_1)\lambda_k(A_2)|>0,\] \[A_1, A_2\in SN_{2p}, p\in[1,\infty),\] where \(\lambda_j(A_1)\) and \(\lambda_k(A_2)\) are the eigenvalues of \(A_1\) and \(A_2\), respectively, with their multiplicities, and \(SN_p\) is the Schatten-von Neumann ideal of operators \(K\) with the finite norm \(N_p(K):=[\)trace\((K^*K)^{p/2}]^{1/p},\) the authors suggest estimates for \[r^{(m)}_{spec}(\Lambda):=\sup_{s\in\sigma_m(\Lambda)}|s|\quad {\text{and}}\quad\omega^{(m)}(\Lambda):=\sup_{s\in\sigma_m(\Lambda)}|(s-\overline{s})/2i|,\] where \(\sigma_m(\Lambda)\) is the set of all \((\mu_1, \mu_2)\) for which (\(\Lambda\)) has a solution.
Reviewer: Petio S. Kelevedjiev (Sliven)A class of the quadratic singularly perturbed boundary value problems with two-fold boundary layer.https://zbmath.org/1449.342052021-01-08T12:24:00+00:00"Xu, Jin"https://zbmath.org/authors/?q=ai:xu.jinSummary: In this paper, a class of the quadratic singularly perturbed boundary value problems with a two-fold boundary layer is studied. Under appropriate conditions, the formal approximation of the problems is constructed by using the method of composite expansions, and the existence of the solution is proved by the theory of second order differential inequalities.A new numerical method for solving fractional delay differential equations.https://zbmath.org/1449.651422021-01-08T12:24:00+00:00"Jhinga, Aman"https://zbmath.org/authors/?q=ai:jhinga.aman"Daftardar-Gejji, Varsha"https://zbmath.org/authors/?q=ai:daftardar-gejji.varshaSummary: We present a new numerical method for solving fractional delay differential equations (FDDEs) along with its error analysis. We illustrate applicability and utility of the method by solving various examples. Further, we compare the method with other existing methods such as fractional Adams method (FAM) and new predictor-corrector method (NPCM) developed by \textit{V. Daftardar-Gejji} et al. [Fract. Calc. Appl. Anal. 18, No. 2, 400--418 (2015; Zbl 1317.34159)]. The order of accuracy is shown to be \(O(h^2)\). It is noted that the new method is more time efficient and works for very small values of the order of the fractional derivative, where FAM as well as NPCM fail.Non-Archimedean stability of nonhomogeneous second order linear differential equations.https://zbmath.org/1449.340442021-01-08T12:24:00+00:00"Majani, Hamid"https://zbmath.org/authors/?q=ai:majani.hamidSummary: Let \((\mathbb{R},|\,|)\) be non-Archimedean normed space of real numbers. In this paper, we prove the Hyers-Ulam stability of nonhomogeneous second order linear differential equations with non-constant coefficients,
\[
y'' +f(x) y' +g(x)y=h(x)
\]
in the non-Archimedean normed space \((\mathbb{R},|\,|)\), where \(f\), \(g\), \(h:(a,b)\subseteq \mathbb{R}\to\mathbb{R}\) are given continuous functions.Fractional differential equations and Volterra-Stieltjes integral equations of the second kind.https://zbmath.org/1449.340092021-01-08T12:24:00+00:00"Asanov, Avyt"https://zbmath.org/authors/?q=ai:asanov.avyt"Almeida, Ricardo"https://zbmath.org/authors/?q=ai:almeida.ricardo"Malinowska, Agnieszka B."https://zbmath.org/authors/?q=ai:malinowska.agnieszka-barbaraSummary: In this paper, we construct a method to find approximate solutions to fractional differential equations involving fractional derivatives with respect to another function. The method is based on an equivalence relation between the fractional differential equation and the Volterra-Stieltjes integral equation of the second kind. The generalized midpoint rule is applied to solve numerically the integral equation and an estimation for the error is given. Results of numerical experiments demonstrate that satisfactory and reliable results could be obtained by the proposed method.Finite-time sliding mode synchronization for Victor-Carmen chaotic systems.https://zbmath.org/1449.341872021-01-08T12:24:00+00:00"Mao, Beixing"https://zbmath.org/authors/?q=ai:mao.beixingSummary: Finite-time synchronization of Victor-Carmen systems is studied in the paper. A new type sliding mode surface is proposed based on sliding mode method and the convergence speed of the sliding mode surface is faster than the old one. Sufficient conditions are acquired for finite-time sliding mode synchronization of Victor-Carmen systems. Conclusions demonstrate that the master-slave system of Victor-Carmen systems is finite-time sliding mode synchronizable under certain conditions.Increasing convergent and divergent solutions to nonlinear delayed differential equations.https://zbmath.org/1449.342332021-01-08T12:24:00+00:00"Diblík, Josef"https://zbmath.org/authors/?q=ai:diblik.josef"Chupáč, Radoslav"https://zbmath.org/authors/?q=ai:chupac.radoslav"Růžičková, Miroslava"https://zbmath.org/authors/?q=ai:ruzickova.miroslavaThe paper deals with the following nonlinear delay differential system \[\dot{y}(t)=\sum_{k=1}^m\beta^k(t)[y(t-\delta_k)-y(t-\tau_k)]^{r_k}.\] It is obtained: criteria guaranteeing the existence of increasing and unbounded solutions, criteria guaranteeing the existence of increasing and convergent solutions, as well as inequalities comparing such solutions with some given increasing functions.
Reviewer: Leonid Berezanski (Beer-Sheva)An investigation on the existence and Ulam stability of solution for an impulsive fractional differential equation.https://zbmath.org/1449.342752021-01-08T12:24:00+00:00"Guo, Yuchen"https://zbmath.org/authors/?q=ai:guo.yuchen"Shu, Xiaobao"https://zbmath.org/authors/?q=ai:shu.xiaobaoSummary: In this paper, we investigate the existence and Ulam stability of solution for impulsive Riemann-Liouville fractional neutral function differential equation with infinite delay of order \(1 < \beta < 2\). Firstly, the solution for the equation is proved. By using the fixed point theorem as well as the Hausdorff measure of noncompactness, the existence results are obtained and the Ulam stability of the solution is proved.Hopf bifurcation analysis of flux neuron model with time delays.https://zbmath.org/1449.343022021-01-08T12:24:00+00:00"Yu, Huanhuan"https://zbmath.org/authors/?q=ai:yu.huanhuan"An, Xinlei"https://zbmath.org/authors/?q=ai:an.xinlei"Lu, Zhengyu"https://zbmath.org/authors/?q=ai:lu.zhengyu"Wang, Wenjing"https://zbmath.org/authors/?q=ai:wang.wenjingSummary: We study the stability, the existence of Hopf bifurcation and the bifurcation direction and bifurcating periodic solution of the double-delay flux neuron model by using stability theory, central manifold theorem and other methods. Some conclusions obtained by numerical simulation are given.Fractional integral inequalities and global solutions of fractional differential equations.https://zbmath.org/1449.340412021-01-08T12:24:00+00:00"Zhu, Tao"https://zbmath.org/authors/?q=ai:zhu.taoSummary: New fractional integral inequalities are established, which generalize some famous inequalities. Then we apply these new fractional integral inequalities to study global existence results for fractional differential equations.Existence of limit cycles for some generalisation of the Liénard equations: the relativistic and the prescribed curvature cases.https://zbmath.org/1449.341042021-01-08T12:24:00+00:00"Carletti, Timoteo"https://zbmath.org/authors/?q=ai:carletti.timoteo"Villari, Gabriele"https://zbmath.org/authors/?q=ai:villari.gabrieleSummary: We study the problem of existence of periodic solutions for some generalisations of the relativistic Liénard equation \[ \frac{d}{dt}\frac{\dot{x}}{\sqrt{1-\dot{x}^{2}}}+\hat{f}(x,\dot{x})\dot{x}+g(x)=0, \] and the prescribed curvature Liénard equation \[ \frac{d}{dt}\frac{\dot{x}}{\sqrt{1+\dot{x}^{2}}}+\hat{f}(x,\dot{x})\dot{x}+g(x)=0, \] where the damping function depends both on the position and the velocity. In the associated phase-plane this corresponds to a term of the form \(f(x; y)\) instead of the standard dependence on \(x\) alone. By controlling the continuability of the solutions, we are able to prove the existence of at least a limit cycle in the associated phase-plane for both cases, moreover we provide results with a prefixed arbitrary number of limit cycles. Some examples are given to show the applicability of these results.Strongly formal Weierstrass non-integrability for polynomial differential systems in \(\mathbb{C}^2\).https://zbmath.org/1449.340012021-01-08T12:24:00+00:00"Giné, Jaume"https://zbmath.org/authors/?q=ai:gine.jaume"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaumeSummary: Recently it has been given a criterion for determining the weakly formal Weierstrass non-integrability of polynomial differential systems in \(\mathbb{C}^2\). Here we extend this criterion for determining the strongly formal Weierstrass non-integrability which includes the weakly formal Weierstrass non-integrability of polynomial differential systems in \(\mathbb{C}^2\). The criterion is based on the solutions of the form \(y=f(x)\) with \(f(x) \in \mathbb{C}[[x]]\) of the differential system whose integrability we are studying. The results are applied to a differential system that contains the famous force-free Duffing and the Duffing-van der Pol oscillators.Research on complex dynamics of a new 4D hyperchaotic system.https://zbmath.org/1449.370672021-01-08T12:24:00+00:00"Hong, Lingling"https://zbmath.org/authors/?q=ai:hong.lingling"Yang, Qigui"https://zbmath.org/authors/?q=ai:yang.qiguiSummary: In this paper, based on the 3D Lorenz-like chaotic system, a linear feedback controller is designed and a new four-dimensional hyperchaos system with only two times nonlinear terms is proposed. This system has simple algebraic structure, but shows complex dynamic behavior. It is proved theoretically that it is not equivalent to hyperchaotic Li system. In order to study the complex dynamics of the system, the stability of the system at the hyperbolic and non-hyperbolic equilibrium points is discussed in detail, and the Hopf bifurcation is strictly analyzed. The approximate expression and stability of the periodic orbit generated by the Hopf bifurcation are obtained. Furthermore, the Lyapunov exponent spectrum, Poincaré map and bifurcation diagram of the system are obtained by numerical simulation with the help of modern mathematical software, and the existence of the hyperchaotic attractor is verified.An ordering on Green's function and a Lyapunov-type inequality for a family of nabla fractional boundary value problems.https://zbmath.org/1449.390182021-01-08T12:24:00+00:00"Jonnalagadda, Jagan Mohan"https://zbmath.org/authors/?q=ai:jonnalagadda.jaganmohanSummary: In this article, we consider a family of two-point Riemann-Liouville type nabla fractional boundary value problems involving a fractional difference boundary condition. We construct the corresponding Green's function and deduce its ordering property. Then, we obtain a Lyapunov-type inequality using the properties of Green's function, and illustrate a few of its applications.Bounded solutions of singular differential equation with asymptotic conditions.https://zbmath.org/1449.340682021-01-08T12:24:00+00:00"Zhao, Jin"https://zbmath.org/authors/?q=ai:zhao.jinSummary: Applying Schauder's fixed point theorem, the author considers a class of second-order singular differential equations with asymptotic conditions, proves the existence of bounded solutions, and generalizes the conclusion of bounded solutions of the arctic circulation model to general second order singular differential equations.On the study of oscillating viscous flows by using the Adomian-Padé approximation.https://zbmath.org/1449.760412021-01-08T12:24:00+00:00"Liu, Chi-Min"https://zbmath.org/authors/?q=ai:liu.chi-minSummary: The Adomian-Padé technique is applied to examine two oscillating viscous flows, the Stokes' second problem and the pressure-driven pulsating flow. Main purposes for studying oscillating flows are not only to verify the accuracy of the approximation solution, but also to provide a basis for analyzing more problems by the present method with the help of Fourier analysis. Results show that the Adomian-Padé approximation presents a very excellent behavior in comparison with the exact solution of Stokes' second problem. For the pulsating flow, only the Adomian decomposition method is required to perform the calculation as the fluid domain is finite where the Padé approximant may not provide a better solution. Based on present results, more problems can be mathematically solved by using the Adomian-Padé technique, the Fourier analysis, and powerful computers.Invariant algebraic surfaces, Hamiltonian structures and dynamic behavior at infinity for three-wave interaction model.https://zbmath.org/1449.341292021-01-08T12:24:00+00:00"Niu, Yanqiu"https://zbmath.org/authors/?q=ai:niu.yanqiu"Yang, Shuangling"https://zbmath.org/authors/?q=ai:yang.shuangling"Xu, Mingxing"https://zbmath.org/authors/?q=ai:xu.mingxingSummary: Firstly, by using the elimination theory in algebraic geometry, we give sufficient conditions for the existence of invariant algebraic surfaces in a three-wave interaction model. Secondly, we construct an infinite number of Hamiltonian-Poisson structures of the system, the system was bi-Hamiltonian. Finally, we use the Poincaré compactification technique in \(\mathbb{R}^3\) to describe the dynamic behavior at infinity of the system.Oscillation criteria for a class of second order neutral generalized Emder-Fowler equations with damping terms.https://zbmath.org/1449.342242021-01-08T12:24:00+00:00"Lin, Wenxian"https://zbmath.org/authors/?q=ai:lin.wenxianSummary: The author considers the oscillation of solutions for a class of second order neutral generalized Emder-Fowler equations with damping terms. In order to achieve the goal, the author first uses the Riccati transformation to deal with the nonlinear terms and damping terms, and then uses Philo's integral average method to establish the oscillation criteria for the solutions of these equations.Stability and Hopf bifurcation in a delayed predator-prey system with herd behavior.https://zbmath.org/1449.920412021-01-08T12:24:00+00:00"Xu, Chaoqun"https://zbmath.org/authors/?q=ai:xu.chaoqun"Yuan, Sanling"https://zbmath.org/authors/?q=ai:yuan.sanlingSummary: A special predator-prey system is investigated in which the prey population exhibits herd behavior in order to provide a self-defense against predators, while the predator is intermediate and its population shows individualistic behavior. Considering the fact that there always exists a time delay in the conversion of the biomass of prey to that of predator in this system, we obtain a delayed predator-prey model with square root functional response and quadratic mortality. For this model, we mainly investigate the stability of positive equilibrium and the existence of Hopf bifurcation by choosing the time delay as a bifurcation parameter.The bifurcation analysis and chaos control of a mixed duopoly model.https://zbmath.org/1449.341772021-01-08T12:24:00+00:00"Zhao, Na"https://zbmath.org/authors/?q=ai:zhao.na"Zhou, Wei"https://zbmath.org/authors/?q=ai:zhou.wei"Wang, Wenrui"https://zbmath.org/authors/?q=ai:wang.wenruiSummary: Under the assumption of bounded rationality, a dynamical mixed duopoly model consisting of a public-private joint firm and a foreign enterprise is established. The existence and stability of an equilibrium point of the system are analyzed, and it is concluded that the equilibrium point cannot loss its stability via Neimark-Sacker bifurcation. The dynamical behaviors of the system when selecting different parameters is numerically simulated by Matlab. The results show that the system will enter the chaotic state through the flip bifurcation. And under some certain parameter conditions, the phenomenon of coexistence of multiple attractors can be found. In addition, it is also found that the speed of adjustment affects the steady state of the built model. When the speed of adjustment is selected large enough, the system is more likely to become unstable and enter a chaotic state. The chaotic state of the system is controlled successfully using the so-called delayed feedback method.Inverse spectral problems of symmetric potential functions.https://zbmath.org/1449.340562021-01-08T12:24:00+00:00"Wang, Wenjing"https://zbmath.org/authors/?q=ai:wang.wenjingSummary: In this paper, we consider the uniqueness problem of inverse Sturm-Liouville (S-L) differential operators defined on the interval \([0, 1]\). By making use of the Weyl function and Marchenko's uniqueness theorem, it is shown that if the potential function \(q (x)\) is multiple, symmetric and is known on some intervals, then the potential function \(q (x)\) on the interval \([0, 1]\) can be uniquely determined in terms of choosing a set of appropriate common eigenvalues.Almost periodic solution of a nonautonomous continuous competitive system.https://zbmath.org/1449.341712021-01-08T12:24:00+00:00"Yu, Shengbin"https://zbmath.org/authors/?q=ai:yu.shengbinSummary: This paper considers the dynamic behaviors of a nonautonomous continuous competitive system with nonlinear inter-inhibition terms. By using the comparison theorem of differential equation and constructing the suitable Lyapunov function, sufficient conditions for the permanence, global attractivity and the existence of a unique positive almost periodic solution of the system are obtained. The results supplement some known ones. Numerical simulations show the feasibility of our results.Analysis on the stability of the static neural networks with time-varying delays based on the convex combination.https://zbmath.org/1449.342472021-01-08T12:24:00+00:00"Mao, Kai"https://zbmath.org/authors/?q=ai:mao.kai"Yang, Shujie"https://zbmath.org/authors/?q=ai:yang.shujie"Liu, Dan"https://zbmath.org/authors/?q=ai:liu.danSummary: The global asymptotic stability of static neural networks with time-varying delay is studied in this paper. By taking more delayed-state variables into account, a newly augmented Lyapunov-Krasovskii functional is constructed. By using the delay partitioning approach together with the free weighting matrix method and Jensen integral inequality, a delay-dependent global asymptotic stability criterion is obtained based on the convex combination method, which is less conservative than some existing ones. An example is provided to show the effectiveness and reduced conservatism of the proposed results.Applications of double well potentials in the collective nuclear motion.https://zbmath.org/1449.810162021-01-08T12:24:00+00:00"Budaca, Radu"https://zbmath.org/authors/?q=ai:budaca.raduSummary: The recent applications of double well potentials in the description of shape coexistence phenomena and chiral symmetry breaking in nuclear physics is discussed with an emphasis on the analytical properties of the corresponding wave functions. By means of the density of probability distribution, the effect of the quantum tunneling on the composition of the wave functions is dully investigated. The results are used to identify the distinctive features between the one-dimensional
and central multidimensional problems.kd-tree based adaptive interpolation algorithm for chemical kinetics problems with interval parameters.https://zbmath.org/1449.800212021-01-08T12:24:00+00:00"Morozov, A. Yu."https://zbmath.org/authors/?q=ai:morozov.alexei-yurievich"Reviznikov, D. L."https://zbmath.org/authors/?q=ai:reviznikov.d-l"Gidaspov, V. Yu."https://zbmath.org/authors/?q=ai:gidaspov.v-yuSummary: In this paper, the simulatiion of chemical processes with uncertainty in the parameters is considered. A new approach is suggested, which consists in building a dynamic structured net based on a kd-tree, over a space formed by the interval parameters. When the algorithm is executed, during each integration step a piecewise constant polynomial function is build, interpolating the connection between the solution and the exact values of interval parameters. The algorithm is tested on chemical kinetics problems, including combustion processes, demonstrating its efficiency and wide area of application.The existence and stability of almost periodic solution for a class of neutral neural networks.https://zbmath.org/1449.342422021-01-08T12:24:00+00:00"Fang, Congna"https://zbmath.org/authors/?q=ai:fang.congna"Xie, Huiqin"https://zbmath.org/authors/?q=ai:xie.huiqinSummary: We study the almost periodic solution for a class of neutral neural networks with distributed time delays and time-varying delays. Using the fixed point theorem in Banach space, we obtain some new results on the existence and uniqueness and stability of the almost periodic solution. Finally, the validity of our results is illustrated by an example.Strong stability of a family enzyme-catalyzed nonlinear dynamic system in microbial batch fermentation.https://zbmath.org/1449.341512021-01-08T12:24:00+00:00"Liu, Yang"https://zbmath.org/authors/?q=ai:liu.yang.19|liu.yang.9|liu.yang.10|liu.yang.6|liu.yang.5|liu.yang.2|liu.yang.12|liu.yang.17|liu.yang.20|liu.yang.3|liu.yang.13|liu.yang.18|liu.yang|liu.yang.23|liu.yang.22|liu.yang.7|liu.yang.11|liu.yang.8|liu.yang.15|liu.yang.16|liu.yang.21|liu.yang.14|liu.yang.1|liu.yang.4"Yang, Qi"https://zbmath.org/authors/?q=ai:yang.qi"Feng, Enmin"https://zbmath.org/authors/?q=ai:feng.enmin"Xiu, Zhilong"https://zbmath.org/authors/?q=ai:xiu.zhilong(no abstract)Hopf bifurcation analysis of a neutral neural network model.https://zbmath.org/1449.342992021-01-08T12:24:00+00:00"Wang, Peiguang"https://zbmath.org/authors/?q=ai:wang.peiguang"Tan, Jun"https://zbmath.org/authors/?q=ai:tan.jun"Cao, Jianzhi"https://zbmath.org/authors/?q=ai:cao.jianzhi"Bao, Junyan"https://zbmath.org/authors/?q=ai:bao.junyanSummary: In this paper, using Lyapunov-Schmidt reduction method and singularity theory, the dynamic behavior of a neutral neural network model is studied. It is proved that Hopf bifurcation occurs at the equilibrium point of the original equation, and the approximate analytical expression of the periodic solution of the bifurcation is obtained, and the error analysis is carried out. Finally, the conclusions obtained in this paper are summarized.Dynamic analysis of Avian Influenza A (H7N9) based on live poultry market.https://zbmath.org/1449.341732021-01-08T12:24:00+00:00"Zhang, Juping"https://zbmath.org/authors/?q=ai:zhang.juping"Sun, Minqi"https://zbmath.org/authors/?q=ai:sun.minqiSummary: H7N9 Avian Influenza virus seriously affects human health and life safety. To study the spread of H7N9, we establish A H7N9 Avian Influenza transmission model based on the interaction between the live poultry market and the population containing the virus in the environment. We calculate the basic reproduction number \({R_0}\) by the next generation matrix method. When \({R_0} < 1\), we prove the global stability of the disease-free equilibrium. When \({R_0} > 1\), we prove the local stability of the endemic equilibrium. At the same time, numerical simulations are carried out. Through sensitivity analysis, it is found that when the closing rate of live poultry market reach a certain value, Avian Influenza A (H7N9) is controlled accordingly, which provides a theoretical basis for effectively preventing and controlling the spread of Avian Influenza A (H7N9).Transmission dynamics of SIR model with rewiring on configuration network.https://zbmath.org/1449.341342021-01-08T12:24:00+00:00"Gao, Shufei"https://zbmath.org/authors/?q=ai:gao.shufei"Yuan, Sanling"https://zbmath.org/authors/?q=ai:yuan.sanlingSummary: Considering the rapid change and heterogeneity of partnerships in disease transmission, in this paper, an edge-based SIR model with rewiring is developed on configuration network. Firstly, the threshold condition to determine whether the disease can be spread on the network is obtained, i.e., the basic reproduction number. When the basic reproduction number is less than 1, the model has a unique and stable disease-free equilibrium point. When the basic reproduction number is greater than 1, the disease-free equilibrium point becomes unstable and emerges a stable endemic equilibrium point. The numerical simulation results show that the solution of the model is in good agreement with the mean value of 100 random simulation results.Stability and local Hopf bifurcation in a delayed predator-prey system with a refuge.https://zbmath.org/1449.343002021-01-08T12:24:00+00:00"Wei, Zhen"https://zbmath.org/authors/?q=ai:wei.zhenSummary: In this paper, we investigate a delayed predator-prey system with a refuge. We consider the stability and the existence of local Hopf bifurcations of the equilibria, and then derive the explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcation by using the normal form theory and center manifold argument.Evolutionary dynamics of environmental pollution model with impulse period disturbance.https://zbmath.org/1449.920352021-01-08T12:24:00+00:00"Song, Le"https://zbmath.org/authors/?q=ai:song.le"Liu, Bing"https://zbmath.org/authors/?q=ai:liu.bing|liu.bing.1"Wang, Xin"https://zbmath.org/authors/?q=ai:wang.xin.3|wang.xin.11|wang.xin.5|wang.xin.12|wang.xin.8|wang.xin.1|wang.xin|wang.xin.7|wang.xin.2|wang.xin.10|wang.xin.6|wang.xin.9|wang.xin.13|wang.xin.4Summary: The effects of pulse pollution on the evolution of phenotypic trait in a single population are studied. Firstly, biological evolution model with phenotypic characteristics is established. Secondly, the conditions for evolutionary stability, convergence stability and evolutionary branch are given by critical function analysis method. Finally, the relationship among evolutionary singularity, pulse period and the amount of toxin input in continuous stability is given by numerical simulation.Some results on difference Riccati equations and delay differential equations.https://zbmath.org/1449.300492021-01-08T12:24:00+00:00"Wang, Qiong"https://zbmath.org/authors/?q=ai:wang.qiong"Long, Fang"https://zbmath.org/authors/?q=ai:long.fang"Wang, Jun"https://zbmath.org/authors/?q=ai:wang.jun.2Summary: We investigate difference Riccati equations with rational coefficients and delay differential equations with constant coefficients. For difference Riccati equations with some relation among coefficients, we prove that every transcendental meromorphic solution is of order no less than one. We also consider the rational solutions for delay differential equations.An HIV-1 virus model with cell-to-cell viral transmissions, discrete delays, humoral immunity and RTI.https://zbmath.org/1449.920462021-01-08T12:24:00+00:00"Pu, Yue"https://zbmath.org/authors/?q=ai:pu.yue"Liu, Xianning"https://zbmath.org/authors/?q=ai:liu.xianningSummary: In this paper, we investigated the dynamical behavior of an HIV-1 virus model with both virus to cell and cell to cell transmissions and two discrete delays using reverse transcriptase inhibitors. We studied the effect of time delay on stability of the equilibria. The study shows that the intracellular delay has no effect on the stability of the equilibria of the infection model. However, the immune delay can change the stability of the immune-activated equilibrium \({E_2}\) and lead to the existence of Hopf bifurcation. Hopf bifurcation occurs when the immune delay passes a critical value. By using suitable Lyapunov functional and the LaSalle's invariance principle, we also established the global stabilities of the two boundary equilibria. If \({R_0} < 1\), the infection-free equilibrium \({R_0}\) is globally asymptotically stable; if \({R_1} < 1 < {R_0}\), the immune-inactivated equilibrium \({E_1}\) is globally asymptotically stable; if \({R_1} > 1\), \({\tau_2} = 0\), the immune-activated equilibrium \({E_2}\) is globally asymptotically stable. In the end, we verified our theorems with numerical simulation.Stability and optimal control strategy for a class of SEIR infectious disease model.https://zbmath.org/1449.341362021-01-08T12:24:00+00:00"Gao, Zhenbin"https://zbmath.org/authors/?q=ai:gao.zhenbin"Guan, Dongyu"https://zbmath.org/authors/?q=ai:guan.dongyuSummary: A class of SEIR infectious disease models are considered which have constant inputs and dual line features for the susceptible and latent. The stability of two equilibria analyzed. If \(R_0 \leq 1\), then the disease free equilibrium is globally asymptotically stable; while if \(R_0 > 1\), the endemic equilibrium is globally asymptotically stable. A control strategy variance is introduced, and the performance index is set for the system. Utilizing the minimum principle based on optimal control theory, the optimal control strategy which satisfies the constraint condition is achieved. Simulating examples demonstrate that the presented method is feasible and effective.Positive solutions for fractional boundary value problems of a class in ordered Banach spaces.https://zbmath.org/1449.340792021-01-08T12:24:00+00:00"Li, Xiaolong"https://zbmath.org/authors/?q=ai:li.xiaolongSummary: The existence of positive solutions for the boundary value problem of a class of fractional differential equations
\[{}^CD_{0^+}^\alpha u (t) = f (t,u (t)), 0 \leq t \leq 1,\, u (0) = u' (1) = u'' (0) = \theta\]
in an ordered Banach space \(E\) is discussed, where \(2<\alpha\leq 3\), and \({}^CD_{0^+}^\alpha\) is the Caputo fractional differentive, \(f:[0,1] \times P \to P\) is continuous, and \(P\) is the cone of positive elements in \(E\). An existence result of positive solutions is obtained by employing a new estimate of noncompactness measure and the fixed point index theory of condensing mapping.Oscillation of second order nonlinear neutral differential equations with distributed delay.https://zbmath.org/1449.342262021-01-08T12:24:00+00:00"Li, Wenjuan"https://zbmath.org/authors/?q=ai:li.wenjuan"Li, Shuhai"https://zbmath.org/authors/?q=ai:li.shuhai"Yu, Yuanhong"https://zbmath.org/authors/?q=ai:yu.yuanhongSummary: In this work, we study the oscillation of the second order nonlinear neutral differential equations with distributed delay \[ (r (t)|z' (t)|^{\alpha - 1}z' (t))' + \int_c^d f (t, x (\sigma (t,\xi))){\mathrm{d}}\xi = 0,\] where \(t \geq {t_0}\), \(z (t) = x (t) + \int_a^b p(t,\xi)x (\tau (t,\xi)){\mathrm{d}}\xi\). We establish some new oscillation criteria for the above equation. These results extend and improve some known results in the cited literature. Also, our results are illustrated with some examples. It is shown that the theorem has some advantages over the existing literature.On the stability of Duffing equations using \({L^p}\) norms.https://zbmath.org/1449.341202021-01-08T12:24:00+00:00"Li, Yuchen"https://zbmath.org/authors/?q=ai:li.yuchen"Zhang, Kangqun"https://zbmath.org/authors/?q=ai:zhang.kangqunSummary: In this paper, the unique existence and local exponential asymptotic stability of a periodic solution of second-order equations of Duffing type are proved. The sharp rate of exponential decay is determined for a solution that is near to the unique periodic solution. Furthermore, the positivity of such a periodic solution under an additional condition is obtained.Existence and finite-time-stability of solutions for a class of fractional order fuzzy Cohen-Grossberg neural networks.https://zbmath.org/1449.342732021-01-08T12:24:00+00:00"Xiang, Hongjun"https://zbmath.org/authors/?q=ai:xiang.hongjun"Wang, Jinhua"https://zbmath.org/authors/?q=ai:wang.jinhua.1Summary: In this paper, a class of fractional-order fuzzy Cohen-Grossberg neural networks is discussed. By combining with the contraction mapping principle, the properties of fractional differential equations and inequality technique, the existence, uniqueness and finite-time-stability of the solutions for this model are studied. Additionally, an example is given to illustrate the main results.Modeling for love dynamics.https://zbmath.org/1449.341432021-01-08T12:24:00+00:00"Hu, Xiao"https://zbmath.org/authors/?q=ai:hu.xiao"Zhu, Zirui"https://zbmath.org/authors/?q=ai:zhu.zirui"Xu, Yancong"https://zbmath.org/authors/?q=ai:xu.yancongSummary: In this paper, the method of mathematical modeling is used to simulate the dynamical behaviors in love, and the model of nonlinear ordinary differential equations is established. By using the knowledge of dynamical systems, the existence of Hopf bifurcation and the existence of stable or unstable limit cycles in the model are proved. The results show that to maintain a long-term love requires a certain sense of distance.Theory of hybrid fractional differential equations with complex order.https://zbmath.org/1449.340332021-01-08T12:24:00+00:00"Vivek, Devaraj"https://zbmath.org/authors/?q=ai:vivek.devaraj"Baghani, Omid"https://zbmath.org/authors/?q=ai:baghani.omid"Kanagarajan, Kuppusamy"https://zbmath.org/authors/?q=ai:kanagarajan.kuppusamySummary: We develop the theory of hybrid fractional differential equations with the complex order \(\theta\in \mathbb{C}\), \(\theta=m+i\alpha\), \(0<m\leq 1\), \(\alpha\in \mathbb{R}\), in Caputo sense. Using Dhage's type fixed point theorem for the product of abstract nonlinear operators in Banach algebra; one of the operators is \(\mathfrak{D}\)-Lipschitzian and the other one is completely continuous, we prove the existence of mild solutions of initial value problems for hybrid fractional differential equations. Finally, an application to solve one-variable linear fractional Schrödinger equation with complex order is given.Global stability of an eco-epidemiological model with Beddington-DeAngelis functional response and delay.https://zbmath.org/1449.342852021-01-08T12:24:00+00:00"Bai, Hongfang"https://zbmath.org/authors/?q=ai:bai.hongfang"Xu, Rui"https://zbmath.org/authors/?q=ai:xu.ruiSummary: In this paper, an eco-epidemiological model with Beddington-DeAngelis functional response and a time delay representing the gestation period of the predator is studied. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are obtained for the global stability of the interior equilibrium and the disease-free equilibrium of the system, respectively.On the solution of Fermat-type differential-difference equations.https://zbmath.org/1449.300692021-01-08T12:24:00+00:00"Liu, Dan"https://zbmath.org/authors/?q=ai:liu.dan"Deng, Bingmao"https://zbmath.org/authors/?q=ai:deng.bingmao"Yang, Degui"https://zbmath.org/authors/?q=ai:yang.deguiSummary: In this paper, we mainly discuss the entire solutions of finite order of the following Fermat type differential-difference equation \[[f^{ (k)} (z)]^2 + [{\Delta_c}f (z)]^2 = 1,\] and the systems of differential-difference equations of the form \[\begin{cases}[{f_1^{ (k)}} (z)]^2 + [{\Delta_c}{f_2} (z)]^2 = 1, \\ [{f_2^{ (k)}} (z)]^2 + [{\Delta_c}{f_1} (z)]^2 = 1.\end{cases}\] Our results can be proved to be the sufficient and necessary solutions to both the equation and the system of equations.Triple positive solutions for a third-order three-point boundary value problem.https://zbmath.org/1449.340912021-01-08T12:24:00+00:00"Wu, Hongping"https://zbmath.org/authors/?q=ai:wu.hongpingSummary: In this paper, we study the existence of triple positive solutions for the nonlinear third-order three-point boundary value problem \[\begin{cases}u''' (t) = -h (t)f (t,u (t),u' (t),u'' (t)),\; 0< t < 1, \\ u (0) = u' (1) = u'' (\eta) = 0,\end{cases}\] where \(\eta\in [0,\frac{1}{2})\) is a constant. By using a fixed-point theorem, we obtain the triple positive solutions to the boundary value problem, and an example is given to illustrate the importance of the result we obtained.Dynamics of a stochastic predator-prey model with pulse input in a polluted environment.https://zbmath.org/1449.341312021-01-08T12:24:00+00:00"Fu, Yingjie"https://zbmath.org/authors/?q=ai:fu.yingjie"Lan, Guijie"https://zbmath.org/authors/?q=ai:lan.guijie"Zhang, Shuwen"https://zbmath.org/authors/?q=ai:zhang.shuwen"Wei, Chunjin"https://zbmath.org/authors/?q=ai:wei.chunjinSummary: In this paper, we show a stochastic predator-prey model with pulse input in a polluted environment. The existence and uniqueness of the positive global solution and the boundedness of expectation of the system are all proved. Sufficient conditions for the existence and boundedness of periodical solution are obtained, and it is globally attractive with probability 1. The threshold of population extinction and persistence in the mean are obtained too. Finally, some numerical simulations are carried out to illustrate the main results.Stability and local bifurcations in the Solow model with delay.https://zbmath.org/1449.342902021-01-08T12:24:00+00:00"Kulikov, D. A."https://zbmath.org/authors/?q=ai:kulikov.dmitrii-anatolevichIn this paper, the author shows that taking into account the delay in a well-known mathematical model of macroeconomics -- the Solow model -- leads to a substantial change in the dynamics of solutions. The delay is introduced in the term characterizing the depreciation of fixed assets. It is shown that taking into account the delay factor leads to a loss of stability of the equilibrium both in the supply and demand model and in the main Solow model. The possibility of subcritical bifurcations for bifurcation parameters close to critical ones is proved. The asymptotic formulae for the corresponding periodic solutions are obtained.
Reviewer: Artyom Andronov (Saransk)Exponential stability of nonlinear neutral delay stochastic differential equation.https://zbmath.org/1449.651442021-01-08T12:24:00+00:00"Song, Meiling"https://zbmath.org/authors/?q=ai:song.meiling"Hu, Liangjian"https://zbmath.org/authors/?q=ai:hu.liangjianSummary: On the exponential stability of numerical solutions of the nonlinear neutral delay stochastic differential equation, the drift term coefficients and the diffusion term coefficients in the equations are generally set to the growth limit conditions separately. In order to reduce the limit on the growth of each coefficient, the diffusion term coefficients and the drift term coefficients in nonlinear neutral delay stochastic differential equations are considered together, that is, limiting the two coefficients in a formula. The sufficient conditions for the exponential stability of the Euler-Maruyama (EM) numerical solution of the nonlinear neutral delay differential equation are given. The results show that for the given sufficient condition, the EM numerical solution of the nonlinear neutral delay differential equation is exponentially stable for any initial value.A pertussis epidemic model with periodic infection rate.https://zbmath.org/1449.341552021-01-08T12:24:00+00:00"Nan, Xi"https://zbmath.org/authors/?q=ai:nan.xi"Liu, Junli"https://zbmath.org/authors/?q=ai:liu.junliSummary: Due to the periodicity of pertussis prevalence, based on the \(S_1I_1 RVS_2I_2\) pertussis model with secondary infection, a pertussis infectious disease model with periodic infection rate is considered. Using the spectral radius of the integral operator, the basic reproduction number \(R_0\) is obtained. \(R_0\) determines the extinction and uniform persistence of pertussis. The uniform persistence of the model is discussed by the Poincaré map. The theoretical results are illustrated by numerical simulations. It is shown that when \(R_0 = 0.2526 < 1\) the disease-free equilibrium of the model is locally asymptotically stable and the disease dies out; when \(R_0= 4.4273 > 1\), the disease-free equilibrium is unstable and the disease persists. The model also has a positive periodic solution.Dynamic behaviors of May type cooperative system with Michaelis-Menten type harvesting.https://zbmath.org/1449.341722021-01-08T12:24:00+00:00"Yu, Xiangqin"https://zbmath.org/authors/?q=ai:yu.xiangqin"Chen, Fengde"https://zbmath.org/authors/?q=ai:chen.fengde"Lai, Liyun"https://zbmath.org/authors/?q=ai:lai.liyunSummary: A traditional May type cooperative model incorporating Michaelis-Menten type harvesting is proposed and studied in this paper. Sufficient conditions which ensure the extinction of the first species and the existence of a unique globally attractive positive equilibrium are obtained, respectively. Numerical simulations are carried out to show the feasibility of the main results.Multiple positive solutions for a class of integral boundary value problem.https://zbmath.org/1449.340922021-01-08T12:24:00+00:00"Yang, Yang"https://zbmath.org/authors/?q=ai:yang.yang.1|yang.yang.5|yang.yang.4|yang.yang.3|yang.yang|yang.yang.2"Yang, Yunrui"https://zbmath.org/authors/?q=ai:yang.yunrui"Liu, Kepan"https://zbmath.org/authors/?q=ai:liu.kepanSummary: In this paper, the existence and multiplicity of positive solutions for a class of non-resonant fourth-order integral boundary value problems
\[\begin{cases}
u^{ (4)} (t)+\beta u'' (t)-\alpha u (t) = f (t,u (t),u'' (t)), \, t \in (0,1),\\
u'' (0) = u'' (1) = 0,\\
u (0) = 0,\, u (1) = \left (\frac{1}{\lambda_2}-\frac{1}{\lambda_1}\right)\int_0^1 q (s)f (s,u (s),u'' (s))\mathrm{d}s,
\end{cases}\]
with two parameters are established by using Guo-Krasnoselskii's fixed point theorem, where \(f\in C ([0,1]\times [0,+\infty)\times [-\infty,0), [0,+\infty))\), \(q (t)\in {L^1}[0,1]\) is nonnegative, \(\alpha\), \(\beta \in \mathbb{R}\) and satisfy \(\beta< 2\pi^2\), \(\alpha > 0\), \(\alpha/\pi^4 +\beta/\pi^2 < 1\), \(\lambda_{1,2} = (-\beta\mp \sqrt{\beta^2+4\alpha})/2\). Corresponding examples illustrate the results we obtained.A class of quasi-fractional Noether's theorems for nonconservative systems in event space.https://zbmath.org/1449.370462021-01-08T12:24:00+00:00"Wang, Ze"https://zbmath.org/authors/?q=ai:wang.ze"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.10|zhang.yi.3|zhang.yi.11|zhang.yi|zhang.yi.8|zhang.yi.2|zhang.yi.7|zhang.yi.12|zhang.yi.5|zhang.yi.1|zhang.yi.9|zhang.yi.4Summary: To study the symmetry and conserved quantity of fractional non-conservative dynamic systems, the Noether theorem based on El-Nabulsi periodic law quasi-fractional model in event space is proposed and studied. Firstly, the fractional order variational problem based on the El-Nabulsi periodic law quasi-fractional model is established in the event space, and the differential equations of the holonomic nonconservative system and the nonholonomic nonconservative system are derived. Secondly, based on the invariance of the action functional under the infinitesimal transformation, the definition and criterion of the Noether symmetric transform and the Noether quasi-symmetric transformation are given. Finally, the Noether theorem based on the El-Nabulsi periodic law quasi-fractional model in the event space is proposed and proved. Two examples are given to illustrate the application of the results.Asymptotic stability of impulsive neutral stochastic functional differential equation driven by fractional Brownian motion.https://zbmath.org/1449.342792021-01-08T12:24:00+00:00"Cui, Jing"https://zbmath.org/authors/?q=ai:cui.jing"Liang, Qiuju"https://zbmath.org/authors/?q=ai:liang.qiuju"Bi, Nana"https://zbmath.org/authors/?q=ai:bi.nanaSummary: In this paper, we consider the asymptotic stability in the \(p\)-th moment of mild solutions of impulsive neutral stochastic functional differential equations driven by fractional Brownian motion in a real separable Hilbert space. A fixed point approach is used to achieve the required result. A practical example is provided to illustrate the viability of the abstract result of this work.Stability of traveling waves in a population dynamic model with delay and quiescent stage.https://zbmath.org/1449.350652021-01-08T12:24:00+00:00"Zhou, Yonghui"https://zbmath.org/authors/?q=ai:zhou.yonghui"Yang, Yunrui"https://zbmath.org/authors/?q=ai:yang.yunrui"Liu, Kepan"https://zbmath.org/authors/?q=ai:liu.kepanSummary: This article is concerned with a population dynamic model with delay and quiescent stage. By using the weighted-energy method combining continuation method, the exponential stability of traveling waves of the model under non-quasi-monotonicity conditions is established. Particularly, the requirement for initial perturbation is weaker and it is uniformly bounded only at \(x = +\infty\) but may not be vanishing.Nonnegativity of solutions of nonlinear fractional differential-algebraic equations.https://zbmath.org/1449.340162021-01-08T12:24:00+00:00"Ding, Xiaoli"https://zbmath.org/authors/?q=ai:ding.xiaoli"Jiang, Yaolin"https://zbmath.org/authors/?q=ai:jiang.yaolinSummary: Nonlinear fractional differential-algebraic equations often arise in simulating integrated circuits with superconductors. How to obtain the nonnegative solutions of the equations is an important scientific problem. As far as we known, the nonnegativity of solutions of the nonlinear fractional differential-algebraic equations is still not studied. In this article, we investigate the nonnegativity of solutions of the equations. Firstly, we discuss the existence of nonnegative solutions of the equations, and then we show that the nonnegative solution can be approached by a monotone waveform relaxation sequence provided the initial iteration is chosen properly. The choice of initial iteration is critical and we give a method of finding it. Finally, we present an example to illustrate the efficiency of our method.On the equation \(a{f^n} + b (f')^m \equiv 1\).https://zbmath.org/1449.300462021-01-08T12:24:00+00:00"Dang, Guoqiang"https://zbmath.org/authors/?q=ai:dang.guoqiang"Chen, Honghui"https://zbmath.org/authors/?q=ai:chen.honghuiSummary: Let \(n, m\) be two positive integers. Some previous researchers proved the existence of meromorphic solutions for the Fermat-type functional equation \({f^n} + (f')^m \equiv 1\) in 2013. This paper extends their results and obtains all general solutions of \(a{f^n} + b (f')^m \equiv 1\).The existence of almost periodic solution for a predator-prey model with nonlinear harvesting.https://zbmath.org/1449.341672021-01-08T12:24:00+00:00"Wei, Zhen"https://zbmath.org/authors/?q=ai:wei.zhenSummary: In this paper, the positive almost periodic solution for a predator-prey model with nonlinear harvesting is considered. Sufficient conditions are established for the existence of almost periodic solution of the system by constructing a suitable Lyapunov function. An example together with its numerical simulation is carried out to show the feasibility of the main results.Solution of the initial problem for a differential equation of fractal oscillator.https://zbmath.org/1449.340112021-01-08T12:24:00+00:00"Beĭbalaev, V. D."https://zbmath.org/authors/?q=ai:beibalaev.vetlugin-dzhabrilovich|beibalaev.vertlugin-dzhabrailovichSummary: This article provides a synthesis of the problem of harmonic oscillator with damping based on differential equations with fractional differentiation and demonstrates the possibility of consideration of nonlinear oscillatory processes on their basis.Oscillation for a class of second-order generalized Emden-Fowler-type functional differential equations.https://zbmath.org/1449.342302021-01-08T12:24:00+00:00"Yu, Qiang"https://zbmath.org/authors/?q=ai:yu.qiangSummary: We analyze the oscillatory behavior of certain second-order generalized Emden-Fowler-type differential equations with a quasilinear neutral term in this article, where the investigated equations are in a noncanonical form, i.e., \(\int_{t_0}^{+\infty} {a^{-1/\beta}} (t){\mathrm{d}}t < + \infty\). By using the generalized Riccati transformation and inequality technique, we establish some new oscillation criteria for the equations.Positive solutions of integral boundary value problem involving derivative boundary conditions.https://zbmath.org/1449.340972021-01-08T12:24:00+00:00"Zhao, Yang"https://zbmath.org/authors/?q=ai:zhao.yang"Wang, Xiamei"https://zbmath.org/authors/?q=ai:wang.xiamei"Yang, Zhilin"https://zbmath.org/authors/?q=ai:yang.zhilinSummary: This paper deals with the positive solutions for the second-order integral boundary value problem \[\begin{cases}u'' + f (t, u) = 0, \\ u (0) = \int_0^1 u (\tau){\mathrm{d}}\alpha (\tau), \\ u' (1) = \int_0^1 u' (\tau){\mathrm{d}}\beta (\tau),\end{cases}\] where \(f \in C ([0, 1] \times \mathbb{R}^+, \mathbb{R}^+)\). Based on a priori estimates, we use fixed-point index theory to establish the existence and multiplicity of positive solutions for the above problem.Multiple positive solutions to a \((2m)\)th-order boundary value problem.https://zbmath.org/1449.340702021-01-08T12:24:00+00:00"Boulaiki, Habiba"https://zbmath.org/authors/?q=ai:boulaiki.habiba"Moussaoui, Toufik"https://zbmath.org/authors/?q=ai:moussaoui.toufik"Precup, Radu"https://zbmath.org/authors/?q=ai:precup.raduSummary: The aim of the present paper is to study the existence, localization and multiplicity of positive solutions for a \((2m)\)th-order boundary value problem subject to the Dirichlet conditions. Our approach is based on critical point theory in conical shells and Harnack type inequalities.Stability of a deterministic and stochastic SIRS epidemic model with saturated incidence rate.https://zbmath.org/1449.341602021-01-08T12:24:00+00:00"Wang, Laiquan"https://zbmath.org/authors/?q=ai:wang.laiquan"Xamxinur, Abdurahman"https://zbmath.org/authors/?q=ai:xamxinur.abdurahmanSummary: We consider a deterministic and stochastic SIRS epidemic model with saturated incidence rate. We calculate the basic reproduction number \(R_0\) for the stochastic model and obtain the global existence and positivity of the unique solution. Under suitable conditions on the intensity of the noise perturbation, we prove the \(p\)-th exponential stability of the disease free equilibrium by using the differential operator and Itô's formula. We also discuss the asymptotic behavior of the solution of the stochastic model around the equilibrium of the deterministic model. Finally, we analyze the effect of the noise perturbation with respect to the stability of the stochastic model.Existence of a unique positive solution for a singular fractional boundary value problem.https://zbmath.org/1449.340762021-01-08T12:24:00+00:00"Karimov, E. T."https://zbmath.org/authors/?q=ai:karimov.erkinjon-tulkinovich"Sadarangani, K."https://zbmath.org/authors/?q=ai:sadarangani.kishin-bSummary: In the present work, we discuss the existence of a unique positive solution of a boundary value problem for a nonlinear fractional order equation with singularity. Precisely, order of equation
\[D^\alpha_0 u+u(t)= f(t,u(t))\]
belongs to \((3,4]\) and \(f\) has a singularity at \(t= 0\) and as boundary conditions we use \(u(0) =u(1) =u'(0) =u'(1) = 0\). Using a fixed point theorem, we prove the existence of unique positive solution of the considered problem.Stability in distribution for a class of neutral stochastic functional differential equations.https://zbmath.org/1449.342812021-01-08T12:24:00+00:00"Yuan, Zhihong"https://zbmath.org/authors/?q=ai:yuan.zhihong"Liu, Guirong"https://zbmath.org/authors/?q=ai:liu.gui-rong.1Summary: This paper considers two classes of stability issues for stochastic delayed neural networks. Different from the usual Lyapunov-Krasovskii functional and linear matrix inequality method, we first introduce a new comparison principle to compare the neural networks (NNs) and stochastic neural networks (SNNs). Then, we apply this new comparison principle to obtain new criteria of which the SNNs adaptive controller can make the controlled system stable in probability and stable in moment. Moreover, an example is given to illustrate the theoretical results well.Global analysis for an epidemic model with the general birth function and stage structure.https://zbmath.org/1449.341652021-01-08T12:24:00+00:00"Wang, Yuping"https://zbmath.org/authors/?q=ai:wang.yuping.1"Li, Jianquan"https://zbmath.org/authors/?q=ai:li.jianquan"Lin, Xiaolin"https://zbmath.org/authors/?q=ai:lin.xiaolinSummary: In this paper, under the assumption that the individual growth of the population consists of two stages: juvenile and adult, and that the infection is transmitted only between the adults, an epidemic model with a general form of birth function of the juveniles is established and investigated. The basic reproduction number of the population determining if the population persists and the basic reproduction number of the infection transmission determining if the disease dies out are found. The global threshold dynamics of the model is obtained by constructing appropriate Lyapunov functions.Controllability for impulsive fractional evolution inclusions with state-dependent delay.https://zbmath.org/1449.342692021-01-08T12:24:00+00:00"Aissani, Khalida"https://zbmath.org/authors/?q=ai:aissani.khalida"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffak"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseSummary: In this paper, sufficient conditions are provided for the controllability of impulsive fractional evolution inclusions with state-dependent delay in Banach spaces. We used a fixed-point theorem for condensing maps due to Bohnenblust-Karlin and the theory of semigroup for the achievement of the results. An illustrative example is presented.Random semilinear system of differential equations with state-dependent delay.https://zbmath.org/1449.342782021-01-08T12:24:00+00:00"Blouhi, Tayeb"https://zbmath.org/authors/?q=ai:blouhi.tayeb"Ferhat, Mohamed"https://zbmath.org/authors/?q=ai:ferhat.mohamedSummary: In this paper, we prove the existence of mild solutions for a first-order semilinear differential equation with state-dependent delay. The existence results are established by means of a new version of Perov's fixed point principles.Solution of the fractional SEIR model of epidemics using residual power series method.https://zbmath.org/1449.340242021-01-08T12:24:00+00:00"Li, Linna"https://zbmath.org/authors/?q=ai:li.linna"Wang, Huan"https://zbmath.org/authors/?q=ai:wang.huan"Huang, Qundan"https://zbmath.org/authors/?q=ai:huang.qundan"Tong, Qiujuan"https://zbmath.org/authors/?q=ai:tong.qiujuanSummary: The SEIR epidemic model has an important application background in the study of infectious diseases and social network information dissemination. The fractional SEIR epidemic model describes the propagation process of these dynamic systems more accurately, but the fractional SEIR model is difficult to solve. This paper presents a residual power series method for solving the model. First, we use the generalized Taylor series to expand the \(S (t)\), \(E (t)\), \(I (t)\) and \(R (t)\) in the fractional SEIR model. Then the expanded expression is brought into the fractional SEIR model. According to the residual being zero, the unknown coefficients are solved. An approximate analytical solution in the form of a series of fractional SEIR models is obtained. Compared with the solution obtained by the homotopy analysis transformation method, the results show that the residual power series method is more effective in solving the fractional SEIR model, and its error is smaller.Finite-time sliding mode synchronization of fractional-order Nadolschi systems.https://zbmath.org/1449.341882021-01-08T12:24:00+00:00"Meng, Xiaoling"https://zbmath.org/authors/?q=ai:meng.xiaolingSummary: Finite-time sliding mode synchronization of fractional-order Nadolschi systems is studied in the paper. Two sufficient conditions are derived for the fractional-order systems getting sliding mode synchronizable in finite time based on Lyapunov stability theory. The research conclusion illustrated that the systems are synchronized in finite time if choosing proper controllers and switching functions.Quasi-asymptotically almost periodic vector-valued generalized functions.https://zbmath.org/1449.460302021-01-08T12:24:00+00:00"Kostić, Marko"https://zbmath.org/authors/?q=ai:kostic.marko"Pilipović, Stevan"https://zbmath.org/authors/?q=ai:pilipovic.stevan-r"Velinov, Daniel"https://zbmath.org/authors/?q=ai:velinov.danielSummary: In this paper are introduced the notions of quasi-asymptotically almost periodic distributions and quasi-asymptotically almost periodic ultradistributions with values in a Banach space, as well as some other generalizations of these concepts. Furthermore, some applications of the introduced concepts in the analysis of systems of ordinary differential equations are provided.Almost sure exponential stability for some neutral partial integro-differential equations.https://zbmath.org/1449.342672021-01-08T12:24:00+00:00"Ramkumar, K."https://zbmath.org/authors/?q=ai:ramkumar.kasinathan"Mohamed, M. S."https://zbmath.org/authors/?q=ai:mohamed.mohamed-salem"Diop, Mamadou Abdoul"https://zbmath.org/authors/?q=ai:diop.mamadou-abdoulSummary: This paper is concerned with the dynamics of a delay stochastic neutral integro-differential equation in Hilbert spaces by using the theory of resolvent operator. After establishing a result ensuring the existence and uniqueness of a mild solution of this class of equations, we investigate the exponential stability of the moments of a mild solution as well as its sample paths. An example is given to illustrate the results.Approximating solutions of third-order nonlinear hybrid differential equations via Dhage iteration principle.https://zbmath.org/1449.340422021-01-08T12:24:00+00:00"Ardjouni, Abdelouaheb"https://zbmath.org/authors/?q=ai:ardjouni.abdelouaheb"Djoudi, Ahcene"https://zbmath.org/authors/?q=ai:djoudi.ahceneSummary: We prove the existence and approximation of solutions of the initial value problems of nonlinear third-order hybrid differential equations. The main tool employed here is the Dhage iteration principle in a partially ordered normed linear space. An example is also given to illustrate the main results.Limit cycles by perturbing a piecewise near-Hamiltonian system with 4 switching lines.https://zbmath.org/1449.341152021-01-08T12:24:00+00:00"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.3|wang.hao.13|wang.hao.7|wang.hao.12|wang.hao.6|wang.hao.11|wang.hao.5|wang.hao.4|wang.hao.2|wang.hao.9|wang.hao.1|wang.hao.10"Liang, Feng"https://zbmath.org/authors/?q=ai:liang.fengSummary: By using the first order Melnikov function method for piecewise near-Hamiltonian systems, we study limit cycle bifurcations by perturbing a compound global center with 4 regions. When the perturbed terms are polynomials with degree \(n\), we give the number of limit cycles bifurcating from the center.Existence of periodic solutions of the first-Order non-Autonomous systems at resonance.https://zbmath.org/1449.341172021-01-08T12:24:00+00:00"Chen, Ruipeng"https://zbmath.org/authors/?q=ai:chen.ruipeng"Li, Xiaoya"https://zbmath.org/authors/?q=ai:li.xiaoyaSummary: This paper studies the existence of periodic solutions of the first-order non-autonomous systems at resonance, where nonlinear terms are periodic continuous functions. Several new existence results are established by means of Miranda's theorem and Schauder's fixed point theorem. Our results enrich and complement those available in the literature.Stability analysis of time-delay SIRS epidemic model with standard incidence rate and information intervention.https://zbmath.org/1449.343042021-01-08T12:24:00+00:00"Zhao, Yingying"https://zbmath.org/authors/?q=ai:zhao.yingying"Hu, Hua"https://zbmath.org/authors/?q=ai:hu.huaSummary: In this paper, a class of time-delay SIRS infectious disease model with standard incidence and information intervention is considered. By analyzing the characteristic equations of the model, the local asymptotic stabilities of the disease-free equilibrium and endemic equilibrium are discussed. The global asymptotic stability of the disease-free equilibrium is proved by using the Halanay inequality. The global asymptotic stability of endemic equilibrium is discussed by constructing a suitable Lyapunov function. Finally, the influence of some important parameters about disease transmission is analyzed by numerical simulation.On a Bagley-Torvik fractional integro-differential inclusion.https://zbmath.org/1449.450112021-01-08T12:24:00+00:00"Cernea, A."https://zbmath.org/authors/?q=ai:cernea.aurelianSummary: Existence of solutions for a Bagley-Torvik fractional integro-differential inclusion is investigated in the case when the values of the set-valued map are not convex.Stability in nonlinear neutral Levin-Nohel integro-dynamic equations.https://zbmath.org/1449.450152021-01-08T12:24:00+00:00"Khelil, Kamil Ali"https://zbmath.org/authors/?q=ai:khelil.kamil-ali"Ardjouni, Abdelouaheb"https://zbmath.org/authors/?q=ai:ardjouni.abdelouaheb"Djoudi, Ahcene"https://zbmath.org/authors/?q=ai:djoudi.ahceneSummary: In this paper we use the Krasnoselskii-Burton's fixed point theorem to obtain asymptotic stability and stability results about the zero solution for the following nonlinear neutral Levin-Nohel integro-dynamic equation
\[
x^\Delta(t)+\int^t_{t-\tau(t)}a(t,s)g(x(s))\Delta s+c(t)x^{\widetilde\Delta}(t-\tau(t))=0.
\]
The results obtained here extend the work of \textit{K. A. Khelil} et al. [Korean J. Math. 25, No. 3, 303--321 (2017; Zbl 07148845)].Quasi-invariant and attractive sets of inertial neural networks with time-varying and infinite distributed delays.https://zbmath.org/1449.342632021-01-08T12:24:00+00:00"Tang, Qian"https://zbmath.org/authors/?q=ai:tang.qian"Jian, Jigui"https://zbmath.org/authors/?q=ai:jian.jiguiSummary: This paper aims at analyzing the quasi-invariant and attractive sets for a class of inertial neural networks with time-varying and infinite distributed delays. By utilizing the properties of nonnegative matrices, a new bidirectional-like delay integral inequality is developed. Some sufficient conditions are obtained for the existence of quasi-invariant and attractive sets of the discussed system according to the bidirectional-like integral inequality. Besides, the framework of the quasi-invariant and attractive sets for the concerned system is provided. Finally, one example is analyzed to illustrate our results.Existence of positive solutions for boundary value problems of second-order systems with nonlinear boundary conditions.https://zbmath.org/1449.340832021-01-08T12:24:00+00:00"Ma, Mantang"https://zbmath.org/authors/?q=ai:ma.mantang"Jia, Kaijun"https://zbmath.org/authors/?q=ai:jia.kaijunSummary: The existence of positive solutions for boundary value problems of second-order singular differential systems with nonlinear boundary conditions
\[\begin{cases}
-u'' = \Lambda G (t)F (u),\, 0 < t < 1, \\
u (0) = 0,\, u' (1) + C (u (1))u (1) = 0
\end{cases}\]
is studied, where \(u = (u_1, u_2, \cdots, u_n)^{\mathrm{T}}\), \(G (t) = \mathrm{diag}[g_1 (t), g_2 (t), \cdots, g_n (t)]\), \(g_i (t) (t = 1, 2, \cdots, n)\), allows singularity at \(t = 0\), \(F (u) = (f^1 (u), f^2 (u), \cdots, f^n (u))^{\mathrm{T}}\), \(C = \mathrm{diag} (c_1, c_2, \cdots, c_n)\), \(\Lambda = \mathrm{diag} (\lambda_1, \lambda_2, \cdots, \lambda_n)\), \(\lambda_i (i = 1, 2, \cdots, n)\) is a positive parameter. Under the condition that the nonlinearity term \(F\) satisfies superlinear, sublinear and asymptotically linear growth respectively, the existence of positive solutions of the problems is obtained by using the fixed-point theorem of cone expansion-compression.Qualitative properties of solution for hybrid nonlinear fractional differential equations.https://zbmath.org/1449.340292021-01-08T12:24:00+00:00"Matar, Mohammed M."https://zbmath.org/authors/?q=ai:matar.mohammed-mSummary: In this article we investigate some qualitative properties for a class of hybrid nonlinear fractional differential equations. The existence, uniqueness, monotonicity and positivity of the solution are studied by the method of upper and lower control functions and using Dhage's fixed point theorem. Some examples are introduced to illustrate the applicability of the results.Noether symmetries for fractional generalized Birkhoffian systems in terms of Caputo derivatives.https://zbmath.org/1449.700252021-01-08T12:24:00+00:00"Zhou, Ying"https://zbmath.org/authors/?q=ai:zhou.ying"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.9|zhang.yi|zhang.yi.1|zhang.yi.3|zhang.yi.2|zhang.yi.5|zhang.yi.11|zhang.yi.4|zhang.yi.12|zhang.yi.8|zhang.yi.7|zhang.yi.10Summary: Noether's theorems for a fractional generalized Birkhoffian system in terms of Caputo derivatives are studied. Firstly, the generalized Pfaff-Birkhoff principle based on Caputo fractional derivatives is established, the fractional generalized Birkhoffian equations are derived. Then, the fractional Noether symmetry and the fractional conserved quantity under special infinitesimal transformations without transforming the time are studied. Noether's theorem for the fractional generalized Birkhoffian system is established. Once more, the fractional Noether symmetry and the fractional conserved quantity under general infinitesimal transformations with transforming the time are studied, Noether's theorem for the fractional generalized Birkhoffian system is established. The proof is given by using the time reparametric method. Finally, an example is given to illustrate its application.On the Hyers-Ulam stability of Riemann-Liouville multi-order fractional differential equations.https://zbmath.org/1449.340152021-01-08T12:24:00+00:00"Cuong, D. X."https://zbmath.org/authors/?q=ai:cuong.d-xSummary: In this paper, by using a Bielecki type norm and the Banach fixed point theorem, we obtain a result on the Hyers-Ulam stability of Riemann-Liouville multi-order fractional differential equations.A class of the stochastic predator-prey model with delay and Lévy jump.https://zbmath.org/1449.342942021-01-08T12:24:00+00:00"Shi, Lili"https://zbmath.org/authors/?q=ai:shi.lili"Liu, Guirong"https://zbmath.org/authors/?q=ai:liu.gui-rong|liu.gui-rong.1Summary: This research focuses on a class of stochastic predator-prey models with delay and Lévy jump. Firstly, the Lyapunov method and Itô formula are used to give the existence and uniqueness of the global positive solution of this model. Then according to Chebyshev's inequality, exponential martingale inequality and Borel-Cantelli lemma, etc., the stochastic ultimate boundedness and extinction are obtained. Finally, the theoretical results are illustrated by numerical simulations.Nonlocal flocking dynamics: learning the fractional order of PDEs from particle simulations.https://zbmath.org/1449.354472021-01-08T12:24:00+00:00"Mao, Zhiping"https://zbmath.org/authors/?q=ai:mao.zhiping"Li, Zhen"https://zbmath.org/authors/?q=ai:li.zhen"Karniadakis, George Em"https://zbmath.org/authors/?q=ai:karniadakis.george-emSummary: Flocking refers to collective behavior of a large number of interacting entities, where the interactions between discrete individuals produce collective motion on the large scale. We employ an agent-based model to describe the microscopic dynamics of each individual in a flock, and use a fractional partial differential equation (fPDE) to model the evolution of macroscopic quantities of interest. The macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic model. Instead of specifying the fPDEs with an ad hoc fractional order for nonlocal flocking dynamics, we learn the effective nonlocal influence function in fPDEs directly from particle trajectories generated by the agent-based simulations. We demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in one- and two-dimensional nonlocal flocking dynamics. In particular, a Cucker-Smale particle model is employed to describe the microscale dynamics of each individual, while Euler equations with nonlocal interaction terms are used to compute the evolution of macroscale quantities. The trajectories generated by the particle simulations mimic the field data of tracking logs that can be obtained experimentally. They can be used to learn the fractional order of the influence function using a Gaussian process regression model implemented with the Bayesian optimization. We show in one- and two-dimensional benchmarks that the numerical solution of the learned Euler equations solved by the finite volume scheme can yield correct density distributions consistent with the collective behavior of the agent-based system solved by the particle method. The proposed method offers new insights into how to scale the discrete agent-based models to the continuum-based PDE models, and could serve as a paradigm on extracting effective governing equations for nonlocal flocking dynamics directly from particle trajectories.Methods to estimate the upper bound of the singular perturbation small parameter.https://zbmath.org/1449.342042021-01-08T12:24:00+00:00"Wang, Cuiling"https://zbmath.org/authors/?q=ai:wang.cuiling"Cheng, Qingjin"https://zbmath.org/authors/?q=ai:cheng.qingjin(no abstract)A Filippov HFMD model with health-education intervention.https://zbmath.org/1449.920312021-01-08T12:24:00+00:00"Wang, Aili"https://zbmath.org/authors/?q=ai:wang.aili"Wang, Yaqiang"https://zbmath.org/authors/?q=ai:wang.yaqiangSummary: A Filippov model is formulated to describe the following real process of HFMD (hand-foot-mouth disease) with health-education intervention: once the case number exceeds the threshold level, the impact of health-education intervention on disease spreading occurs; otherwise, no impact exists for the case number below the threshold level. The sliding mode, sliding dynamics and all possible equilibria are examined. The main results show that the Filippov system will stabilize at one of the two endemic equilibria of the subsystems, i.e., \({E_1}\) or \({E_2}\), or the pseudo-equilibrium \({E_s}\) for \({R_0} > 1\). It is indicated that the number of infected individuals can be stabilized at a previously given level if we implement health-education intervention properly.Complexity of one-prey multi-predator system with impulsive effect and incomplete trophic transfer.https://zbmath.org/1449.920372021-01-08T12:24:00+00:00"Cheng, Xian"https://zbmath.org/authors/?q=ai:cheng.xian"Yan, Ping"https://zbmath.org/authors/?q=ai:yan.ping"Liu, Liping"https://zbmath.org/authors/?q=ai:liu.liping"Zhang, Changqin"https://zbmath.org/authors/?q=ai:zhang.changqinSummary: A new one-prey multi-predator system with impulsive effect and incomplete trophic transfer was proposed. This system used a different rate of trophic absorption of predators from the rate of the conversion of consumed prey to predator in Ivlev-type functional responses. The extinction and permanence of the system with impulsive perturbation on the predators at fixed moments were investigated. The conditions for asymptotically stable and permanence of the system were given by using Floquet theory and comparison theorem. Finally, numerical simulations demonstrated the obtained conclusions.Strong convergence of the truncated Euler-Maruyama method for neutral stochastic differential delay equation.https://zbmath.org/1449.650092021-01-08T12:24:00+00:00"Wang, Bei"https://zbmath.org/authors/?q=ai:wang.bei"Hu, Liangjian"https://zbmath.org/authors/?q=ai:hu.liangjianSummary: In order to study the convergence of numerical scheme for neutral stochastic differential delay equations (NSDDEs) with highly nonlinear coefficients, by Itô formula, Gronwall lemma and some inequalities, the truncated Euler-Maruyama method for NSDDEs is proved to be strongly convergent (namely in \({L^q}, q \ge 1\)) under generalized Khasminskii-type conditions.On Taylor differential transform method for the first Painlevé equation.https://zbmath.org/1449.651532021-01-08T12:24:00+00:00"Sakka, A. H."https://zbmath.org/authors/?q=ai:sakka.ayman-hashem"Sulayh, A. M."https://zbmath.org/authors/?q=ai:sulayh.a-mSummary: We apply the Taylor differential transform method (TDTM) to the initial value problem of the first Painlevé equation. We use the deviation to calculate the accuracy of the solutions and the results are compared with the known results. Four cases of initial values, two of them were not considered before, are considered to illustrate the effectiveness of the method.Ratio-integral sliding mode synchronization of fractional-order atmospheric chaotic system.https://zbmath.org/1449.341902021-01-08T12:24:00+00:00"Wang, Dongxiao"https://zbmath.org/authors/?q=ai:wang.dongxiaoSummary: The problem of sliding mode synchronization of fractional-order atmospheric chaotic systems is studied. The fractional-order sliding mode function and control laws are designed and sufficient conditions are obtained for fractional-order atmospheric chaotic systems acquiring ratio integral sliding mode synchronization. It is proved that fractional-order atmospheric chaotic systems are ratio sliding mode synchronizable under certain conditions. Matlab is used for numerical simulation, and the validity of the design scheme is illustrated.Estimation of the number of limit cycles for a class of piecewise linear Hamilton systems.https://zbmath.org/1449.341072021-01-08T12:24:00+00:00"Deng, Rui"https://zbmath.org/authors/?q=ai:deng.rui"Li, Baoyi"https://zbmath.org/authors/?q=ai:li.baoyi"Zhang, Yongkang"https://zbmath.org/authors/?q=ai:zhang.yongkangSummary: In this paper, a plane is divided into three sectors, and on this basis, the number of limit cycles of a class of piecewise linear Hamilton systems under the polynomial perturbation of degree \(n\) is studied. By calculating the first-order Melnikov function \(M_1(h)\), it is proved that the piecewise linear Hamilton system can generate at least \(2n + 2[ (n + 1)/2] + 2\) limit cycles under polynomial perturbation of degree \(n\) when \(M_1(h)\)is not constant at 0.Global stability and simulation of a class of recurrent neural networks with proportional delays.https://zbmath.org/1449.343012021-01-08T12:24:00+00:00"Xing, Lin"https://zbmath.org/authors/?q=ai:xing.lin"Zhou, Liqun"https://zbmath.org/authors/?q=ai:zhou.liqunSummary: For a class of recurrent neural networks with proportional delays, the sufficient conditions for global asymptotic stability and global exponential stability of the system are obtained by using diagonal (semi) stability matrix, Lyapunov stability theory and constructing delay differential inequality. A numerical example is given, and the correctness and validity of the conclusions are illustrated through the variation of the bias input and the corresponding simulations.Maximal order of accuracy of \((m, 1)\)-methods for solving stiff problems.https://zbmath.org/1449.651512021-01-08T12:24:00+00:00"Novikov, Evgeniĭ Aleksandrovich"https://zbmath.org/authors/?q=ai:novikov.evgeny-aSummary: We investigate \((m, 1)\)-methods for solving stiff problems in which the right part of system of the differential equations is calculated one times on each step. It is shown that the maximal order of accuracy of the \(L\)-stability \((m, 1)\)-method is equal to two, and the method of the maximal order is constructed.\(p\)-moment stability with general decay rate of impulsive stochastic functional differential equations.https://zbmath.org/1449.342822021-01-08T12:24:00+00:00"Zhang, Xiuying"https://zbmath.org/authors/?q=ai:zhang.xiuying"Su, Chunhua"https://zbmath.org/authors/?q=ai:su.chunhuaSummary: This paper investigates the \(p\)-moment stability with general decay rate of impulsive stochastic functional differential equations. Based on the Lyapunov functional method, stochastic analysis theory and the impulsive differential inequality established in this paper, some sufficient conditions for \(p\)-moment stability and almost sure stability with general decay rate are derived. The obtained results are more general and simple, and are used to deal with impulsive stochastic delay differential equations. Finally, two numerical examples are given to demonstrate the effectiveness of the obtained results.Existence results of self-similar solutions to the Caputo-type's space-fractional heat equation.https://zbmath.org/1449.354292021-01-08T12:24:00+00:00"Basti, Bilal"https://zbmath.org/authors/?q=ai:basti.bilal"Benhamidouche, Noureddine"https://zbmath.org/authors/?q=ai:benhamidouche.noureddineSummary: This paper investigates the problem of existence and uniqueness of solutions under the self-similar forms to the space-fractional heat equation. By applying the properties of Banach's fixed point theorems, Schauder's fixed point theorem and the nonlinear alternative of Leray-Schauder type, we establish several results on the existence and uniqueness of self-similar solutions to this equation.On the numerical solution convergence of optimal control problems for Leontief type system.https://zbmath.org/1449.490092021-01-08T12:24:00+00:00"Sviridyuk, Georgiĭ Anatol'evich"https://zbmath.org/authors/?q=ai:sviridyuk.georgii-anatolevich"Keller, Alevtina Viktorovna"https://zbmath.org/authors/?q=ai:keller.alevtina-viktorovnaSummary: This article contains proving the convergence of numerical solving of optimal control problem for degenerate linear systems of ordinary differential equations with constant coefficients. Considering different appendixes of such systems, they belong to Leontief type system, as in the first time such systems were investigated as a dynamic Leontief input-output model with noninvertible operator on derivative. By using the initial condition of Showalter-Sidorov we gain an ability to extend the range of practical applicability for this model. The article includes existence and uniqueness theorem of numerical solution of the investigated problem, his kind, and results of numerical experiment for dynamic input-output model, which was offered by \textit{W. Leontief} [Input-output economics. Oxford University Press, New York (1986)].The sufficient conditions for polystability of solutions of nonlinear systems of ordinary differential equations.https://zbmath.org/1449.341972021-01-08T12:24:00+00:00"Shamanaev, P. A."https://zbmath.org/authors/?q=ai:shamanaev.pavel-a|shamanaev.p-a"Yazovtseva, O. S."https://zbmath.org/authors/?q=ai:yazovtseva.o-sPolystability of ODEs' system may be described as ``stability of different types with respect to different variables''. The authors of the paper suppose to replace the nonlinear system of ODEs by its linear approximation in order to investigate its polystability; then the spectrum of linear approximation must be explored. To transfer stability properties from nonlinear system to linear one the notion of Brauer local componentwise asymptotic equivalence is used. To prove this equivalence the authors construct an integral operator connecting the solutions of the two systems and satisfying the conditions of Schauder's principle. At the final part of the article particular examples are given that illustrate applicability of the obtained stability conditions.
Reviewer: Aleksey Syromyasov (Saransk)The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems.https://zbmath.org/1449.341122021-01-08T12:24:00+00:00"Nikonov, V. I."https://zbmath.org/authors/?q=ai:nikonov.vasilii-ivanovichThe author investigates partial stability of the zero equilibrium position of ODEs' nonlinear systems using algebras and Lie groups. The approach is based on the results of \textit{Yu. A. Mitropol'skij} and \textit{A. K. Lopatin} [The group theoretical approach in asymptotic methods of nonlinear mechanics. (Teoretiko-gruppovoj podkhod v asimptoticheskikh metodakh nelinejnoj mekhaniki). Ed. by O. S. Parasyuk (Russian). Institut Mathematiki AN USSR, Kiev (USSR). Kiev: Naukova Dumka (1988; Zbl 0722.22011)]. The authors suggest that the system under study has a group of transformations that is invariant with respect to partial stability. In this case, the transformation found leads to decomposition of the considered system, and the question of partial stability is reduced to the study of the selected subsystem. Examples of nonlinear systems are presented for which the question of the partial stability of the zero equilibrium position reduces to the study of the partial stability of linear systems' zero equilibrium position.
Reviewer: Pavel A. Shamanaev (Saransk)On the analytical solution of one creep problem.https://zbmath.org/1449.340022021-01-08T12:24:00+00:00"Kuznetsov, E. B."https://zbmath.org/authors/?q=ai:kuznetsov.e-b"Leonov, S. S."https://zbmath.org/authors/?q=ai:leonov.s-sIn the article, the authors explore the possibility of an analytical solution to the problem of deformation of specimens made of X18H10T steel at constant uniaxial tensile load and temperature under creep conditions. The problem is described by a system of two ordinary differential equations with homogeneous initial conditions. To obtain necessary and sufficient integrability conditions for this problem, the authors use the Chebyshev theorem on the integration of a binomial differential. In addition, the article provides recommendations on the use of numerical methods for solving creep problems that cannot be integrated.
Reviewer: Tatuana Badokina (Saransk)On the birth of a common-mode limit cycle in an ensemble of excitingly coupled FitzHugh-Nagumo elements.https://zbmath.org/1449.341462021-01-08T12:24:00+00:00"Korotkov, A. G."https://zbmath.org/authors/?q=ai:korotkov.alexander-g"Levanova, T. A."https://zbmath.org/authors/?q=ai:levanova.tatiana-aThe authors consider an ensemble of two excitable neuron-like FitzHugh-Nagumo elements connected by symmetric exciting bonds. Studying the emerging system of differential equations describing such an ensemble, they derive the conditions for the existence of the supercritical Andronov-Hopf bifurcation and the birth of the common-mode limit cycle. A bifurcation diagram is presented, a region corresponding to the existence of a stable common-mode limit cycle is indicated.
Reviewer: Artyom Andronov (Saransk)Analysis of a curved bimetallic beam.https://zbmath.org/1449.740292021-01-08T12:24:00+00:00"Gönczi, Dávid"https://zbmath.org/authors/?q=ai:gonczi.davidSummary: This paper deals with the determination of stresses and displacements in a curved bimetallic beam which has uniform curvature. The two curved beam components of different materials have common displacements at their interface. The thermal load is derived from uniform temperature change. Two models are considered. The first one is based on the theory of the generalized plane stress state of elasticity and the second one uses a strength of materials approach. The results obtained by these models are verified by a comparison with finite element analysis.Analytical investigations of temperature effects on creep strain relaxation of biomaterials using homotopy perturbation and differential transform methods.https://zbmath.org/1449.800082021-01-08T12:24:00+00:00"Adeleye, Olurotimi"https://zbmath.org/authors/?q=ai:adeleye.olurotimi"Abdulkareem, Olakanla"https://zbmath.org/authors/?q=ai:abdulkareem.olakanla"Yinusa, Ahmed"https://zbmath.org/authors/?q=ai:yinusa.ahmed-a"Sobamowo, Gbeminiyi"https://zbmath.org/authors/?q=ai:sobamowo.gbeminiyi-mSummary: In this paper, the effect of temperature on relaxation of creep strain in biomaterials is modeled and analyzed with homotopy perturbation and differential transform methods. Polymeric biomaterials used as implants undergo both geometric and material nonlinear deformation when subjected to different loading conditions. The present study is concerned with the effects of temperature on the geometric nonlinear deformation of the relaxation of creep strain in these materials. Polymeric biomaterials exhibit time dependent response as observed in viscoelastic materials and this is represented by a one-dimensional rheological material model with constant material parameters. This model is then extended to capture the effects of temperature and the resulting final governing model is a nonlinear differential equation which cannot be easily solved by the standard analytic techniques. Here, two efficient special nonlinear analytic techniques, the homotopy perturbation and differential transform methods, are applied to obtain the solution of the developed nonlinear differential equation. The obtained analytical solutions are validated with the fourth-order Runge-Kutta numerical method. An error analysis shows that good agreement exists between the solutions obtained with these methods. The effects of some parameters on the model were investigated. As observed from the study, it can be shown that an increase in thermal conductivity and viscosity resulted in an increase in resistance to deformation of the material, while an increase in the material stiffness resulted in an increase in the rate of deformation and relaxation.The assymmetric Lorenz attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems.https://zbmath.org/1449.340452021-01-08T12:24:00+00:00"Kazakov, A. O."https://zbmath.org/authors/?q=ai:kazakov.alexey-o"Kozlov, A. D."https://zbmath.org/authors/?q=ai:kozlov.a-dThe authors consider a class of three-dimensional systems of the form \[\begin{cases} \dot{x}=y+g_1(x,y,z),\\ \dot{y}=z+g_2(x,y,z),\\ \dot{z}=Ax+By+Cz+g_3(x,y,z), \end{cases}\] where \(A\), \(B\) and \(C\) are parameters of the system; \(g_i\), \(i=1,2,3\) are nonlinear terms, satisfying the relations \(g_i(0,0,0)=\dfrac{\partial g_i}{\partial x}(0,0,0)=\dfrac{\partial g_i}{\partial y}(0,0,0)=\dfrac{\partial g_i}{\partial z}(0,0,0)\), \(i=1,2,3\). A new method for constructing three-dimensional systems with various, including pseudohyperbolic, attractors is presented. Using the proposed method, the authors construct an example of a three-dimensional system with a pseudohyperbolic attractor of a new type, an asymmetric Lorenz attractor. It differs from the classical one in the absence of symmetry in any of the coordinates. To search for the asymmetric Lorenz attractor in the class of three-dimensional systems, the authors use the saddle chart method, and to check its pseudohyperbolicity, they use the LMP-method.
Reviewer: Artyom Andronov (Saransk)Sufficient conditions for the existence of an asymptotic quiescent position in time-delay systems.https://zbmath.org/1449.342502021-01-08T12:24:00+00:00"Zaranik, U. P."https://zbmath.org/authors/?q=ai:zaranik.u-p"Kuptsova, S. E."https://zbmath.org/authors/?q=ai:kuptsova.s-e"Stepenko, N. A."https://zbmath.org/authors/?q=ai:stepenko.n-aThe article is focused on the study of the asymptotic quiescent position for trajectories of a system of differential equations with delay \[ \dot{x}=f(t,x(t),x(t-h)). \eqno{(1)} \] Here \(x(t)=(x_1(t),\ldots,x_n(t))^\top\) is unknown \(n\)-dimensional vector; \(h>0\) is delay; \(f(t,x,y)=(f_1,\ldots,f_n)^\top\) is \(n\)-dimensional vector function, which is defined and continuous on the set \(t\geq 0\), \(x\in \mathbb{R}^n\), \(y\in \mathbb{R}^n\) and satisfies the Lipschitz condition in all arguments, starting with the second one. The authors give sufficient conditions under which \(x=0\) is an asymptotic quiescent position for the trajectories of system (1). The research method is based on the construction of the classical Lyapunov function, but the derivative of this function, by virtue of the system, is estimated not on the entire set of integral curves, but on its subset (the Razumikhin method). The rest of the article contains examples of first-order differential equations with delay, for which the position \(x=0\) is an asymptotic quiescent position.
Reviewer: Pavel A. Shamanaev (Saransk)Existence of periodic solutions for a class of damped vibration problems.https://zbmath.org/1449.341162021-01-08T12:24:00+00:00"Chen, Mengxi"https://zbmath.org/authors/?q=ai:chen.mengxi"Wang, Zhiyong"https://zbmath.org/authors/?q=ai:wang.zhiyong.1|wang.zhiyong.2|wang.zhiyongSummary: This paper is dedicated to study the periodic solutions for a class of damped vibration problems. By virtue of an auxiliary function, we obtain some new superquadratic growth and asymptotically quadratic growth conditions. Using the minimax methods in critical point theory, we establish some existence results, which unify and generalize some known results in the literature.Stability analysis of an SIRS epidemic model with information intervention.https://zbmath.org/1449.341532021-01-08T12:24:00+00:00"Li, Xiaoni"https://zbmath.org/authors/?q=ai:li.xiaoni"Zhang, Qimin"https://zbmath.org/authors/?q=ai:zhang.qiminSummary: The asymptotic behavior of an SIRS epidemic model containing information intervention and vaccination is studied. The results indicate that the basic reproduction number \({\mathfrak{R}_0}\) is the threshold of disease persistence and extinction. If \({\mathfrak{R}_0} < 1\), the system has an disease-free equilibrium which is globally asymptotically stable, while if \({\mathfrak{R}_0} > 1\), there exists an epidemic equilibrium which is globally asymptotically stable. At last, some numerical examples are given to illustrate the results.On the dynamics of an SEIR epidemic model.https://zbmath.org/1449.920422021-01-08T12:24:00+00:00"Bernoussi, Amine"https://zbmath.org/authors/?q=ai:bernoussi.amine"Kaddar, Abdelilah"https://zbmath.org/authors/?q=ai:kaddar.abdelilah"Asserda, Said"https://zbmath.org/authors/?q=ai:asserda.saidSummary: In this work, we propose a delayed SEIR epidemic model. The time delay, $\tau$ is introduced to model the latent period. The resulting model has two possible equilibria (free disease equilibrium and endemic equilibrium). Our main contribution affirms the existence of non constant periodic solutions which bifurcate from the endemic equilibrium when the delay crosses some critical values. Also we propose a comparison of a delayed SEIR model and its corresponding delayed SIR model. Furthermore, some numerical simulations are presented to illustrate our theoretical results.Different types of solutions for nonlinear fractional integral boundary value problems with two parameters.https://zbmath.org/1449.340352021-01-08T12:24:00+00:00"Wang, Wenxia"https://zbmath.org/authors/?q=ai:wang.wenxia"Mi, Fang"https://zbmath.org/authors/?q=ai:mi.fangSummary: This paper is concerned with the existence of different types of solutions for a class of nonlinear fractional differential equations with two parameters under integral boundary conditions. By using a fixed point theorem and analytic technique, we divide the range of these parameters for the existence of positive solutions, negative solutions and sign-changing solutions for the boundary value problem and obtain some new results.Dynamical analysis on a class of cross ecosystem resources-phytoplankton-zooplankton model.https://zbmath.org/1449.341422021-01-08T12:24:00+00:00"Huang, Zhiwei"https://zbmath.org/authors/?q=ai:huang.zhiwei"He, Guofeng"https://zbmath.org/authors/?q=ai:he.guofeng"Huang, Gang"https://zbmath.org/authors/?q=ai:huang.gangSummary: In this paper, a cross ecosystem resources-phytoplankton-zooplankton model is proposed and studied. By constructing a Lyapunov function and using the LaSalle invariance principle, the global stability of the equilibria is obtained. Based on the theoretical results of the model, we find that the cross ecosystem resources have a large impact on the stability of the marine ecosystem. Furthermore, it may cause the extinction of the phytoplankton. Some suggestions on controlling the phytoplankton blooms are proposed. In particular, the results of our model fit well with the hypothesis related to the cross ecosystem resources.A global bifurcation theorem for a multiparameter positone problem and its application to the one-dimensional perturbed Gelfand problem.https://zbmath.org/1449.340752021-01-08T12:24:00+00:00"Huang, Shao-Yuan"https://zbmath.org/authors/?q=ai:huang.shaoyuan"Hung, Kuo-Chih"https://zbmath.org/authors/?q=ai:hung.kuo-chih"Wang, Shin-Hwa"https://zbmath.org/authors/?q=ai:wang.shin-hwa.1|wang.shin-hwa.2Summary: We study the global bifurcation and exact multiplicity of positive solutions for
\[u'' (x)+\lambda f_\varepsilon (u)=0,\, -1<x<1,\]
\[u(-1)=u(1)=0,\]
where \(\lambda >0\) is a bifurcation parameter, \(\varepsilon \in \Theta\) is an evolution parameter, and \(\Theta \equiv (\sigma_1,\sigma_2)\) is an open interval with \(0\leq \sigma_1<\sigma_2 \leq \infty\). Under some suitable hypotheses on \(f_\varepsilon\), we prove that there exists \(\varepsilon_0 \in \Theta\) such that, on the \((\lambda,\Vert u\Vert_\infty)\)-plane, the bifurcation curve is S-shaped for \(\sigma_1 <\varepsilon <\varepsilon_0\) and is monotone increasing for \(\varepsilon_0 \leq \varepsilon <\sigma_2\). We give an application to prove the global bifurcation of bifurcation curves for the one-dimensional perturbed Gelfand problem.Positive solutions of a derivative dependent second-order problem subject to Stieltjes integral boundary conditions.https://zbmath.org/1449.340842021-01-08T12:24:00+00:00"Ming, Zhongyang"https://zbmath.org/authors/?q=ai:ming.zhongyang"Zhang, Guowei"https://zbmath.org/authors/?q=ai:zhang.guowei|zhang.guowei.1"Li, Hongyu"https://zbmath.org/authors/?q=ai:li.hongyuSummary: In this paper, we investigate the derivative dependent second-order problem subject to Stieltjes integral boundary conditions
\[-u''(t)=f(t,u(t),u'(t)),\quad t\in[0,1],\]
\[au(0)-bu'(0)=\alpha[u],\, cu(1)+du'(1)=\beta[u],\]
where \(f\): \([0,1]\times \mathbb{R}^+\times \mathbb{R}\rightarrow \mathbb{R}^+\) is continuous, \(\alpha[u]\) and \(\beta[u]\) are linear functionals involving Stieltjes integrals. Some conditions on the nonlinearity \(f\) and the spectral radius of the linear operator are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index. The conditions allow that \(f(t,x_1,x_2)\) has superlinear or sublinear growth in \(x_1,x_2\). Two examples are provided to illustrate the theorems under multi-point and integral boundary conditions with sign-changing coefficients.Inertial manifolds and limit cycles of dynamical systems in \({\mathbb{R}}^{n}\).https://zbmath.org/1449.341272021-01-08T12:24:00+00:00"Kondratieva, Liudmila"https://zbmath.org/authors/?q=ai:kondratieva.liudmila"Romanov, Aleksandr"https://zbmath.org/authors/?q=ai:romanov.aleksandr-sergeevichSummary: We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in \({\mathbb{R}}^{n}\) permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincaré-Bendixson theory. In the case \(n=3\) we implement such a scenario for a model of a satellite rotation around a celestial body of small mass and for a biochemical model.On the composition conjecture for a class of rigid systems.https://zbmath.org/1449.341062021-01-08T12:24:00+00:00"Zhou, Zhengxin"https://zbmath.org/authors/?q=ai:zhou.zhengxin"Yan, Yuexin"https://zbmath.org/authors/?q=ai:yan.yuexinSummary: In this paper, we prove that for a class of rigid systems the composition conjecture is correct. We show that the moments condition is a sufficient and necessary conditions for these rigid systems to have a center at the origin. By the obtained conclusions we can derive all the focal values of these higher order polynomial differential systems.Noether's theorems based on El-Nabulsi extended exponentially quasi-fractional models in event space.https://zbmath.org/1449.370452021-01-08T12:24:00+00:00"Wang, Ze"https://zbmath.org/authors/?q=ai:wang.ze"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.1|zhang.yi.8|zhang.yi.11|zhang.yi.5|zhang.yi.7|zhang.yi.12|zhang.yi|zhang.yi.3|zhang.yi.10|zhang.yi.9|zhang.yi.2|zhang.yi.4Summary: In order to further study the dynamic behavior of non-conservative systems and reveal the relationship between the symmetries and conserved quantities of dynamic systems, we proposed and investigated the Noether theorems based on El-Nabulsi extended exponentially quasi fractional models in event space. Firstly, we put forward the El-Nabulsi quasi fractional variational problem based on the extended exponentially fractional integral, and established the differential equations of motion for holonomic systems and nonholonomic systems. Secondly, we provided the definition and criterion of Noether symmetric transformation and Noether quasi-symmetric transformation based on the invariance of the action functional under the infinitesimal transformations. Finally, we presented and proved the Noether theorems based on El-Nabulsi extended exponentially quasi fractional models in event space. Two examples were given to illustrate the application of the results.\(C^1\)-smooth dependence on initial conditions and delay: spaces of initial histories of Sobolev type, and differentiability of translation in \(L^p\).https://zbmath.org/1449.342152021-01-08T12:24:00+00:00"Nishiguchi, Junya"https://zbmath.org/authors/?q=ai:nishiguchi.junyaSummary: The objective of this paper is to clarify the relationship between the \(C^1\)-smooth dependence of solutions to delay differential equations (DDEs) on initial histories (i.e., initial conditions) and delay parameters. For this purpose, we consider a class of DDEs which include a constant discrete delay. The problem of \(C^1\)-smooth dependence is fundamental from the viewpoint of the theory of differential equations. However, the above mentioned relationship is not obvious because the corresponding functional differential equations have the less regularity with respect to the delay parameter. In this paper, we prove that the \(C^1\)-smooth dependence on initial histories and delay holds by adopting spaces of initial histories of Sobolev type, where the differentiability of translation in \(L^p\) plays an important role.On singular \(p\)-Laplacian boundary value problems involving integral boundary conditions.https://zbmath.org/1449.340672021-01-08T12:24:00+00:00"Dang Dinh, Hai"https://zbmath.org/authors/?q=ai:dang-dinh.hai"Wang, Xiao"https://zbmath.org/authors/?q=ai:wang.xiao.1|wang.xiaoSummary: We prove the existence of positive solutions for the \(p\)-Laplacian equations
\[-(\phi (u'))' =\lambda f(t,u),\quad t\in (0,1)\]
with integral boundary conditions. Here, \(\lambda\) is a positive parameter, \(\phi (s)=\vert s\vert^{p-2}s\), \(p>1\), \(f:(0,1)\times (0,\infty)\rightarrow \mathbb{R}\) is \(p\)-superlinear or \(p\)-sublinear at \(\infty\) and is allowed to be singular at \(t=0,1\) and \(u=0\).Oscillatory behavior of the second order noncanonical differential equations.https://zbmath.org/1449.342202021-01-08T12:24:00+00:00"Baculiková, Blanka"https://zbmath.org/authors/?q=ai:baculikova.blankaSummary: Establishing monotone properties of nonoscillatory solutions we introduce new oscillatory criteria for the second order noncanonical differential equation with delay/advanced argument \[ (r(t)y'(t))'+p(t)y(\tau(t))=0.\] Our oscillatory results essentially extend the earlier ones. The progress is illustrated via the Euler differential equation.Existence of positive solutions for a class of second order periodic boundary value problems with one parameter.https://zbmath.org/1449.340802021-01-08T12:24:00+00:00"Li, Zhaoqian"https://zbmath.org/authors/?q=ai:li.zhaoqianSummary: In this paper, we consider the existence of positive solutions for the following nonlinear second-order ordinary differential equation with periodic boundary values:
\[\begin{cases}
u'' + a (t,u)u = \lambda g (t)f (u),\; t \in [0, T],\\
u (0) = u (T), u' (0) = u' (T),
\end{cases}\]
where \(\lambda\) is an positive parameter, \(a:[0, T] \times [0, \infty) \to \mathbb{R}^+\) is a \(L^p\)-Caratheodory function, \(g:[0, T] \to [0, \infty)\), \(f:[0, \infty) \to [0, \infty)\) are continuous functions. The proof of the main results is based on the fixed point index theory on cones.Existence of positive solutions for a class of singular second-order ordinary differential equations with nonlinear boundary condition.https://zbmath.org/1449.340872021-01-08T12:24:00+00:00"Su, Xiaoxiao"https://zbmath.org/authors/?q=ai:su.xiaoxiaoSummary: In this paper, we study the existence of positive solutions of a class of singular second-order ordinary differential equations with nonlinear boundary conditions
\[\begin{cases}
u'' + \rho^2 u = \lambda g (t)f (u),\, t \in [0, 2\pi],\\
u (0) = h (u (2\pi))u (2\pi),\, u' (0) = u' (2\pi),
\end{cases}\]
where \(\rho \in (0, 1/4)\), \(\lambda > 0\) is a parameter, \(g:(0,2\pi] \to (0,\infty)\), and \(f: (0,\infty) \to \mathbb{R}\), \(h:[0, \infty) \to [1, \infty)\) are continuous functions, \(f\) may be singular at 0 and superlinear at \(\infty\). The proof of the main results is based on the Krasnoselskii's fixed point theorem.Existence and multiplicity of positive solutions for a class of second-order boundary value problems.https://zbmath.org/1449.340822021-01-08T12:24:00+00:00"Ma, Mantang"https://zbmath.org/authors/?q=ai:ma.mantang"Jia, Kaijun"https://zbmath.org/authors/?q=ai:jia.kaijunSummary: In this paper, existence and multiplicity of positive solutions of the nonlinear second-order boundary value problems
\[\begin{cases}
(q (t)u' (t))' + f (u' (t)) = 0,\, t \in (0, 1), \\
q (0)u' (0) = 0, cu (1) + dq (1)u' (1) = 0
\end{cases}\]
are considered, where \(f: (-\infty, 0] \to [0, \infty)\), \(q:[0, 1] \to (0, \infty)\) are continuous functions, \(c > 0\), \(d \geq 0\) are constants. When the nonlinear term \(f\) satisfies superlinear growth condition or sublinear growth condition, we show that there exists at least one positive solution to the problem. When the nonlinear term \(f\) satisfies \(f_0:= \lim\limits_{s\to 0^-} \frac{f (s)}{s} = f_\infty:= \lim\limits_{s\to -\infty} \frac{f (s)}{s} = 0\) or \(f_0:= \lim\limits_{s \to 0^-} \frac{f (s)}{s} = f_\infty: = \lim\limits_{s \to -\infty} \frac{f (s)}{s} = \infty\), we show that there are at least two positive solutions to the problem. The proof is based on the fixed point theorem on cones.Existence of positive solutions for a class of nonlinear fourth-order ordinary differential equations with boundary values.https://zbmath.org/1449.340952021-01-08T12:24:00+00:00"Zhang, Yali"https://zbmath.org/authors/?q=ai:zhang.yaliSummary: In this paper, we study the existence of positive solutions for a class of nonlinear fourth-order ordinary differential equations with boundary values:
\[\begin{cases}
u^{ (4)} (t) = \lambda f (t, u (t)),\, t \in (0, 1),\\
u (0) = u'' (0) = u''' (1) = 0,\\
u' (1) + C (u (1))u (1) = 0,
\end{cases}\]
where \(\lambda\) is a positive parameter, \(f:[0,1] \times \mathbb{R} \to [0,\infty)\) satisfies \({L^1}\)-Caratheodory conditions, \(C:[0, \infty) \to [0, \infty)\) is continuous. The proof of the main results is based on the fixed-point theorem of cone expansion-compression.Barriers in impulsive antiperiodic problems.https://zbmath.org/1449.340502021-01-08T12:24:00+00:00"Rachůnková, Irena"https://zbmath.org/authors/?q=ai:rachunkova.irena"Rachůnek, Lukáš"https://zbmath.org/authors/?q=ai:rachunek.lukasSummary: Some real world models are described by means of impulse control of nonlinear BVPs, where the time instants of impulse actions depend on intersection points of solutions with given barriers. For \(i=1,\dots, m\), and \([a,b]\subset \mathbb{R}\), continuous functions \(\gamma_i:\mathbb{R} \to [a,b]\) determine barriers \(\Gamma_i=\{(t,z): t=\gamma_i(z), z\in \mathbb{R} \}\). A solution \((x,y)\) of a planar BVP on \([a,b]\) is searched such that the graph of its first component \(x(t)\) has exactly one intersection point with each barrier, i.e. for each \(i\in \{1,\dots,m\}\) there exists a unique root \(t=t_{ix}\in [a,b]\) of the equation \(t=\gamma_i(x(t))\). The second component \(y(t)\) of the solution has impulses (jumps) at the points \(t_{1x},\dots, t_{mx}\). Since the size of jumps and especially the points \(t_{1x},\dots,t_{mx}\) depend on \(x\), impulses are called state-dependent. Here, we focus our attention on an antiperiodic solution \((x,y)\) of the van der Pol equation with a positive parameter \(\mu\) and a Lebesgue integrable antiperiodic function \(f\) \[ x'(t) = y(t), \ y'(t) = \mu \left(x(t) - \frac{x^3(t)}{3}\right)' - x(t) + f(t)\] for a.e. \( t \in \mathbb{R},\ t\not\in \{t_{1x},.\dots, t_{mx}\},\) where \(y\) has impulses at the points from the set \(\{t_{1x},\dots,t_{mx}\}\), \[ y(t+)-y(t-) = \mathcal{J}_i(x), \quad t=t_{ix}, \quad i=1,\dots,m, \] and \(\mathcal{J}_i\) are continuous functionals defining a size of jumps. Previous results in the literature for this antiperiodic problem assume that the impulse points are values of given continuous functionals. Such formulation is a certain handicap for applications to real world problems where impulse instants depend on barriers. The paper presents conditions which enable to find such functionals from given barriers. Consequently, the existence results for impulsive antiperiodic problem to the van der Pol equation formulated in terms of barriers are reached.Stable manifolds for non-instantaneous impulsive nonautonomous differential equations.https://zbmath.org/1449.341282021-01-08T12:24:00+00:00"Li, Mengmeng"https://zbmath.org/authors/?q=ai:li.mengmeng"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrong"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper, we study stable invariant manifolds for a class of nonautonomous non-instantaneous impulsive equations where the homogeneous part has a nonuniform exponential dichotomy. We establish a stable invariant manifold result for sufficiently small perturbations by constructing stable and unstable invariant manifolds and we also show that the stable invariant manifolds are of class \(C^{1}\) outside the jumping times using the continuous Fiber contraction principle technique.Generalization of coarse-grained models with introduction of three-dimensional space.https://zbmath.org/1449.700112021-01-08T12:24:00+00:00"Nazarov, Maksim Nikolaevich"https://zbmath.org/authors/?q=ai:nazarov.m-nSummary: As a candidate for generalization in the scope of this work we consider any model derived from a system of first order ordinary differential equations for quantities of abstract bulk objects. The main objective of this work is to construct a universal scheme for generalization of such models with introduction of three-dimensional space and regard for migration of objects without switching to partial derivatives.Existence of positive periodic solutions of second-order differential equation with weak singularity.https://zbmath.org/1449.341212021-01-08T12:24:00+00:00"Miao, Liangying"https://zbmath.org/authors/?q=ai:miao.liangying"Liu, Xilan"https://zbmath.org/authors/?q=ai:liu.xilan"He, Zhiqian"https://zbmath.org/authors/?q=ai:he.zhiqianSummary: The existence of positive periodic solutions of the following second-order differential equation
\[u'' + a (t)u = f (t, u) + c (t)\]
is considered via Schauder's fixed point theorem, where \(a \in L^1(\mathbb{R}/T\mathbb{Z}; \mathbb{R}_+)\), \(c \in L^1 (\mathbb{R}/T\mathbb{Z}; \mathbb{R})\), \(f\) is a Carathéodory function. Our main results generalize some known results.Existence of solutions for a class of periodic boundary value problem of third-order nonlinear ordinary differential equations.https://zbmath.org/1449.340642021-01-08T12:24:00+00:00"Deng, Zhengping"https://zbmath.org/authors/?q=ai:deng.zhengping"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: In this paper, the existence of solutions for the periodic boundary value problem of the following third-order ordinary differential equation
\[\begin{cases}Lu (t) = f (t,u (t),u' (t),u'' (t)),\; t\in[0,\omega], \\ u^{ (k)} (0) = u^{ (k)} (\omega), k = 0, 1, 2\end{cases}\]
is considered, where \(Lu (t) = u''' (t)+{a_2}u'' (t) + {a_1}u' (t) + {a_0}u (t)\) is a third-order ordinary differential operator, \(f:[0,\omega]\times \mathbb{R}^3\to \mathbb{R}\) is continuous. Applying the Fourier analysis method and Leray-Schauder fixed point theorem, we obtain the existence and uniqueness of the solutions of the equation when the nonlinear term \(f\) satisfies some proper growth conditions.Positive periodic solutions for a class of differential equations involving parameter and multiple delays.https://zbmath.org/1449.342382021-01-08T12:24:00+00:00"Zhang, Lu"https://zbmath.org/authors/?q=ai:zhang.lu"Yang, He"https://zbmath.org/authors/?q=ai:yang.heSummary: In this paper, we use Krasnoselskii's fixed point theorem on cones to study the existence of positive \(\omega \)-periodic solutions for a class of differential equations involving a parameter and multiple delays. We prove some theorems about the multiplicity and a theorem on nonexistence of positive \(\omega\)-periodic solutions.The discreteness of the spectrum of \(2N\)-th order one term vector differential operators.https://zbmath.org/1449.343092021-01-08T12:24:00+00:00"Qian, Zhixiang"https://zbmath.org/authors/?q=ai:qian.zhixiangSummary: The discreteness of the spectrum of vector differential operators generated by the \(2N\)-th order one term differential expression with matrix coefficients is considered and some sufficient conditions are obtained for ensuring the discreteness of the spectrum of these operators in the cases of self-adjoint and \(J\)-self-adjoint, respectively.Integrating factors and conserved quantities for Birkhoffian systems on time scales.https://zbmath.org/1449.370352021-01-08T12:24:00+00:00"Yang, Lixia"https://zbmath.org/authors/?q=ai:yang.lixia"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.8|zhang.yi.11|zhang.yi.10|zhang.yi.2|zhang.yi.9|zhang.yi.12|zhang.yi.5|zhang.yi.3|zhang.yi|zhang.yi.7|zhang.yi.4|zhang.yi.1Summary: This paper studied the conserved quantities of Birkhoffian systems on time scales. The method of integrating factors was proposed to find the conserved quantities of Birkhoffian systems on time scales. And the energy equation of Birkhoff's equations was established on time scales. The integrating factors and conservation theorems for Birkhoffian systems on time scales were investigated. The integrating factors and conservation theorems for Hamiltonian systems and Lagrangian systems on time scales were special cases of Birkhoffian systems on time scales. Finally, an example was given to illustrate the application of the results.Establishment and analysis of a rumor propagation model based on differential equations.https://zbmath.org/1449.341762021-01-08T12:24:00+00:00"Zhao, Min"https://zbmath.org/authors/?q=ai:zhao.min"Chen, Wenxia"https://zbmath.org/authors/?q=ai:chen.wenxiaSummary: The spread and control of rumors have attracted the attention of academic and management departments. Considering the influence of official media propaganda on the susceptible persons of rumor in the process of rumor dissemination, the rumor susceptible people are divided into two categories. According to the different nature of the two types of people, a new rumor propagation model is formulated. By using Hurwitz criterion, Lyapunov function and LaSalle invariance principle, sufficient conditions of stability are obtained and carried out by numerical simulation. Media coverage can not only reduce the rate of rumor infection, but also expand the number of rumor mongering and disseminators and reduce the harm caused by rumors to society. In addition, moderate media coverage can change the rumor from widespread to rapid disappearance.An SIQRS epidemic model with constant input and Markovian switching.https://zbmath.org/1449.341792021-01-08T12:24:00+00:00"Zhao, Xiaojing"https://zbmath.org/authors/?q=ai:zhao.xiaojing"Zhang, Long"https://zbmath.org/authors/?q=ai:zhang.long"Zhang, Deting"https://zbmath.org/authors/?q=ai:zhang.detingSummary: In this paper, we consider a class of SIQRS epidemic models with constant input and Markovian switching. We analyze its global positivity and boundedness, and establish the threshold value \({R_0}\). The disease is eradicated almost surely if \({R_0} < 1\), while the disease persists almost surely if \({R_0} >1\). Finally, the results are illustrated by numerical simulation.Eventual \(\varphi_0\)-stability of differential systems in terms of two measures by perturbing Lyapunov functions.https://zbmath.org/1449.341962021-01-08T12:24:00+00:00"Russinov, Ivan K."https://zbmath.org/authors/?q=ai:russinov.ivan-kSummary: The notation of eventual \(\varphi_0\)-stability of nonlinear systems of ordinary differential equations in terms of two measures is introduced. Our technique depends on Lyapunov direct method. Perturbed cone-valued Lyapunov functions have been applied.On the number of zeros for a kind of Abel integrals.https://zbmath.org/1449.341102021-01-08T12:24:00+00:00"Wang, Xihong"https://zbmath.org/authors/?q=ai:wang.xihongSummary: In this paper, we study the number of zeros for Abel integrals \[\mathscr{J} (h) = \oint\limits_{\Gamma_h} {\frac{\alpha_0 + \alpha_1x + \alpha_2x^2 + \alpha_3x^3}{y}}{\mathrm{d}}x,\] where \(\alpha_i \in \mathbb{R}\), \(i = 0, 1, 2, 3\), \(\Gamma_h = \{H (x,y) = h,\; h \in (-\frac{1}{20}, 0)\}\) are closed curves and \(H (x,y) = \frac{1}{2}{y^2} - \frac{1}{4}{x^4} + \frac{1}{5}{x^5}\). The exact number of zeros in \( (-\frac{1}{20}, 0)\) for \(\mathscr{J} (h)\) is obtained when one of the four parameters \({\alpha_i} (i = 0, 1, 2, 3)\) vanishes.The cyclicity of period annuli of a Hamilton system with nilpotent saddle points.https://zbmath.org/1449.341092021-01-08T12:24:00+00:00"Li, Huimin"https://zbmath.org/authors/?q=ai:li.huimin"Zhang, Erli"https://zbmath.org/authors/?q=ai:zhang.erliSummary: In this paper, we study the cyclicity of period annuli of the following cubic Hamilton system with nilpotent saddle points \[\frac{{\mathrm{d}}x}{{\mathrm{d}}t} = 4{x^2}y + 4{y^3} - y,\; \frac{{\mathrm{d}}y}{{\mathrm{d}}t} = 4{x^2} - 4x{y^2} + x.\] By using the first order Melnikov function and Picard-Fuchs equation, we obtain that the above system under perturbations of real polynomials with degree \(n\) can bifurcate at most \(4n + 10\) limit cycles (taking into account the multiplicity).Synchronization of chaos in simultaneous time-frequency domain.https://zbmath.org/1449.370682021-01-08T12:24:00+00:00"Liu, Meng-Kun"https://zbmath.org/authors/?q=ai:liu.mengkun"Suh, C. Steve"https://zbmath.org/authors/?q=ai:suh.c-steveSummary: Synchronization of chaos presents many challenges for controller design. The novel notion of exerting concurrent control in the joint time-frequency domain is applied to formulate a chaos synchronization scheme that requires no linearization or heuristic trial-and-errors for nonlinear controller design. The concept is conceived through recognizing the basic attributes inherent of all chaotic systems, including the simultaneous deterioration of dynamics in both the time and frequency domains when bifurcates, nonstationarity, and sensitivity to initial conditions. Having its philosophical bases established in simultaneous time-frequency control, on-line system identification, and adaptive control, the chaos synchronization scheme incorporates multiresolution analysis, adaptive filters, and filtered-x Least Mean Square algorithm as its physical features. Without \textit{A priori} knowledge of the driven system parameters, synchronization is invariably achieved regardless of the initial and forcing conditions the response system is subjected to. In addition, driving and driven trajectories are seen robustly synchronized with negligible errors in spite of the infliction of high frequency noise.\(L_p\)-equivalence between two ordinary impulse differential equations with bounded linear impulse operators in a Banach Space.https://zbmath.org/1449.340482021-01-08T12:24:00+00:00"Kostadinov, G."https://zbmath.org/authors/?q=ai:kostadinov.g-d|kostadinov.georgi"Zahariev, A."https://zbmath.org/authors/?q=ai:zahariev.andrey-ivanovSummary: By means of the fixed point principle of Schauder-Tychonoff and Banach there are found sufficient conditions for the existence of \(L_p\)-equivalence between two ordinary impulse differential equations with bounded linear impulse operators in an arbitrary Banach space.Existence of \(L_p(\varphi,\psi)\)-solutions of linear differential equations with generalized dichotomy in a Banach space.https://zbmath.org/1449.342072021-01-08T12:24:00+00:00"Kiskinov, H."https://zbmath.org/authors/?q=ai:kiskinov.hristo"Kostadinov, S."https://zbmath.org/authors/?q=ai:kostadinov.stepan-iSummary: A generalization of the well known dichotomies for a class of homogeneous linear differential equations in an arbitrary Banach space is used. By the help of them there are found sufficient conditions for the existence of \(L_p(\varphi,\psi)\)-solutions of the nonhomogeneous equation.Multiple solutions of a Dirichlet problem in one-dimensional billiard space.https://zbmath.org/1449.341012021-01-08T12:24:00+00:00"Tomeček, Jan"https://zbmath.org/authors/?q=ai:tomecek.janSummary: The paper gives multiplicity results for the impulsive boundary value problem
\[ x'' = f(t,x)\text{ for all }t \in [0,T] \text{ such that }x(t) \in \text{int }K,\]
\[x'(s+) = -x'(s-),\text{ if }s \in (0,T),\ x(s) \in \partial{K},\]
\[x(0) = A, \quad x(T) = B,\] where \(K\subset \mathbb{R}\) is a compact interal, \(f\) is Carathéodory function on \([0,T]\times K\) and \(A,B \in\) int\(K\). This problem can be understood as a problem in one-dimensional billiard space and it is also a generalization of an oscillator with obstacles from below and from above and absolutely elastic impacts. A simple condition for the existence of solution with exact number of impacts is given, as well as the multiplicity result. The results are obtained by a transformation into problem without impulses (without impacts) and using Schauder's fixed point theorem.Nonlocal boundary value problem for a Lykov's type system of first-order.https://zbmath.org/1449.353002021-01-08T12:24:00+00:00"Repin, Oleg Aleksandrovich"https://zbmath.org/authors/?q=ai:repin.oleg-aleksandrovich"Kumykova, Svetlana Kanshubievna"https://zbmath.org/authors/?q=ai:kumykova.svetlana-kanshubievnaSummary: In this paper we prove the unique solution of the problem with a shift to a Lykov's type system of differential equations of first order. The proof is given for different values of the generalized operators of fractional integro-differentiation included in the boundary condition.Emergence of consensus of multi-agents systems on time scales.https://zbmath.org/1449.343232021-01-08T12:24:00+00:00"Schmeidel, Ewa"https://zbmath.org/authors/?q=ai:schmeidel.ewa-l"Ostaszewska, Urszula"https://zbmath.org/authors/?q=ai:ostaszewska.urszula"Zdanowicz, Małgorzata"https://zbmath.org/authors/?q=ai:zdanowicz.malgorzataSummary: In this paper an emergence of leader-following consensus on arbitrary time scales is investigated. It means that the step size is not necessarily constant but it is a function of time. We propose and prove conditions ensuring a leader-following consensus for discrete time scales. The presented results are illustrated by numerical examples.Qualitative properties of solutions for mixed type functional-differential equations with maxima.https://zbmath.org/1449.342162021-01-08T12:24:00+00:00"Otrocol, Diana"https://zbmath.org/authors/?q=ai:otrocol.dianaSummary: In this paper, we study some properties of the solutions of a second order system of functional-differential equations with maxima, of mixed type, with ``boundary'' conditions. We use the weakly Picard operator technique.A study on the uniform convergence of spectral expansions for continuous functions on a Sturm-Liouville problem.https://zbmath.org/1449.343102021-01-08T12:24:00+00:00"Maris, Emir Ali"https://zbmath.org/authors/?q=ai:maris.emir-ali"Goktas, Sertac"https://zbmath.org/authors/?q=ai:goktas.sertacSummary: The paper is about investigating the uniform convergence conditions of spectral expansions of continuous functions in terms of root functions of a spectral problem with the same eigenparameter in the second-order differential equation and depending on quadratically in one of the boundary conditions on a closed interval.Uniform stability of fractional-order fuzzy cellular neural networks with delay.https://zbmath.org/1449.342722021-01-08T12:24:00+00:00"Ma, Weijun"https://zbmath.org/authors/?q=ai:ma.weijun"Li, Xining"https://zbmath.org/authors/?q=ai:li.xining"Gao, Jianguo"https://zbmath.org/authors/?q=ai:gao.jianguoSummary: In this paper, a class of fractional-order fuzzy cellular neural networks with delay is investigated. A condition for uniform stability is established by using inequality technique for this networks. Moreover, existence, uniqueness and stability of its equilibrium point for this model based on a fixed point theorem are proved. An example is given to illustrate the main results.On local resolvability of a certain class of the first-order partial differential equations.https://zbmath.org/1449.350112021-01-08T12:24:00+00:00"Alekseenko, S. N."https://zbmath.org/authors/?q=ai:alekseenko.sergey-n"Platonova, L. E."https://zbmath.org/authors/?q=ai:platonova.l-eIn this paper the Cauchy problem for some quasilinear PDE of the first order with two independent variables is examined. The curve where initial data are given may be described by parametric or explicit equations; also it may have finite or infinite length. By means of additional argument method for all these cases the Cauchy problem is reduced to the system of integral equations. The authors prove local resolvability of this system and deduce that problem's solution has the same smoothness as the initial data.
Reviewer: Aleksey Syromyasov (Saransk)Almost automorphic solutions for shunting inhibitory cellular neural networks with leakage delays on time scales.https://zbmath.org/1449.343182021-01-08T12:24:00+00:00"Dai, Lihua"https://zbmath.org/authors/?q=ai:dai.lihua"Hui, Yuanxian"https://zbmath.org/authors/?q=ai:hui.yuanxianSummary: Shunting inhibitory cellular neural networks with time-varying delays in the leakage term and continuously distributed delays on a time scale \(T\) are proposed. Based on the exponential dichotomy of linear dynamic equation on time scales, fixed point theorems on time scales, we obtain some new sufficient conditions for the existence and global exponential stability of almost automorphic solution for the class of neural networks. Moreover, we give convictive numerical examples to show the feasibility of our results. This paper studies several classes of functional differential equations, including the existence of solutions and the stability of this solution on time scales.Local existence and uniqueness of positive solutions for a Sturm-Liouville boundary value problem of second order differential equations.https://zbmath.org/1449.340992021-01-08T12:24:00+00:00"Zhu, Xiaolin"https://zbmath.org/authors/?q=ai:zhu.xiaolin"Zhai, Chengbo"https://zbmath.org/authors/?q=ai:zhai.chengboSummary: The positive solution of a class of second-order nonlinear differential equations with Sturm-Liouville boundary value conditions is studied. By using fixed point theorems in ordered Banach spaces, the local existence and uniqueness of positive solutions are established. Finally, two applied examples are established.Existence of positive solutions to a semipositone second-order boundary value problem.https://zbmath.org/1449.340902021-01-08T12:24:00+00:00"Wei, Jinying"https://zbmath.org/authors/?q=ai:wei.jinying"Wang, Suyun"https://zbmath.org/authors/?q=ai:wang.suyun"Li, Yongjun"https://zbmath.org/authors/?q=ai:li.yongjunSummary: We consider the existence of positive solutions to the boundary value problem
\[u'' + c (t)u + \lambda f (t, u) = 0,\, 0 < t < 1,\, u (0) = u (1) = 0,\]
where \(\lambda > 0\), \(c (\cdot) \in C[0, 1]\) satisfies \(-\infty < c (t) < \pi^2\) for \(t \in [0, 1]\), \(f:[0, 1] \times \mathbb{R}^+ \to \mathbb{R}\) is continuous function and satisfies \(f \geq -L\), \(L > 0\) is a constant. By investigating the sign property of the Green function of the associated linear boundary value problem, we show the existence of positive solutions of semipositone problems. The proof of the main result is based on the Krasnosel'skii fixed point theorems in cone.Study on nonlinear HIV /AIDS model with two infection stages treatment and incidence rate.https://zbmath.org/1449.341592021-01-08T12:24:00+00:00"Wang, Fei"https://zbmath.org/authors/?q=ai:wang.fei.1|wang.fei.2"Yang, Yali"https://zbmath.org/authors/?q=ai:yang.yali"Jin, Yingji"https://zbmath.org/authors/?q=ai:jin.yingji"Cao, Shumiao"https://zbmath.org/authors/?q=ai:cao.shumiaoSummary: Based on the actual transmission and treatment, a mathematical model of HIV /AIDS with two infection stages, treatment and nonlinear incidence is established in this paper. Then the range of the feasible region of the system is discussed by using the limit theory. Secondly, the basic regeneration number is obtained by constructing regeneration matrix, and the range of the basic regeneration number is discussed. The existence and number of equilibrium points are obtained. Finally, the local and global properties of equilibrium point are proved by constructing Lyapunov function, using Lasalle invariant set, Busenberg theorem and Van den Lacy principle.Existence of positive solutions for fractional nonhomogeneous boundary value problem with \(p\)-Laplacian.https://zbmath.org/1449.340852021-01-08T12:24:00+00:00"Song, Junqiu"https://zbmath.org/authors/?q=ai:song.junqiu"Jia, Mei"https://zbmath.org/authors/?q=ai:jia.mei"Liu, Xiping"https://zbmath.org/authors/?q=ai:liu.xiping"Li, Lin"https://zbmath.org/authors/?q=ai:li.lin|li.lin.1|li.lin.2Summary: We study the existence of positive solutions for an integral boundary value problem of a fractional \(p\)-Laplacian equation with disturbance parameters. According to the properties of an integral kernel and using the cone expansion and cone compression fixed point theorem and the super-linear and sub-linear conditions, sufficient conditions of existence and nonexistence of positive solutions for the boundary value problem are obtained. The conclusions show the impact of parameters on the existence of positive solutions. Finally, we give some examples to illustrate our main results.Eigenvalues asymptotic formula and trace formula for a class of impulsive Sturm-Liouville operators.https://zbmath.org/1449.343132021-01-08T12:24:00+00:00"Ji, Jie"https://zbmath.org/authors/?q=ai:ji.jieSummary: A class of impulsive Sturm-Liouville boundary value problems on a finite interval is discussed. There are discontinuities inside the interval and the discontinuity coefficient on the right side of the equation. The asymptotic formula of the eigenvalues and the trace formula are obtained by using the residue theorem.Existence of positive solutions for boundary value problems of third-order delay differential equations.https://zbmath.org/1449.342182021-01-08T12:24:00+00:00"Luo, Qiang"https://zbmath.org/authors/?q=ai:luo.qiang"Han, Xiaoling"https://zbmath.org/authors/?q=ai:han.xiaoling"Yang, Zhonggui"https://zbmath.org/authors/?q=ai:yang.zhongguiSummary: By applying the fixed point theorem on the cone, this paper studies the existence of positive solutions for boundary value problems of third-order delay differential equation \[\begin{cases}
u''' (t) + \lambda a (t)f (t, u (t-\tau)) = 0, & t \in (0, 1),\, \tau > 0, \\
u (t) = 0, & -\tau \leq t \leq 0, \\
u (0) = u'' (0) = 0,\, u (1) = \alpha u (\eta),
\end{cases}\]
where \(\lambda\) is parameter, and \(0 < \eta < 1\), \(0 < \alpha < \frac{1}{\eta}\), \(f:[0,1] \times [0, \infty] \to [0, \infty)\) is continuous.Periodicity and almost periodicity for solutions of third-order differential equations with piecewise constant argument.https://zbmath.org/1449.342372021-01-08T12:24:00+00:00"Zhang, Bao"https://zbmath.org/authors/?q=ai:zhang.bao"Li, Hongxu"https://zbmath.org/authors/?q=ai:li.hong-xuSummary: In this paper, we consider the following third-order differential equation with piecewise constant argument \[x''' (t) - {a^2}x' (t) = bx\left (2\left[\frac{t+1}{2}\right]\right).\] We argument the form of the solution in term of the solution of the corresponding difference equation. Then we give some results on the periodicity and almost periodicity for the solutions of the equation.Periodic solutions for a class of Kirchhoff-type differential systems.https://zbmath.org/1449.341242021-01-08T12:24:00+00:00"Zhang, Shengui"https://zbmath.org/authors/?q=ai:zhang.shenguiSummary: By using variational principle, the author studies periodic solutions for a class of superlinear Kirchhoff-type \(p (t)\)-Laplacian systems. Under the condition of no Ambrosetti-Rabinowitz-type growth, some results for the existence of periodic solutions are obtained by means of a variant mountain pass type theorem.Caputo fractional differential inclusions of arbitrary order with nonlocal integro-multipoint boundary conditions.https://zbmath.org/1449.340082021-01-08T12:24:00+00:00"Ahmad, Bashir"https://zbmath.org/authors/?q=ai:ahmad.bashir.2"Garout, Doa'a"https://zbmath.org/authors/?q=ai:garout.doaa"Ntouyas, Sotiris K."https://zbmath.org/authors/?q=ai:ntouyas.sotiris-k"Alsaedi, Ahmed"https://zbmath.org/authors/?q=ai:alsaedi.ahmedSummary: We study a new class of boundary value problems of Caputo type fractional differential inclusions supplemented with nonlocal integro-multipoint boundary conditions. An existence result for the problem with convex valued (multivalued) map is obtained via nonlinear alternative of Leray-Schauder type, while the existence of solutions for the problem involving nonconvex valued map is established by means of Wegrzyk's fixed point theorem. Our results are well illustrated with examples.Existence of fast positive semi-wavefront solutions to monostable integro-differential equations with delay.https://zbmath.org/1449.342112021-01-08T12:24:00+00:00"Aguerrea, Maitere"https://zbmath.org/authors/?q=ai:aguerrea.maitere"Hakl, Robert"https://zbmath.org/authors/?q=ai:hakl.robertSummary: We establish the existence of fast positive semi-wavefront solutions to a delay integro-differential problem \[ cu'(t)=J\star u(t)-u(t)+f(u(t-h)),\quad t\in\mathbb{R},\quad u(-\infty)=0,\] where the asymmetric kernel \(J\) is exponentially bounded, the nonlinearity \(f\in C^1([0,+\infty);\mathbb{R})\) is monostable, \(h\geq 0\), and \(c>0\).Existence and uniqueness of solutions for boundary value problems of fractional Langevin equation.https://zbmath.org/1449.340342021-01-08T12:24:00+00:00"Wang, Wenqian"https://zbmath.org/authors/?q=ai:wang.wenqian"Sun, Rui"https://zbmath.org/authors/?q=ai:sun.rui"Chai, Jianhong"https://zbmath.org/authors/?q=ai:chai.jianhong"Zhou, Yuqun"https://zbmath.org/authors/?q=ai:zhou.yuqunSummary: In this paper, the existence and uniqueness of solutions for boundary value problems of fractional Langevin equations are studied by using Leray-Schauder fixed point theorem. Then, the existence theorem of solutions is obtained.New oscillation criterion for conformable fractional differential equations with damping.https://zbmath.org/1449.340272021-01-08T12:24:00+00:00"Li, Wenjie"https://zbmath.org/authors/?q=ai:li.wenjie"Hou, Wei"https://zbmath.org/authors/?q=ai:hou.wei"Zheng, Zhaowen"https://zbmath.org/authors/?q=ai:zheng.zhaowenSummary: In this paper, a new oscillation criterion for conformable fractional differential equations with damping terms of the form \[ (p (t)y^{(\alpha)} (t))^{(\alpha)} + r (t)y^{(\alpha)} (t) + q (t)y (t) = 0,\, t \geq 0,\, 0 < \alpha \leq 1,\] is derived. Examples are given to illustrate the effectiveness of the main result.Extinction in two species nonlinear competitive system with time delays.https://zbmath.org/1449.342972021-01-08T12:24:00+00:00"Wang, Lili"https://zbmath.org/authors/?q=ai:wang.lili.1|wang.lili|wang.lili.4|wang.lili.8|wang.lili.2|wang.lili.5|wang.lili.6|wang.lili.3|wang.lili.7Summary: A two species nonlinear competitive system is studied. Sufficient conditions are derived for the extinction of the second species of the system. The results generalize and improve the conclusion of related references.Global asymptotic stability for a class of Hopfield neural networks with proportional delays.https://zbmath.org/1449.342532021-01-08T12:24:00+00:00"Zhou, Rui"https://zbmath.org/authors/?q=ai:zhou.rui"Zhou, Liqun"https://zbmath.org/authors/?q=ai:zhou.liqunSummary: The stability of a class of delayed Hopfield neural networks is studied. Here the delay is a kind of unbounded proportional delay, which is different from the unbounded distribution delay. Applying Lyapunov stability theory and the Barbalat lemma, two novel sufficient conditions guaranteeing the global asymptotic stability of Hopfield neural networks are given. Finally, the obtained results are illustrated by numerical examples and simulations. The obtained results can lay a theoretical foundation for the further application of Hopfield neural networks with proportional delays.Entire function solutions of two types of Fermat type \(q\)-difference differential equations.https://zbmath.org/1449.300532021-01-08T12:24:00+00:00"Fan, Bo"https://zbmath.org/authors/?q=ai:fan.bo"Ding, Jie"https://zbmath.org/authors/?q=ai:ding.jieSummary: In this paper, using Nevanlinna's value distribution theory and the complex differential equations theory, the existence of finite order transcendental entire function solutions for two types of Fermat type \(q\)-difference differential equations of the following form \[{f^2} (qz+c) + ({f^{ (k)}} (z))^2 = 1,\; [f (qz+c) - f (z)]^2 + ({f^{ (k)}} (z))^2 = 1\] is investigated. Moreover, the precise expression of the solutions is obtained under some assumptions.General solutions of a higher order impulsive fractional differential equation involving the Riemann-Liouville fractional derivatives.https://zbmath.org/1449.340252021-01-08T12:24:00+00:00"Liu, Yuji"https://zbmath.org/authors/?q=ai:liu.yuji.1|liu.yujiSummary: We present general solutions (explicit solutions) of a class of multi-term impulsive fractional differential equations involving the Riemann-Liouville fractional derivatives.Finite-time stability of fractional-order singular impulsive systems with Caputo derivative.https://zbmath.org/1449.342492021-01-08T12:24:00+00:00"Wu, Tong"https://zbmath.org/authors/?q=ai:wu.tong"Zhang, Zhixing"https://zbmath.org/authors/?q=ai:zhang.zhixing"Jiang, Wei"https://zbmath.org/authors/?q=ai:jiang.wei.1Summary: Finite-time stability problem of fractional order linear singular systems in the sense of Caputo with both disturbance and impulses is studied in this paper by constructing a new Lyapunov functional, using the related properties for Caputo fractional derivative and the generalized Gronwall inequality. The results obtained extend conclusions of relevant literature. Finally, some numerical examples are given for different cases to illustrate of the theorems.Theoretical analysis of predator-prey system considering cooperative hunting.https://zbmath.org/1449.341802021-01-08T12:24:00+00:00"Zhao, Yeqing"https://zbmath.org/authors/?q=ai:zhao.yeqing"Li, Guihua"https://zbmath.org/authors/?q=ai:li.guihuaSummary: In the ecosystem, the cooperative behavior in populations could promote the sustainable survival of the natural species. We theoretically analyze a predator-prey system proposed in 2017, and it was found that hunting intensity of the predator affected the survival of the population, and the intensity of their cooperative hunting may be strengthened or weakened by changing the initial value of the predator density. Thus the coexistence state of the predator and the predator will be adjusted, so that the system tends to be stable. By changing the existence threshold value of the predator, some conditions of coexisting equilibria are given. Furthermore, the behavior of the equilibria are analyzed. It was found that the system undergoes Hopf bifurcation, and its stability condition was given.Stability and Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations.https://zbmath.org/1449.341982021-01-08T12:24:00+00:00"Wang, Jinbin"https://zbmath.org/authors/?q=ai:wang.jinbin"Zhang, Rui"https://zbmath.org/authors/?q=ai:zhang.rui.1|zhang.rui.4|zhang.rui.2|zhang.rui|zhang.rui.5|zhang.rui.3Summary: We study the stability of composite laminated piezoelectric plate subject to external and internal excitations. The qualitative theory of ordinary differential equations is applied to analyze the stability of the system's trivial equilibrium point. By using the bifurcation theory of differential equation, the existence of Hopf bifurcation and the stability of bifurcation direction and periodic solution are obtained. Moreover, numerical simulations are presented to demonstrate the applicability of the theoretical results.Existence of positive solutions of fractional differential equations with integral boundary conditions.https://zbmath.org/1449.340732021-01-08T12:24:00+00:00"He, Xingyue"https://zbmath.org/authors/?q=ai:he.xingyue"Gao, Chenghua"https://zbmath.org/authors/?q=ai:gao.chenghuaSummary: Based on the fixed-point index theory in a cone, by constructing a cone and the properties of the Green function, we give the existence, multiplicity and nonexistence of positive solutions for the following nonlinear boundary value problem
\[\begin{cases}
{}^CD^\alpha u (t) +\lambda f (t,u(t)) = 0, \quad t \in (0,1),\\
u (0) = u'' (0) = 0, \quad u (1) = \mu\int_0^1 u (s)\mathrm{d}s,
\end{cases}\]
with two parameters under different growth conditions, where \(2<\alpha<3\). \(0<\mu<2\) and \(\lambda>0\) are two parameters.Stability and Hopf bifurcation of asymmetric double-ring structured neural network.https://zbmath.org/1449.343052021-01-08T12:24:00+00:00"Zhou, Shuai"https://zbmath.org/authors/?q=ai:zhou.shuai"Xiao, Min"https://zbmath.org/authors/?q=ai:xiao.min"Xing, Ruitao"https://zbmath.org/authors/?q=ai:xing.ruitao"Zhang, Yuezhong"https://zbmath.org/authors/?q=ai:zhang.yuezhong"Cheng, Zunshui"https://zbmath.org/authors/?q=ai:cheng.zunshuiSummary: Ring structures are widely used in neural networks, and previous researches with respect to neural dynamic bifurcation were confined to models with single-ring structure. Notably, neural networks are composed of thousands of neurons coupled together, and these structures are so complex that they cannot be accurately described through only a single-ring structure, and therefore it is more practical for investigating neural network models with multiple ring topology. In this paper, an asymmetric model with double-ring neural structure is proposed, and the stability and Hopf bifurcation of the model are investigated. Numerical simulations are subsequently actualized to illustrate the theoretical results.Finite-time combination synchronization of multiple chaotic systems with multi-switching mode.https://zbmath.org/1449.341842021-01-08T12:24:00+00:00"Li, Chao"https://zbmath.org/authors/?q=ai:li.chao.1|li.chao.2|li.chao.3|li.chao.5"Zhang, Bin"https://zbmath.org/authors/?q=ai:zhang.bin.2|zhang.bin.1|zhang.bin.4|zhang.bin.3"Chen, Xiangyong"https://zbmath.org/authors/?q=ai:chen.xiangyong"Li, Tianze"https://zbmath.org/authors/?q=ai:li.tianze"Dong, Hefu"https://zbmath.org/authors/?q=ai:dong.hefuSummary: This paper mainly investigates finite-time synchronization control of chaotic systems with multi-switching mode. For multiple real chaotic systems with different orders, their multi-switching synchronization behavior is investigated and finite-time combination multi-switching synchronization is defined. A class of finite-time control schemes is designed, which can realize fast synchronization, when sufficient conditions for finite-time stability of error systems are provided. Finally, simulation results show that the proposed control scheme has fast convergence and reasonable validity.Synchronization of two-layer heterogeneous networks with stochastic perturbations and mixed delays.https://zbmath.org/1449.342552021-01-08T12:24:00+00:00"Jin, Xin"https://zbmath.org/authors/?q=ai:jin.xin.1|jin.xin"Yang, Huihui"https://zbmath.org/authors/?q=ai:yang.huihui"Wang, Zhengxin"https://zbmath.org/authors/?q=ai:wang.zhengxinSummary: In a real world scenario, several complex systems are represented by a group of interdependent network systems, and not by a single network. This paper focuses on the synchronization control of corresponding nodes in two-layer heterogeneous networks with stochastic perturbations and mixed delays. Based on the LaSalle-type invariance principle and the Lyapunov stability theory, the paper derives sufficient conditions for global asymptotic synchronization by applying the pinning control, which only controls a small fraction of the nodes. To reduce the gain of feedback control, the synchronization conditions of two-layer heterogeneous networks are further weakened by adopting the adaptive control scheme. Finally, the effectiveness of the theoretical results is verified by numerical simulations.Positive periodic solutions of general nonlinear third-order ordinary differential equations.https://zbmath.org/1449.341192021-01-08T12:24:00+00:00"Deng, Zhengping"https://zbmath.org/authors/?q=ai:deng.zhengping"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: Using the fixed point index theory of cones, we consider the existence of positive \(2\pi\)-periodic solutions for general third-order ordinary differential equation
\[Lu (t) = f (t, u (t), u' (t), u'' (t)) (t\in\mathbb{R}),\]
where
\[Lu (t) = u''' (t) + a_2 u'' (t) + {a_1}u' (t) + {a_0}u (t)\]
is a third-order ordinary differential operator, \(f:\mathbb{R} \times [0,\infty) \times {\mathbb{R}^2} \to [0,\infty)\) is a continuous function and \(f (t, x, y, z)\) is \(2\pi\)-periodic with respect to \(t\). Under the conditions that the nonlinear term \(f\) satisfies some easily verifiable inequalities, some existence results for positive \(2\pi\)-periodic solutions of the equation are obtained that allow \(f (t, x, y, z)\) satisfies superlinear or sublinear growth with respect to \(x\), \(y\), \(z\).Fixed-time mixed outer synchronization of complex networks.https://zbmath.org/1449.341892021-01-08T12:24:00+00:00"Nie, Pingping"https://zbmath.org/authors/?q=ai:nie.pingping"Li, Wang"https://zbmath.org/authors/?q=ai:li.wang"Shi, Hongjun"https://zbmath.org/authors/?q=ai:shi.hongjunSummary: In this paper, the fixed-time mixed outer synchronization between two complex dynamical networks is studied. By using suitable controllers, we achieve the fixed-time mixed outer synchronization between two complex networks based on the fixed-time stability theory. Finally, numerical simulations are performed to illustrate the effectiveness and feasibility of the proposed control approach.Synchronization analysis for general linear complex networks via event-based aperiodically intermittent pinning control.https://zbmath.org/1449.341922021-01-08T12:24:00+00:00"Wang, Ruifeng"https://zbmath.org/authors/?q=ai:wang.ruifeng"Feng, Jianwen"https://zbmath.org/authors/?q=ai:feng.jianwenSummary: In this paper, the global synchronization problem of general linear complex networks is analyzed by combining the event-triggered control strategy with the aperiodically intermittent pinning control scheme. By designing an event-triggered condition and the rules for selecting the controlled nodes, the pinning node set can be updated under certain conditions, which not only greatly improves the efficiency of networks synchronization, but also avoids the deficiencies caused by random selection of controlled node set in existing research. Furthermore, a simple Lyapunov function is constructed, and the stability theory and differential inequality are applied. Some sufficient conditions for the asymptotically exponential synchronization of the networks are obtained through rigorous mathematical analysis. At the same time, the infinitely fast switching of the pinning node set is also avoided.A new exact solution of a damped quadratic non-linear oscillator.https://zbmath.org/1449.340032021-01-08T12:24:00+00:00"Zhu, Jin-wen"https://zbmath.org/authors/?q=ai:zhu.jin-wenSummary: In this paper, we derive a new exact solution of the damped quadratic nonlinear oscillator (Helmholtz oscillator) based on the developed solution for the undamped case by the Jacobi elliptic functions. It is interesting to see that both of the damped Duffing oscillator and Helmholtz oscillator possess solutions that follow closely to the undamped case, and even the solution procedures are almost the same.Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations.https://zbmath.org/1449.340102021-01-08T12:24:00+00:00"Atabakzadeh, M. H."https://zbmath.org/authors/?q=ai:atabakzadeh.m-h"Akrami, M. H."https://zbmath.org/authors/?q=ai:akrami.mohammad-hussian|akrami.mohammad-hossein"Erjaee, G. H."https://zbmath.org/authors/?q=ai:erjaee.gholam-hussain|erjaee.gholam-hussian|erjaee.g-hossianSummary: The aim of this article is to present an analytical approximation solution for linear and nonlinear multi-order fractional differential equations (FDEs) by extending the application of the shifted Chebyshev operational matrix. For this purpose, we convert FDE into a counterpart system and then using proposed method to solve the resultant system. Our results in solving four different linear and nonlinear FDE, confirm the accuracy of proposed method.Population dynamics models based on the transmission mechanism of mcr-1 without immigration.https://zbmath.org/1449.920402021-01-08T12:24:00+00:00"Qu, Leilei"https://zbmath.org/authors/?q=ai:qu.leilei"Kang, Baolin"https://zbmath.org/authors/?q=ai:kang.baolin"He, Mingfeng"https://zbmath.org/authors/?q=ai:he.mingfeng"Pan, Qiuhui"https://zbmath.org/authors/?q=ai:pan.qiuhuiSummary: Antimicrobial resistance is now considered as one of the most serious global threats to human health in the 21st century. Recently, the discovery of a plasmid-borne colistin resistance gene, mcr-1, in China heralds the emergence of truly pan-drug-resistant bacteria. The gene has been found primarily in \textit{Escherichia coli} but has also been identified in other members of the Enterobacteriaceae in animal, food, human, and environmental samples on every continent. Until now, the articles published have appeared to fairly use and interpret experimental studies. The aim of this paper is to develop mathematical models that quantitatively describe the dynamic of the population of domestic animals and humans. We propose two ordinary differential equation models for the transmission dynamic of mcr-1 gene. The models are analyzed using stability theory of differential equations. Positive equilibrium points of the system are investigated and their stability analyses are carried out. Moreover, the numerical simulations of the proposed model, which supports the theoretical findings, are also performed. Our results show that the fraction of animals used for human consumption is the most effective parameters of controlling the spread of mcr-1 gene in comparison with other parameters.Existence of positive solutions for a kind of nonlinear fractional differential equation with nonlinear boundary value conditions.https://zbmath.org/1449.340782021-01-08T12:24:00+00:00"Li, Mengting"https://zbmath.org/authors/?q=ai:li.mengting"Zhang, Kemei"https://zbmath.org/authors/?q=ai:zhang.kemeiSummary: In this paper, we use mixed monotone operator method to deal with the existence and uniqueness of positive solutions to a class of boundary value problems of the nonlinear fractional order differential equation. Our analysis is based on a new fixed point theorem for mixed monotone operators. As an application, we also give a simple example to illustrate our main conclusions.Solvability for fractional \(p\)-Laplacian differential equation with integral boundary conditions at resonance on infinite interval.https://zbmath.org/1449.340262021-01-08T12:24:00+00:00"Liu, Zongbao"https://zbmath.org/authors/?q=ai:liu.zongbao"Liu, Wenbin"https://zbmath.org/authors/?q=ai:liu.wenbin.1"Zhang, Wei"https://zbmath.org/authors/?q=ai:zhang.wei.12|zhang.wei.9|zhang.wei.7|zhang.wei.3|zhang.wei.6|zhang.wei.15|zhang.wei.2|zhang.wei.18|zhang.wei.4|zhang.wei.5|zhang.wei.17|zhang.wei.11|zhang.wei.1|zhang.wei.13|zhang.wei.10|zhang.wei.14|zhang.wei.16Summary: In this paper, we investigate the existence of solutions for a class of fractional integral boundary value problems with \(p\)-Laplacian operator at resonance on infinite interval by using Mawhin's continuation theorem. An example is given to show the applicability of our main result.Noether theorem on time scales for Lagrangian systems in event space.https://zbmath.org/1449.700242021-01-08T12:24:00+00:00"Shi, Yufei"https://zbmath.org/authors/?q=ai:shi.yufei"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yiSummary: The Noether symmetry and the conserved quantity on time scales in event space are studied in this paper. Firstly, the Lagrangian of parameter forms on time scales in event space are established. The Euler-Lagrange equations and the second Euler-Lagrange equations of variational calculus on time scales in event space are established. Secondly, based upon the invariance of the Hamilton action on time scales in event space under the infinitesimal transformations of a group, the Noether symmetry and the conserved quantity on time scales in event space are established. Finally, an example is given to illustrate the method and results.Adiabatic invariant for dynamic systems on time scale.https://zbmath.org/1449.700152021-01-08T12:24:00+00:00"Song, Chuanjing"https://zbmath.org/authors/?q=ai:song.chuanjingSummary: Perturbation to Noether symmetry and adiabatic invariants for Birkhoffian system, Hamiltonian system and Lagrangian system with delta derivative are investigated, respectively. Firstly, the definition and some related properties of time scale calculus are listed simply as preliminaries. Secondly, the Birkhoffian system with delta derivative is studied. Based on the differential equation of motion as well as Noether symmetry and conserved quantity, perturbation to Noether symmetry and adiabatic invariant are investigated. Thirdly, adiabatic invariants for the Hamiltonian system and the Lagrangian system are presented through some transformations. Finally, an example is given to illustrate the methods and results.Sliding mode synchronization of fractional-order hyperchaotic financial systems with uncertainty and outer disturbance.https://zbmath.org/1449.341862021-01-08T12:24:00+00:00"Mao, Beixing"https://zbmath.org/authors/?q=ai:mao.beixingSummary: In this paper, the synchronization problem of hyperchaotic fractional-order financial systems with uncertainty and outer disturbances is studied based on the sliding mode control and integral sliding mode control techniques. Two sufficient conditions are established to ensure the fractional-order uncertain hyperchaotic financial systems acquiring sliding mode and integral sliding mode synchronization by designing sliding mode functions and controllers through founding adaptive rules using fractional-order integral and calculus. The conclusions are verified by Matlab numerical simulations. The sliding mode functions, controllers and adaptive rules which are designed in this paper can be planted for studying sliding mode synchronization of integer-order chaotic systems. The approaches used in this paper supply the techniques to study the fractional-order systems and can be extended to the integer-order systems synchronization problems.Change in criticality of Hopf bifurcation in a time-delayed cancer model.https://zbmath.org/1449.342932021-01-08T12:24:00+00:00"Ncube, Israel"https://zbmath.org/authors/?q=ai:ncube.israel"Martin, Kiara"https://zbmath.org/authors/?q=ai:martin.kiaraSummary: The main goal of this work is to conduct a rigorous study of a mathematical model that was first proposed by \textit{M. Gałach} [Int. J. Appl. Math. Comput. Sci. 13, No. 3, 395--406 (2003; Zbl 1035.92019)]. The model itself is an adaptation of an earlier model proposed by \textit{V. A. Kuznetsov} et al. [Bull. Math. Biol. 56, No. 2, 295--321 (1994; Zbl 0789.92019)], and attempts to describe the interaction that exists between immunogenic tumour cells and the immune system. The particular adaptation due to \textit{M. Gałach} consists of replacing the Michaelis-Menten function of \textit{V. A. Kuznetsov} et al. by a Lotka-Volterra form instead, and incorporating a single discrete time delay in the latter to account for the biophysical fact that the immune system takes finite, non-zero time to mount a response to the presence of immunogenic tumour cells in the body. In this work, we perform a linear stability analysis of the model's three equilibria, and formulate a local Hopf bifurcation theorem for one of the two endemic equilibria. Furthermore, using centre manifold reduction and normal form theory, we characterise the criticality of the Hopf bifurcation. Our theoretical results are supported by some sample numerical plots of the Poincaré-Lyapunov constant in an appropriate parameter space. In a sense, our work in this article complements and significantly extends the work initiated by \textit{M. Gałach}.The solving integro-differential equations of fractional order with the ultraspherical functions.https://zbmath.org/1449.653662021-01-08T12:24:00+00:00"Panahy, Saeid"https://zbmath.org/authors/?q=ai:panahy.saeid"Khani, Ali"https://zbmath.org/authors/?q=ai:khani.aliSummary: In this paper, an integration method is presented based on using ultraspherical polynomials for solving a class of linear fractional integro-differential equations of Volterra types. This method is based on a new investigation of ultraspherical integration to approximate the highest order derivative in the equations and generate approximations to the lower order derivatives through integration of the higher-order derivatives. Numerical example illustrate the efficiency and accuracy of the method.A study on functional fractional integro-differential equations of Hammerstein type.https://zbmath.org/1449.653672021-01-08T12:24:00+00:00"Saeedi, Leila"https://zbmath.org/authors/?q=ai:saeedi.leila"Tari, Abolfazl"https://zbmath.org/authors/?q=ai:tari.abolfazl"Babolian, Esmail"https://zbmath.org/authors/?q=ai:babolian.esmailSummary: In this paper, functional Hammerstein integro-differential equations of fractional order is studied. Here, the existence and uniqueness of the solution is proved. A numerical method to approximate the solution of problem is also presented which is based on an improvement of the successive approximations method. Error estimation of the method is analyzed and error bound is obtained. The convergence and stability of the method are proved. At the end, application of the method is revealed by presenting some examples.Neumann method for solving conformable fractional Volterra integral equations.https://zbmath.org/1449.450032021-01-08T12:24:00+00:00"Ilie, Mousa"https://zbmath.org/authors/?q=ai:ilie.mousa"Biazar, Jafar"https://zbmath.org/authors/?q=ai:biazar.jafar"Ayati, Zainab"https://zbmath.org/authors/?q=ai:ayati.zainabSummary: This paper deals with the solution of a class of Volterra integral equations in the sense of the conformable fractional derivative. For this goal, the well-organized Neumann method is developed and some theorems related to existence, uniqueness, and sufficient condition of convergence are presented. Some illustrative examples are provided to demonstrate the efficiency of the method in solving conformable fractional Volterra integral equations.Oscillatory behavior for nonlinear delay differential equation with several non-monotone arguments.https://zbmath.org/1449.342272021-01-08T12:24:00+00:00"Öcalan, Özkan"https://zbmath.org/authors/?q=ai:ocalan.ozkan"Kılıç, Nurten"https://zbmath.org/authors/?q=ai:kilic.nurten"Özkan, Umut Mutlu"https://zbmath.org/authors/?q=ai:ozkan.umut-mutlu"Öztürk, Sermin"https://zbmath.org/authors/?q=ai:ozturk.sermin-sahinSummary: This paper is devoted to obtaining some new sufficient conditions for the oscillation of all solutions of first order nonlinear differential equations with several deviating arguments. Finally, an illustrative example related to our results is given.The inverse Sturm-Liouville problem by three spectra.https://zbmath.org/1449.340552021-01-08T12:24:00+00:00"Liu, Yao"https://zbmath.org/authors/?q=ai:liu.yaoSummary: The inverse Sturm-Liouville problem associated with three spectra is considered. It is shown that if the given two sequences can be divided into three sequences under conditions, which can be the corresponding parts of eigenvalues of three Sturm-Liouville problems defined on the interval \([0, a]\), then the potential function \(q (x)\) on the interval \([0, a]\) can be uniquely determined by the corresponding parts of the three spectra.Adaptive cluster synchronization in networks with time-varying and distributed coupling delays.https://zbmath.org/1449.342562021-01-08T12:24:00+00:00"Li, Kezan"https://zbmath.org/authors/?q=ai:li.kezan"Zhou, Jin"https://zbmath.org/authors/?q=ai:zhou.jin"Yu, Wenwu"https://zbmath.org/authors/?q=ai:yu.wenwu"Small, Michael"https://zbmath.org/authors/?q=ai:small.michael"Fu, Xinchu"https://zbmath.org/authors/?q=ai:fu.xinchuSummary: This paper studies the adaptive cluster synchronization of a generalized linearly coupled network with time-varying delay and distributed delays. This network includes nonidentical nodes displaying different local dynamical behaviors, while for each cluster of that network the internal dynamics is uniform (such as chaotic, periodic, or stable behavior). In particular, the generalized coupling matrix of this network can be asymmetric and weighted. Two different adaptive laws of time-varying coupling strength and a linear feedback control are designed to achieve the cluster synchronization of this network. Some sufficient conditions to ensure the cluster synchronization are obtained by using the invariant principle of functional differential equations and linear matrix inequality (LMI). Numerical simulations verify the efficiency of our proposed adaptive control method.Dynamics of single population model with transient and non-transient impulsive harvesting in polluted environment.https://zbmath.org/1449.341442021-01-08T12:24:00+00:00"Jiao, Jianjun"https://zbmath.org/authors/?q=ai:jiao.jianjun"Chen, Lansun"https://zbmath.org/authors/?q=ai:chen.lansun"Li, Limei"https://zbmath.org/authors/?q=ai:li.limeiSummary: Using the theories of impulsive differential equations, we investigate the dynamics of a single population model with transient and non-transient impulsive harvesting in polluted environment, and give sufficient conditions for the permanence of the population. The results show that the amount of transient impulsive harvesting and the non-transient impulsive harvesting interval length play important roles in the permanence of the population.Asymptotic solution to a class of turbulent Lorenz system for atmospheric physics.https://zbmath.org/1449.341822021-01-08T12:24:00+00:00"Xu, Jianzhong"https://zbmath.org/authors/?q=ai:xu.jianzhong"Wang, Weigang"https://zbmath.org/authors/?q=ai:wang.weigang"Mo, Jiaqi"https://zbmath.org/authors/?q=ai:mo.jiaqi(no abstract)Existence of positive solutions of a class of first-order singular differential equations with nonlinear boundary conditions.https://zbmath.org/1449.340692021-01-08T12:24:00+00:00"Zhu, Yan"https://zbmath.org/authors/?q=ai:zhu.yanSummary: By using the Krasnoselskii fixed point theorem, the author proves the existence of positive solutions of the first-order differential equation with nonlinear boundary conditions: \[\begin{cases}u' (t) + a (t)u (t) = \lambda h (t)f (u (t)),\; t \in (0, 1), \\ u (0) = c (u (1))u (1),\end{cases}\] where \(\lambda\) is a positive parameter. \(a \in C ([0, 1], [0, \infty))\) and \(\int_0^1 a (t){\mathrm{d}}t > 0, h \in C ([0, 1], (0, \infty)), c \in C ([0, \infty), [1, \infty))\) and \(c < \exp \left\{\int_0^1 a (\theta){\mathrm{d}}\theta\right\}\), \(f: (0, \infty) \to \mathbb{R}\) is continuous, superlinear at \(\infty\) and is allowed to be singular at 0.Hopf bifurcation analysis of time-delay Rössler system.https://zbmath.org/1449.342442021-01-08T12:24:00+00:00"Zhao, Shaoqing"https://zbmath.org/authors/?q=ai:zhao.shaoqing"Cui, Yan"https://zbmath.org/authors/?q=ai:cui.yan"Zhou, Liuyuan"https://zbmath.org/authors/?q=ai:zhou.liuyuan"Sun, Guan"https://zbmath.org/authors/?q=ai:sun.guan"He, Hongjun"https://zbmath.org/authors/?q=ai:he.hongjunSummary: The Hopf bifurcation problem of time-delay Rössler system is studied. Combining canonical form with Hopf bifurcation theory, the occurrence condition of Hopf bifurcation of time-delay Rössler system is given. The Hopf bifurcation point of time-delay parameter of the system is obtained and the stability of the system near the time delay bifurcation point is analyzed. The simulation images of the system with different time-delay parameters are drawn by MATLAB software.Symmetric nonlinear functional differential equations at resonance.https://zbmath.org/1449.342122021-01-08T12:24:00+00:00"Dilna, Nataliya"https://zbmath.org/authors/?q=ai:dilna.nataliya"Feckan, Michal"https://zbmath.org/authors/?q=ai:feckan.michal"Solovyov, Mykola"https://zbmath.org/authors/?q=ai:solovyov.mykola"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrongSummary: It is shown that a class of symmetric solutions of scalar nonlinear functional differential equations at resonance with deviations from \(\mathbb{R}\to\mathbb{R}\) can be investigated by using the theory of boundary-value problems. Conditions on a solvability and unique solvability are established. Examples are presented to illustrate given results.Existence of extremal solutions for second-order interval-valued differential equations.https://zbmath.org/1449.340462021-01-08T12:24:00+00:00"Liu, Xuelong"https://zbmath.org/authors/?q=ai:liu.xuelong"Ye, Guoju"https://zbmath.org/authors/?q=ai:ye.guoju"Liu, Wei"https://zbmath.org/authors/?q=ai:liu.wei.7"Zhao, Dafang"https://zbmath.org/authors/?q=ai:zhao.dafangSummary: In this paper, we discuss a second-order interval-valued differential equations under generalized differentiability, and use the monotone iterative technique combining with the method of upper and lower solutions to investigate the existence of extremal solutions for the interval-valued differential equation.On the fractional-order extended Kalman filter and its application to chaotic cryptography in noisy environment.https://zbmath.org/1449.370712021-01-08T12:24:00+00:00"Sadeghian, Hoda"https://zbmath.org/authors/?q=ai:sadeghian.hoda"Salarieh, Hassan"https://zbmath.org/authors/?q=ai:salarieh.hassan"Alasty, Aria"https://zbmath.org/authors/?q=ai:alasty.aria"Meghdari, Ali"https://zbmath.org/authors/?q=ai:meghdari.aliSummary: In this paper via a novel method of discretized continuous-time Kalman filter, the problem of synchronization and cryptography in fractional-order systems has been investigated in presence of noisy environment for process and output signals. The fractional-order Kalman filter equation, applicable for linear systems, and its extension called the extended Kalman filter, which can be used for nonlinear systems, are derived. The result is utilized for chaos synchronization with the aim of cryptography while the transmitter system is fractional-order, and both the transmitter and transmission channel are noisy. The fractional-order stochastic chaotic Chen system is then presented to apply the proposed method for chaotic signal cryptography. The results show the effectiveness of the proposed method.Phase-response synchronization in coupled neuronal oscillator population with time delay.https://zbmath.org/1449.342622021-01-08T12:24:00+00:00"Zhang, Zhaokun"https://zbmath.org/authors/?q=ai:zhang.zhaokun"Jiao, Xianfa"https://zbmath.org/authors/?q=ai:jiao.xianfaSummary: In the nervous system, there is a time delay in the information transmission from the neuronal of pre-synapse to the neuronal of post-synapse. In this paper, a dynamical evolution model of the population of coupled neuronal oscillators with time delay in the presence of external periodic stimulus and noise is proposed. The average number density is used to describe the synchronous activity of the population of neuronal oscillators, and the evolution equation of average number density is derived. Numerical simulations indicate that the time delay influences the synchronous activity of the neuronal oscillator population. As the time delay increases, the neuronal population shows stable periodic synchronous oscillation and the degree of synchronization decreased. As the stimulus intensity increases, the synchronization degree of the neuronal population increases. As the stimulus frequency increases, the synchronization degree of the neuronal population decreases rapidly. As the stimulus frequency is close to the population characteristic frequency, the synchronization degree of the neuronal population decreases. As the noise intensity increases, the synchronization degree of the neuronal population increases.Combining fractional differential transform method and reproducing kernel Hilbert space method to solve fuzzy impulsive fractional differential equations.https://zbmath.org/1449.340062021-01-08T12:24:00+00:00"Najafi, Nematallah"https://zbmath.org/authors/?q=ai:najafi.nematallah"Allahviranloo, Tofigh"https://zbmath.org/authors/?q=ai:allahviranloo.tofighSummary: The aim of this paper is to use the combination of Reproducing kernel Hilbert space method (RKHSM) and fractional differential transform method (FDTM) to solve the linear and nonlinear fuzzy impulsive fractional differential equations. Finding the numerical solution of this class of equations is a difficult topic to analyze. In this study, convergence analysis, estimations error and bounds errors are discussed in detail under some hypotheses which provide the theoretical basis of the proposed algorithm. Some numerical examples indicate that this method is an efficient one to solve the mentioned equations.Dynamics behavior for second-order neutral Clifford differential equations: inertial neural networks with mixed delays.https://zbmath.org/1449.342412021-01-08T12:24:00+00:00"Aouiti, Chaouki"https://zbmath.org/authors/?q=ai:aouiti.chaouki"Ben Gharbia, Imen"https://zbmath.org/authors/?q=ai:ben-gharbia.imenSummary: In this paper, Clifford-valued inertial neutral neural networks with time-varying delays and infinite distributed delay are investigated. With the help of the pseudo almost periodic function theory, Banach's fixed point theorem, and the differential inequality theory, a set of sufficient conditions that guarantee the existence and the global exponential stability of unique pseudo-almost periodic solutions of Clifford-valued inertial neutral neural networks with mixed delays are established. Our results are new and complement some previously known ones. Moreover, numerical simulations are carried out to illustrate our theoretical results.The number of limit cycles for a class of quasi-homogeneous polynomial centers.https://zbmath.org/1449.341082021-01-08T12:24:00+00:00"He, Zecen"https://zbmath.org/authors/?q=ai:he.zecen"Liang, Haihua"https://zbmath.org/authors/?q=ai:liang.haihuaSummary: This paper considers a class of \( (m, 1)\)-quasi-homogeneous polynomial planar differential systems with global center. By studying the number of zeros of abelian integrals, we obtain the number of limit cycles bifurcating from the period annulus of the center of the system, separately under the perturbation of polynomial of degree \(n\), and the \( (n, 1)\)-quasi-homogeneous polynomial. Moreover, the upper bound of these numbers is given and proved to be reachable.Homotopy analysis solution for solving boundary value problems on the second order nonlinear differential equation.https://zbmath.org/1449.340542021-01-08T12:24:00+00:00"Luo, Jiongxing"https://zbmath.org/authors/?q=ai:luo.jiongxingSummary: This paper studies the application of homotopy analysis method combined with variable substitution to solve two points boundary value problems of second order nonlinear differential equation. Three examples of solving second order nonlinear problems by homotopy analysis are given. The effectiveness of the proposed homotopy analysis method in solving nonlinear problems is shown.Modeling and nonlinear control of a flexible-link manipulator.https://zbmath.org/1449.932012021-01-08T12:24:00+00:00"Shawky, Alaa"https://zbmath.org/authors/?q=ai:shawky.alaa"Zydek, Dawid"https://zbmath.org/authors/?q=ai:zydek.dawid"Elhalwagy, Yehia Z."https://zbmath.org/authors/?q=ai:elhalwagy.yehia-z"Ordys, Andrzej"https://zbmath.org/authors/?q=ai:ordys.andrzej-wSummary: The problem of modeling and controlling the tip position of a one-link flexible manipulator is considered. The proposed model has been used to investigate the effect of the open-loop control torque profile, and the payload. The control strategy is based on the nonlinear state dependent Riccati equation (SDRE) design method in the context of application to robotics and manufacturing systems. In this paper, an experimental test-bed was developed to demonstrate the concept of end-point position feedback on a single-link elastic manipulator, and the control strategy for a single-link flexible manipulator. The controller is designed based on the nonlinear SDRE developed by the authors and applied to a flexible manipulator. The experimental results are compared with conventional PD controller strategy. The results reveal that the nonlinear SDRE controller is near optimal and robustly; and its performance is improved comparing to the PD control scheme.A note on the solutions of a second-order evolution inclusion in non separable Banach spaces.https://zbmath.org/1449.342092021-01-08T12:24:00+00:00"Cernea, Aurelian"https://zbmath.org/authors/?q=ai:cernea.aurelianSummary: We consider a Cauchy problem associated to a second-order evolution inclusion in non separable Banach spaces under Filippov type assumptions and we prove the existence of mild solutions.Growth of meromorphic solutions of some delay differential equations.https://zbmath.org/1449.343172021-01-08T12:24:00+00:00"Long, Fang"https://zbmath.org/authors/?q=ai:long.fang"Wang, Jun"https://zbmath.org/authors/?q=ai:wang.jun.2Summary: The growth property of meromorphic solutions of delay differential equations with rational coefficients is studied. The fact that every transcendental meromorphic solution is of order no less than one is proved under some conditions.Existence of solutions for a third-order two-point boundary value problem.https://zbmath.org/1449.340662021-01-08T12:24:00+00:00"He, Xingyue"https://zbmath.org/authors/?q=ai:he.xingyueSummary: Using the monotone iterative method, we demonstrate the existence of nontrivial solutions for a nonlinear third-order two-point boundary value problem by constructing two monotone iterative sequences.Analysis of stability of the equilibria for stochastic SIQR epidemic models with vaccination.https://zbmath.org/1449.341632021-01-08T12:24:00+00:00"Wang, Suxia"https://zbmath.org/authors/?q=ai:wang.suxia"Dong, Lingzhen"https://zbmath.org/authors/?q=ai:dong.lingzhen"Wang, Xiaoyan"https://zbmath.org/authors/?q=ai:wang.xiaoyanSummary: Based on the deterministic SIQR model with vaccination, we introduce the random perturbations, and establish the stochastic SIQR epidemic models with vaccination. By constructing suitable Lyapunov functions and using Itô's formula, it is proven that equilibria of the stochastic SIQR epidemic models under certain conditions are stochastically asymptotically stable. Further, we conjecture that the stability of equilibria is destroyed when the intensity of random perturbations is more larger. Finally, numerical simulations are presented to illustrate our conclusions and conjectures.Oscillation criteria for second order half-linear differential equations with sublinear neutral term.https://zbmath.org/1449.342252021-01-08T12:24:00+00:00"Li, Wenjuan"https://zbmath.org/authors/?q=ai:li.wenjuan"Li, Shuhai"https://zbmath.org/authors/?q=ai:li.shuhaiSummary: In this paper, we study the oscillation of the second order half-linear differential equations with sublinear neutral term. We establish some new oscillation criteria for the equations above, which generalize and improve the results of some literatures.Hidden attractors of a class of Van der Pol-Duffing oscillator.https://zbmath.org/1449.342002021-01-08T12:24:00+00:00"Nie, Jiasheng"https://zbmath.org/authors/?q=ai:nie.jiasheng"Li, Shumin"https://zbmath.org/authors/?q=ai:li.shuminSummary: In order to analyze and study the problem of hidden attractors in the Van der Pol-Duffing system, some new research results are obtained. The Routh-Hurwitz criterion, Hopf bifurcation theory, the harmonic linearization method and the analytical-numerical method are used to study the stability of the equilibrium points and the existence of hidden attractors in the system. There are hidden attractors in the system, and there are phenomena in which hidden attractors coexist with stable equilibrium points, stable periodic orbits, and chaotic attractors.Stability of the solution semigroup for neutral delay differential equations.https://zbmath.org/1449.342452021-01-08T12:24:00+00:00"Fabiano, Richard"https://zbmath.org/authors/?q=ai:fabiano.richard-h"Payne, Catherine"https://zbmath.org/authors/?q=ai:payne.catherineSummary: We derive a new condition for delay-independent stability of systems of linear neutral delay differential equations. The method applies ideas from linear semigroup theory, and involves renorming the underlying Hilbert space to obtain a dissipative inequality on the infinitesimal generator of the solution semigroup. The new stability condition is shown to either improve upon or be independent of existing stability conditions.Modified function projective synchronization by sliding mode control for a class of fractional-order hyper chaotic systems.https://zbmath.org/1449.341832021-01-08T12:24:00+00:00"Geng, Yanfeng"https://zbmath.org/authors/?q=ai:geng.yanfeng"Wang, Lizhi"https://zbmath.org/authors/?q=ai:wang.lizhi"Liu, Fang"https://zbmath.org/authors/?q=ai:liu.fang|liu.fang.1Summary: Modified function projective synchronization for a class of fractional-order hyper chaotic system with unknown parameters is investigated. By designing the compensator of the response system, the error system of modified function projective synchronization is obtained. Based on the theory of adaptive sliding mode control and the stability theory of fractional-order differential systems, an adaptive projective synchronization control scheme is designed. By selecting the adaptive sliding mode controller and adaptive control laws of the parameter, the modified function projective synchronization between the master system and the response system is implemented, and the uncertain parameters of the master system could be estimated. Finally, the fractional-order hyper chaotic Lü system that is taken as an example with simulations via Adams-Bashforth-Moultom algorithm is used to demonstrate the validity and feasibility of the proposed method.Noether theorem of second-order linear nonholonomic controllable mechanical systems in phase space on time scales.https://zbmath.org/1449.700232021-01-08T12:24:00+00:00"Ji, Xiaohui"https://zbmath.org/authors/?q=ai:ji.xiaohui"Zhu, Jianqing"https://zbmath.org/authors/?q=ai:zhu.jianqingSummary: This paper mainly investigated Noether symmetry and conserved quantity of second-order linear nonholonomic controllable mechanical systems in phase space on time scales. The Hamilton equation for second-order linear nonholonomic controllable mechanical systems on the time scale was established. The definitions and criteria of the generalized Noether quasi-symmetry were offered, and conserved quantities deduced from the generalized Noether quasi-symmetry were obtained. In the end of the paper, an example was given to illustrate the application of the results.A new \({\phi_0}\) concept of stability and stability criteria of nonlinear differential equations.https://zbmath.org/1449.341932021-01-08T12:24:00+00:00"Bao, Junyan"https://zbmath.org/authors/?q=ai:bao.junyan"Tan, Jun"https://zbmath.org/authors/?q=ai:tan.junSummary: In this paper, we present a new \(\tau\)-\({\phi_0}\)-stable concept and develop a new comparison principle. By using the cone-valued Lyapunov functions and the new comparison principle, we obtain the stability criterion.Sliding mode synchronization of a class of integer-order and fractional-order happiness models.https://zbmath.org/1449.341912021-01-08T12:24:00+00:00"Wang, Dongxiao"https://zbmath.org/authors/?q=ai:wang.dongxiao"Wang, Zhanwei"https://zbmath.org/authors/?q=ai:wang.zhanweiSummary: Based on Lyapunov stability theory and fractional order calculus, using the sliding mode control, synchronization problem of a class of integer order and fractional order 2-dimensional and 3-dimensional happiness models is studied. Both the fractional order and the integer order system could be synchronized. Numerical simulations results show that the approach is feasible and effective.Melnikov method of impulsive system and its application to chaos prediction.https://zbmath.org/1449.340492021-01-08T12:24:00+00:00"Niu, Yujun"https://zbmath.org/authors/?q=ai:niu.yujun"Hu, Shuangnian"https://zbmath.org/authors/?q=ai:hu.shuangnianSummary: A necessary condition of chaos appearance in impulsive systems is derived. The Duffing system with impulsive signals is employed to show the efficiency of this method.Existence of multiple positive solutions for a class of fractional differential equations with integral boundary value conditions.https://zbmath.org/1449.340862021-01-08T12:24:00+00:00"Sun, Rui"https://zbmath.org/authors/?q=ai:sun.rui"Zhou, Wenxue"https://zbmath.org/authors/?q=ai:zhou.wenxueSummary: In this paper, using the fixed point theorem of Guo-Krasnosellskii's cone expansion and cone compression, the existence of multiple positive solutions of fractional differential equations with integral boundary value conditions is studied in case of uniform fractional derivative.An accurate numerical method for solving the generalized time-fractional diffusion equation.https://zbmath.org/1449.354492021-01-08T12:24:00+00:00"Syam, Muhammed"https://zbmath.org/authors/?q=ai:syam.muhammed-i"Al-Subaihi, Ibrahim"https://zbmath.org/authors/?q=ai:al-subaihi.ibrahim-aSummary: In this paper, a formulation for the fractional Legendre functions is constructed to solve a class of time-fractional diffusion equation. The fractional derivative is described in the Caputo sense. The method is based on the collection Legendre. Analysis for the presented method is given and numerical results are presented.Sturm-Liouville operators with complex singular coefficients.https://zbmath.org/1449.470812021-01-08T12:24:00+00:00"Goryunov, A. S."https://zbmath.org/authors/?q=ai:goryunov.a-sSummary: We consider on a finite interval the Sturm-Liouville differential expression \(l(y)= -(py')' + qy + i((ry)' + ry')\) with coefficients satisfying the condition: \(q= Q'\), \(1/\sqrt{| p|}\), \(Q/\sqrt{| p|}\), \(r/\sqrt{| p|}\in L_2\), where the derivative of function \(Q\) is understood in the sense of distributions. The corresponding operators are correctly defined as quasi-differential. Conditions for the
minimal operator to be symmetric are obtained and all its self-adjoint, maximal dissipative and maximal accumulative extensions are described in terms of boundary conditions.Global stability of an SLI AIDS model with staged progression.https://zbmath.org/1449.341642021-01-08T12:24:00+00:00"Wang, Xiaohong"https://zbmath.org/authors/?q=ai:wang.xiaohong"Liu, Guirong"https://zbmath.org/authors/?q=ai:liu.gui-rong.1|liu.gui-rongSummary: The incubation period of AIDS is very long, and it usually goes through several latent stages, so it is not reasonable to think of it as a constant. This paper formulates a mathematical model with a realistic distribution, i.e., it divides the latent period into \(n\) stages. For a general \(n\)-stage stage-progression model with bilinear incidence, we analyze its dynamic behavior. Firstly we give the basic reproduction number. Then we obtain that, if the basic reproduction number is less than 1, the disease-free equilibrium is globally asymptotically stable and the disease always dies out; if the basic reproduction number is more than 1, the unique endemic equilibrium is globally asymptotically stable and the disease persists at the endemic equilibrium.The homotopy perturbation renormalization group method to solve the WKB problem with turn points.https://zbmath.org/1449.342062021-01-08T12:24:00+00:00"Lu, Yue"https://zbmath.org/authors/?q=ai:lu.yue"Zhao, Xutong"https://zbmath.org/authors/?q=ai:zhao.xutong"Liu, Mingji"https://zbmath.org/authors/?q=ai:liu.mingji(no abstract)The existence of positive periodic solutions for a kind of generalized Liénard equation.https://zbmath.org/1449.341182021-01-08T12:24:00+00:00"Cui, Xiaoxiao"https://zbmath.org/authors/?q=ai:cui.xiaoxiao"Cheng, Zhibo"https://zbmath.org/authors/?q=ai:cheng.zhibo"Yao, Shaowen"https://zbmath.org/authors/?q=ai:yao.shaowenSummary: In this paper, by application of the Manasevich-Mawhin continuation theorem, we prove the existence and asymptotic stability of positive periodic solutions for a class of generalized Liénard equation, where the non-autonomous function satisfies superlinear condition. At last, two examples and numerical solutions (phase portraits and time series portraits) are given to illustrate our conclusions.Mathematical modeling and computer simulation on hand, foot and mouth disease in children.https://zbmath.org/1449.920452021-01-08T12:24:00+00:00"Li, Hua"https://zbmath.org/authors/?q=ai:li.hua"Liu, Sanhong"https://zbmath.org/authors/?q=ai:liu.sanhong"Fang, Yile"https://zbmath.org/authors/?q=ai:fang.yile"Zhang, Xing'an"https://zbmath.org/authors/?q=ai:zhang.xinganSummary: Hand, foot and mouth disease (HFMD) is a contagious disease mainly caused by the enterovirus 71 (EV71) and coxsackievirus A 16 (CoxA16). The infectious of HFMD is mainly children. In this paper, we construct an SEIHRS model, simulate the HFMD data of infectious from 2012 to 2016, estimate the basic reproductive number each year from 2012 to 2016, predict the infectious number of HFMD in year 2017, compute the domain of vaccination rate on children and put forward the preventive measures and control strategy. The results of this paper can provide a theory basis for disease control and prevention of HFMD.Global and bifurcation analysis of an HIV pathogenesis model with saturated reverse function.https://zbmath.org/1449.341522021-01-08T12:24:00+00:00"Liu, Yongqi"https://zbmath.org/authors/?q=ai:liu.yongqi"Meng, Xiaoying"https://zbmath.org/authors/?q=ai:meng.xiaoyingSummary: In this paper, an HIV dynamics model with the proliferation of CD4 T cells is proposed. We consider nonnegativity, boundedness, global asymptotic stability of the solutions and bifurcation properties of the steady states. It is proved that the virus is cleared from the host under some conditions if the basic reproduction number \({R_0}\) is less than unity. Meanwhile, the model exhibits the phenomenon of backward bifurcation. We also obtain that one equilibrium is semi-stable by using center manifold theory. It is proved that the endemic equilibrium is globally asymptotically stable under some conditions if \({R_0}\) is greater than unity. It is also proved that the model undergoes Hopf bifurcation from the endemic equilibrium under some conditions. It is a novelty that the model exhibits two famous bifurcations, backward bifurcation and Hopf bifurcation. The model is extended to incorporate the specific Cytotoxic T Lymphocytes (CTLs) immune response. Stabilities of equilibria and Hopf bifurcation are considered accordingly. In addition, some numerical simulations for justifying the theoretical analysis results are also given in this paper.Runge-Kutta convolution quadrature methods for solving fractional differential equations with delay.https://zbmath.org/1449.651482021-01-08T12:24:00+00:00"Zhu, Rui"https://zbmath.org/authors/?q=ai:zhu.rui"Zhang, Gengen"https://zbmath.org/authors/?q=ai:zhang.gengen"Xiao, Feiyan"https://zbmath.org/authors/?q=ai:xiao.feiyan"Lan, Haifeng"https://zbmath.org/authors/?q=ai:lan.haifengSummary: In this paper, a strongly A-stable Runge-Kutta method is constructed to solve a class of nonlinear fractional differential equation with delay and Caputo fractional derivative. Stability and error analysis of the numerical algorithm are given. Numerical experiments demonstrate the validity of the proposed numerical algorithm and related theoretical results.Stability and Hopf bifurcation in a time-delayed predator-prey system with stage structures for both predator and prey.https://zbmath.org/1449.343062021-01-08T12:24:00+00:00"Zhu, Huan"https://zbmath.org/authors/?q=ai:zhu.huan"Gao, Debao"https://zbmath.org/authors/?q=ai:gao.debaoSummary: In nature, population growth often has a process of growing and development. At different age stages, both predator and prey will show different growth characteristics. In addition, the delay has a great influence on the topological structure of differential equation solutions. In many cases, the change of the delay will destroy the stability of the positive equilibrium point and produce Hopf bifurcation. Therefore, this paper takes the growth time from young predator to adult predator as the delay, constructs a time-delayed predator-prey system with stage structure for both predator and prey. Using the persistence theory for infinite-dimensional systems and Hurwitz criterion, permanent persistence condition of this system and the local stability condition of the system's coexistence equilibrium are given. Choosing the delay as a bifurcation parameter, we derive the existence of the Hopf bifurcation in this system. Then using normal form theory and center manifold arguments, we discuss the direction of the Hopf bifurcation and the stability of period solutions bifurcating from the Hopf bifurcations. Finally, the critical value \(\tau_{0 n}\) that causes Hopf bifurcation is obtained by choosing the qualified parameters satisfying the theorem conditions, and numerical results are presented to illustrate the theoretical conclusion.Boundary value problems for fractional differential inclusions with Hadamard type derivatives in Banach spaces.https://zbmath.org/1449.340172021-01-08T12:24:00+00:00"Graef, John R."https://zbmath.org/authors/?q=ai:graef.john-r"Guerraiche, Nassim"https://zbmath.org/authors/?q=ai:guerraiche.nassim"Hamani, Samira"https://zbmath.org/authors/?q=ai:hamani.samiraSummary: The authors establish sufficient conditions for the existence of solutions to boundary value problems for fractional differential inclusions involving the Hadamard type fractional derivative of order \(\alpha\in(1, 2]\) in Banach spaces. Their approach uses Mönch's fixed point theorem and the Kuratowski measure of noncompacteness.Stability of a stochastic SIRI model with saturated incidence and relapse.https://zbmath.org/1449.341542021-01-08T12:24:00+00:00"Mu, Yuguang"https://zbmath.org/authors/?q=ai:mu.yuguang"Xu, Rui"https://zbmath.org/authors/?q=ai:xu.rui|xu.rui.2|xu.rui.3|xu.rui.1Summary: In this paper, a stochastic SIRI epidemiological model with saturation incidence and relapse is investigated. Firstly, we show that there exists a unique global positive solution of the stochastic system. Then we discuss the stability of the disease-free equilibrium state and show the extinction of epidemics by using Lyapunov functions. Subsequently, a sufficient condition for persistence is obtained in the mean of the disease. Finally, some numerical simulations are carried out to confirm the analytical results.Heteroclinic and traveling wave solutions for an SIR epidemic model with nonlocal response.https://zbmath.org/1449.341252021-01-08T12:24:00+00:00"Wang, Zongyi"https://zbmath.org/authors/?q=ai:wang.zongyiSummary: The existence of positive heteroclinic solutions is proved for a class of SIR epidemic model with nonlocal interaction and non-monotone property. Applying the theory of Fredholm operator decomposition and nonlinear perturbation, we study a connection between traveling wave solutions for the reaction-diffusion system and heteroclinic solutions of the associated differential equations. Existence and dynamics of wavefront profile are obtained as a consequence.A note on stage structure predator-prey model with prey refuge.https://zbmath.org/1449.341682021-01-08T12:24:00+00:00"Yang, Wensheng"https://zbmath.org/authors/?q=ai:yang.wensheng"Zheng, Yanhong"https://zbmath.org/authors/?q=ai:zheng.yanhongSummary: A stage structure predator-prey model with prey refuge and Beddington-DeAngelis functional response is considered in this work. A sufficient condition which ensures persistence of the system is obtained by using a comparison method. Furthermore, sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the system are derived by constructing suitable Lyapunov function.Matrix representations of Sturm-Liouville problems with distribution potentials on time scales.https://zbmath.org/1449.343212021-01-08T12:24:00+00:00"Liu, Nana"https://zbmath.org/authors/?q=ai:liu.nana"Ao, Jijun"https://zbmath.org/authors/?q=ai:ao.jijunSummary: The matrix representations of second order Sturm-Liouville problems with distribution potentials on bounded time scales are investigated. The corresponding equivalences between Sturm-Liouville problems with distribution potentials on time scales and a certain kind of matrix eigenvalue problems are obtained. Both of the separated and coupled self-adjoint boundary conditions are considered to obtain the main results.Exponential stability of ISFDEs with Markov switching.https://zbmath.org/1449.342802021-01-08T12:24:00+00:00"Yang, Shujie"https://zbmath.org/authors/?q=ai:yang.shujie"Mao, Kai"https://zbmath.org/authors/?q=ai:mao.kai"Chen, Han"https://zbmath.org/authors/?q=ai:chen.hanSummary: A class of impulsive stochastic functional differential equations with Markov switching is investigated. Based on Itô formula and by introducing a class of special Lyapunov functions, Lyapunov-Razumikhin method and combing with inequality technique, some novel sufficient conditions are derived to ensure the \(p\)th moment exponential stability of the trivial solution. A numerical example is given to illustrate the effectiveness and lower conservation.Sufficient conditions of oscillation for certain second-order functional dynamic equations on time scales.https://zbmath.org/1449.343202021-01-08T12:24:00+00:00"Li, Jimeng"https://zbmath.org/authors/?q=ai:li.jimeng"Yang, Jiashan"https://zbmath.org/authors/?q=ai:yang.jiashanSummary: This paper is concerned with the oscillatory behavior of the following second-order Emden-Fowler variable delay neutral functional dynamic equation
\[\{a (t)\varphi ([x (t) + p(t)g (x (\tau (t)))]^\Delta)\}^\Delta + {q_1} (t){f_1} ({\varphi_1} (x ({\delta_1} (t)))) + {q_2} (t){f_2} ({\varphi_2} (x ({\delta_2} (t)))) = 0\]
on a time scale \(\mathbb{T}\), where \(\varphi (u) = |u|^{\alpha-1}u (\alpha > 0)\), \({\varphi_1} (u) = |u|^{\beta-1}u (\beta > 0)\) and \({\varphi_2} (u) = |u|^{\gamma-1}u (\gamma > 0)\). By using the time scales theory and the Riccati transformation as well as the inequality technique, we establish some new sufficient conditions of oscillation for the equation. Our results deal with some cases not covered by the existing results in the literature. Finally, some interesting examples are given to illustrate the versatility of our results.Fuzzy logic embedding of fractional order sliding mode and state feedback controllers for synchronization of uncertain fractional chaotic systems.https://zbmath.org/1449.931642021-01-08T12:24:00+00:00"Pahnehkolaei, Seyed Mehdi Abedi"https://zbmath.org/authors/?q=ai:pahnehkolaei.seyed-mehdi-abedi"Alfi, Alireza"https://zbmath.org/authors/?q=ai:alfi.alireza"Machado, J. A. Tenreiro"https://zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiroSummary: This paper studies the synchronization of a class of uncertain fractional order (FO) chaotic systems that is applicable in secure communication. A novel hybrid FO controller, based on sliding mode and state feedback techniques combined with fuzzy logic, is developed. The algorithm, derived via the fractional Lyapunov theory, guarantees the stability of the overall system and the convergence of the synchronization errors toward a small residual set. Simulations demonstrate the capability of the proposed control algorithm in secure communications, not only in terms of speed of response, but also by reducing the chattering phenomenon.Lower bounds for unbounded operators and semigroups.https://zbmath.org/1449.470232021-01-08T12:24:00+00:00"Batty, Charles J. K."https://zbmath.org/authors/?q=ai:batty.charles-j-k"Geyer, Felix"https://zbmath.org/authors/?q=ai:geyer.felixSummary: Let \(A\) be an unbounded operator on a Banach space \(X\). It is sometimes useful to improve the operator \(A\) by extending it to an operator \(B\) on a larger Banach space \(Y\) with smaller spectrum. It would be preferable to do this with some estimates for the resolvent of \(B\), and also to extend bounded operators related to \(A\), for example a semigroup generated by \(A\). When \(X\) is a Hilbert space, one may also want \(Y\) to be Hilbert space. Results of this type for bounded operators have been given by Arens, Read, Müller and Badea, and we give some extensions of their results to unbounded operators and we raise some open questions. A related problem is to improve properties of a \(C_0\)-semigroup satisfying lower bounds by extending it to a \(C_0\)-group on a larger space or by finding left-inverses. Results of this type for Hilbert spaces have been obtained by \textit{J. C. Louis} and \textit{D. Wexler} [J. Differ. Equations 49, 258--269 (1983; Zbl 0477.49022)], and by \textit{H. Zwart} [J. Evol. Equ. 13, No. 2, 335--342 (2013; Zbl 1288.47040)], and we give some additional results.On the existence of multiple solutions for a three-point nonlinear boundary value problem of \(p\)-Laplacian type.https://zbmath.org/1449.352372021-01-08T12:24:00+00:00"Abaspour, S."https://zbmath.org/authors/?q=ai:abaspour.s"Khademloo, S."https://zbmath.org/authors/?q=ai:khademloo.somayeh|khademloo.somaye"Rasouli, S. H."https://zbmath.org/authors/?q=ai:rasouli.sayyed-hahsem|rasouli.sayyed-hasem|rasouli.sayyed-hashem|rasouli.sayyyed-h|rasouli.seyed-h|rasouli.sayed-hashemSummary: Using variational methods, we establish the existence of at least three solutions for a three-point nonlinear boundary value problem. Our technical approach is based on critical point theory.Asymptotic properties of a stochastic mutualism model with a saturation term and Lévy jumps.https://zbmath.org/1449.341782021-01-08T12:24:00+00:00"Zhao, Xiaodan"https://zbmath.org/authors/?q=ai:zhao.xiaodan"Zhao, Aimin"https://zbmath.org/authors/?q=ai:zhao.aimin"Liu, Guirong"https://zbmath.org/authors/?q=ai:liu.gui-rong.1Summary: This paper investigates a stochastic mutualism model with a saturation term and Lévy jumps. It chooses a suitable Lyapunov function to demonstrate the existence and uniqueness of global positive solutions. Using Itô formula, sufficient conditions for the extinction of each species are established. The results in this paper extend results of the existing literature. Finally, some numerical simulations are given to illustrate the theoretical results.On the theorem for a generalized concave operator in differential equations involving a fractional order and impulsive boundary conditions.https://zbmath.org/1449.341032021-01-08T12:24:00+00:00"Zheng, Fengxia"https://zbmath.org/authors/?q=ai:zheng.fengxia"Xiao, Weizhong"https://zbmath.org/authors/?q=ai:xiao.weizhong"Xie, Maosen"https://zbmath.org/authors/?q=ai:xie.maosenSummary: By using a fixed point theorem for a generalized concave operator, a new criterion for the existence and uniqueness of solutions involving a fractional order and impulsive boundary conditions is established. Finally, an example is given to illustrate the main results.Stability analysis of quaternion-valued neutral neural networks with time delay.https://zbmath.org/1449.342482021-01-08T12:24:00+00:00"Shu, Jinlong"https://zbmath.org/authors/?q=ai:shu.jinlong"Xiong, Lianglin"https://zbmath.org/authors/?q=ai:xiong.lianglin"Wu, Tao"https://zbmath.org/authors/?q=ai:wu.tao"Zheng, Yingli"https://zbmath.org/authors/?q=ai:zheng.yingliSummary: This paper discusses the global-\(\mu\) stability problem of quaternion-valued neutral neural networks (QVNTNN) with time delay. Firstly, in order to reduce the computational complexity caused by the non-commutability of quaternion matrix multiplication, the QVNTNN system is converted into two complex-valued systems by using the complex decomposition method. Secondly, the existence and uniqueness of the solution of the QVNTNN system with time delay is proved by the theory of homeomorphism mapping. Then, a new Lyapunov functional is constructed, and the inequality is scaled by the inequality technique and the free weight matrix. We obtain global-\(\mu\) stability criterion for the QVNTNN system with time-delay. Finally, a numerical example is given to illustrate the method.Interval observers for linear functions of states and unknown inputs of nonlinear fractional-order systems with time delays.https://zbmath.org/1449.342702021-01-08T12:24:00+00:00"Huong, Dinh Cong"https://zbmath.org/authors/?q=ai:huong.dinh-cong"Yen, Dao Thi Hai"https://zbmath.org/authors/?q=ai:yen.dao-thi-haiSummary: The main objective of this paper is to design interval observers using the output signals and the delayed output signals to estimate the state vector and unknown input vector of nonlinear fractional-order systems with time delay. We first propose and design two new functional observers, such that they bound the set of all admissible values of a linear function of the state vector and the unknown input vector at each instant of time. We then derive conditions for the existence of interval functional observers and provide an effective design algorithm for computing unknown observer matrices. Finally, an example is given to show the effectiveness of the proposed design approach.Global stability of an epidemic hemorrhagic fever model.https://zbmath.org/1449.341472021-01-08T12:24:00+00:00"Li, Feng"https://zbmath.org/authors/?q=ai:li.feng.2|li.feng.1"Liu, Junli"https://zbmath.org/authors/?q=ai:liu.junliSummary: In this paper, an epidemic hemorrhagic fever model is formulated and investigated. The basic reproduction number \({R_0}\) for the model is identified. Local and global stability of the equilibria are discussed by using the Routh-Hurwitz criterion, Lyapunov function and LaSalle invariant set principle, and the theory of cooperate systems. The results show that if \({R_0} < 1\), there is only a disease-free equilibrium, which is globally asymptotically stable; while if \({R_0} > 1\), the disease-free equilibrium is unstable, there also exists a unique endemic equilibrium, which is globally asymptotically stable.The method of coupled upper and lower solutions for boundary value problems of fractional functional differential equations.https://zbmath.org/1449.342172021-01-08T12:24:00+00:00"Jian, Xingyue"https://zbmath.org/authors/?q=ai:jian.xingyue"Liu, Xiping"https://zbmath.org/authors/?q=ai:liu.xiping"Jia, Mei"https://zbmath.org/authors/?q=ai:jia.mei"Luo, Zeyu"https://zbmath.org/authors/?q=ai:luo.zeyuSummary: In this paper, a class of boundary value problems of fractional functional differential equations with time delays is studied. Firstly, the problems studied in this paper are transformed into integral equations. The existence and uniqueness theorems of solutions for boundary value problems are proved by using nonlinear analysis theory. The monotone iterative sequences for solving the solutions of boundary value problems are generated and the error estimates are given. Secondly, by using the generalized monotone iteration technique and the coupled upper and lower solutions method, sufficient conditions for the existence and uniqueness of solutions of boundary value problems are obtained, and the range of solutions is determined. Finally, some examples are given to illustrate the wide applicability of our main results.Analysis of a toxic producing phytoplankton-zooplankton interaction with delay.https://zbmath.org/1449.342922021-01-08T12:24:00+00:00"Mehbuba, Rehim"https://zbmath.org/authors/?q=ai:mehbuba.rehim"Li, Xiaona"https://zbmath.org/authors/?q=ai:li.xiaonaSummary: The present paper aims to investigate a toxic producing phytoplankton-zooplankton (a prey-predator interaction) system with delay. The delay in the zooplankton predation is considered and its effect on the overall dynamics of phytoplankton-zooplankton interaction is studied. Firstly, the nonnegativity and boundedness of solutions are given. Then the existence and stability of the equilibrium are investigated. Furthermore, the occurrence of local Hopf bifurcation is established as the delay crosses a threshold value. The system is modeled via a Tissiet type functional response. Analytical methods and numerical simulations are used to obtain information about the qualitative behavior of the models.Some problems of linear differential equations on abstract spaces and unbounded perturbations of linear operator semigroup.https://zbmath.org/1449.342082021-01-08T12:24:00+00:00"Xu, Genqi"https://zbmath.org/authors/?q=ai:xu.gen-qiSummary: This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the \({L^p}\) linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedback operator. After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations. As a principle, we propose a concept of admissible state feedback operator for system \( (A, B)\). Finally we give an existence result of admissible state feedback operators, including semigroup generation and the equivalent conditions of admissibility of state feedback operators, for an \({L^p}\) well-posed system.Explicit iterative sequences of positive solutions for a class of fractional differential equations on an infinite interval.https://zbmath.org/1449.340942021-01-08T12:24:00+00:00"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Li, Yaohong"https://zbmath.org/authors/?q=ai:li.yaohongSummary: By applying the monotone iterative method, this study develops two explicit monotone iterative sequences for approximating the minimal and maximal positive solutions. At the same time, by applying the Banach fixed-point theory, an explicit iterative sequence and error estimate for approximating the unique positive solution are obtained. Some examples are given to illustrate the application of the results.Noether symmetry and conserved quantity of nonholonomic controllable mechanical systems in phase space.https://zbmath.org/1449.700222021-01-08T12:24:00+00:00"Ji, Xiaohui"https://zbmath.org/authors/?q=ai:ji.xiaohui"Zhu, Jianqing"https://zbmath.org/authors/?q=ai:zhu.jianqingSummary: This paper mainly investigated Noether symmetry and conserved quantity of nonholonomic controllable mechanical systems in phase space on time scales. The Hamilton equation for nonholonomic controllable mechanical systems on the time scale was established. We offered the definitions and criteria of the generalized Noether quasi-symmetry, and conserved quantities deduced from the generalized Noether quasi-symmetry were obtained. In the end of the paper, an example was given to illustrate the application of the results.On solvability of inhomogeneous boundary-value problems in Sobolev-Slobodetskiy spaces.https://zbmath.org/1449.471332021-01-08T12:24:00+00:00"Mikhailets, V. A."https://zbmath.org/authors/?q=ai:mikhailets.vladimir-a|mikhailets.volodymyr-a"Skorobohach, T. B."https://zbmath.org/authors/?q=ai:skorobohach.t-bSummary: We investigate the most general class of Fredholm one-dimensional boundary-value problems in the Sobolev-Slobodetskiy spaces. Boundary conditions of these problems may contain a derivative of the whole or fractional order. It is established that each of these boundary-value problems corresponds to a certain rectangular numerical characteristic matrix with kernel and cokernel having the same dimension as the kernel and cokernel of the boundary-value problem. Sufficient conditions for the sequence of the characteristic matrices of a specified boundary-value problems to converge are found.Optimal control applied to investigate the effect of glucose tolerance test of diabetes mellitus in human body.https://zbmath.org/1449.920152021-01-08T12:24:00+00:00"Sardar, Anadi Kumar"https://zbmath.org/authors/?q=ai:sardar.anadi-kumar"Sahani, Santosh Kumar"https://zbmath.org/authors/?q=ai:sahani.santosh-kumar"Islam, Md. Sirajul"https://zbmath.org/authors/?q=ai:islam.md-sirajul"Biswas, Md. Haider Ali"https://zbmath.org/authors/?q=ai:biswas.md-haider-aliSummary: Diabetes mellitus is a metabolic disorder in which the body is unable to respond properly to the consumption of carbohydrates, sugars and starches leading to increased levels of glucose in the blood and urine. In glucose tolerance test (GTT), a patient fasts overnight; the patient is then given a large dose of glucose and the concentration of glucose in the body is monitored for the next three to five hours. In this paper, a mathematical model of optimal control of glucose tolerance test has been discussed. In this control model of GTT, glucose and insulin concentrations have been described for time durations of test. The control model has been analyzed and investigated through analytically and numerically. It is found that the perfect time for GTT is about 4.8 hours and glucose concentration decreases steadily almost linearly while insulin level remains positive.A piecewise spectral-collocation method for solving fractional Riccati differential equation in large domains.https://zbmath.org/1449.651642021-01-08T12:24:00+00:00"Azin, H."https://zbmath.org/authors/?q=ai:azin.h"Mohammadi, F."https://zbmath.org/authors/?q=ai:mohammadi.farah|mohammadi.fakhrodin|mohammadi.fatemeh|mohammadi.farin|mohammadi.faezeh|mohammadi.farid"Machado, J. A. Tenreiro"https://zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiroSummary: This paper addresses the approximate solution of the fractional Riccati differential equation (FRDE) in large domains. First, the solution interval is divided into a finite number of subintervals. Then, the Legendre-Gauss-Radau points along with the Lagrange interpolation method are employed to approximate the FRDE solution in each subinterval. The method has the advantage of providing the approximate solutions in large intervals. Additionally, the convergence analysis of the numerical algorithm is also provided. Three illustrative examples are given to illustrate the efficiency and applicability of the proposed method.Stability analysis of the zero solution for time-periodic linear system with small perturbations.https://zbmath.org/1449.341952021-01-08T12:24:00+00:00"Li, Zhuo"https://zbmath.org/authors/?q=ai:li.zhuo"Li, Xia"https://zbmath.org/authors/?q=ai:li.xiaSummary: In this paper, we study the stability of the zero solution for time-periodic linear perturbed systems. After investigating the properties of periodic linear systems, we apply the method of finding nonsingular and differentiable periodic matrices to the study of the properties of linear systems with constant coefficients. Then we extend the zero solution's stability of the constant coefficient linear system with small perturbation to the periodic linear system with small perturbations.A periodic solution of the coupled matrix Riccati differential equations.https://zbmath.org/1449.340652021-01-08T12:24:00+00:00"Goodarzi, Zahra"https://zbmath.org/authors/?q=ai:goodarzi.zahra"Razani, Abdolrahman"https://zbmath.org/authors/?q=ai:razani.abdolrahman"Mokhtarzadeh, M. R."https://zbmath.org/authors/?q=ai:mokhtarzadeh.mohammad-rezaSummary: Here, we generalize the martix Riccati differential equation to the coupled matrix Riccati differential equation. Using Schauder's fixed point theorem, the existence of at least one periodic solution of the coupled matrix Riccati equation with \(n\times n\) matrix coefficients is proved. Finally, two numerical examples are presented.Dynamical properties of a discontinuous models with signal molecules regulation.https://zbmath.org/1449.341622021-01-08T12:24:00+00:00"Wang, Ru"https://zbmath.org/authors/?q=ai:wang.ru"Zhang, Zhonghua"https://zbmath.org/authors/?q=ai:zhang.zhonghua"Liu, Yeling"https://zbmath.org/authors/?q=ai:liu.yelingSummary: In this paper, the concentration of signal molecules is used as an indicator to establish a discontinuous model that describes the regulation of signal molecules concentration by the quorum sensing mechanism of pathogens. The existence of the sliding module region, the existence and stability of the true (false) equilibrium point and the pseudo equilibrium point are discussed. In particular, the existence of the crossing limit cycles is proved. Finally, the Matlab software is used to carry out numerical simulations to support the theoretical results.A revisit on multiple-pulse homoclinic solutions in a generalized Gierer-Meinhardt equation.https://zbmath.org/1449.341262021-01-08T12:24:00+00:00"Zhu, Kun"https://zbmath.org/authors/?q=ai:zhu.kun"Shen, Jianhe"https://zbmath.org/authors/?q=ai:shen.jianheSummary: In a previous paper, the authors studied the existence and stability of multiple-pulse homoclinic solutions in a generalized Gierer-Meinhardt equation. However, a general integral measuring the distance of the stable and unstable manifolds of the critical manifold of the layer system, i.e., the Melnikov integral, was not computed explicitly. So we have two aims in this manuscript. Firstly, we give an elementary method to solve a second-order nonlinear conservative system and hence obtain the explicit representation of the homoclinic orbit. Secondly, we substitute the explicit representation of the homoclinic orbit into the Melnikov integral. By computing such a general integral, we obtain a more explicitly parametric condition on the existence of multiple-pulse homoclinic solutions in such a generalized Gierer-Meinhardt equation.On the existence of positive solutions to a type of Riemann-Liouville fractional differential equations.https://zbmath.org/1449.340392021-01-08T12:24:00+00:00"Xue, Yimin"https://zbmath.org/authors/?q=ai:xue.yimin"Dai, Zhenxiang"https://zbmath.org/authors/?q=ai:dai.zhenxiang"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie|liu.jie.7|liu.jie.2|liu.jie.3|liu.jie.1|liu.jie.5|liu.jie.4Summary: Using the properties of Green's function and Guo-Krasnosel'skii's fixed point theorem, the boundary value problem for the existence of positive solutions to a type of Riemann-Liouville fractional differential equations is studied: \[\begin{cases}{D^\alpha}u (t) + f (t,u (t)) = 0,\; (0 < t < 1), \\ u (0) = {D^\beta}u (0) = {D^\beta}u (1) = 0, \end{cases}\] where \(2 <\alpha \le 3\), \(1 < \beta \le 2\), \(1 + \beta \le \alpha\), \(f\in C ([0,1] \times [0,\infty), [0,\infty))\), \({D^\alpha}\) and \({D^\beta}\) are the standard Riemann-Liouville fractional derivative of order \(\alpha\) and \(\beta\), respectively. Two sufficient conditions for the existence of positive solutions are obtained, and one example is given to illustrate the applicability of the main result.Stability of age-structured with proportion of infected groups or enter the latent epidemiological model with varying population size.https://zbmath.org/1449.342952021-01-08T12:24:00+00:00"Wang, Gaixia"https://zbmath.org/authors/?q=ai:wang.gaixia"Liu, Jixuan"https://zbmath.org/authors/?q=ai:liu.jixuan"Li, Xuezhi"https://zbmath.org/authors/?q=ai:li.xuezhiSummary: This paper studies an age-structure infectious disease model in which the proportion of population change enters the latent or infected population. The expression of threshold parameter related to population growth index is obtained. According to this parameter, the existence and local asymptotic stability conditions of disease-free equilibrium and endemic equilibrium are discussed. These conditions have important theoretical and practical significance to control the spread of disease.Dynamical properties of a delayed epidemic model with vaccination and saturation incidence.https://zbmath.org/1449.343032021-01-08T12:24:00+00:00"Zhang, Xinzhe"https://zbmath.org/authors/?q=ai:zhang.xinzhe"He, Guofeng"https://zbmath.org/authors/?q=ai:he.guofeng"Huang, Gang"https://zbmath.org/authors/?q=ai:huang.gangSummary: In this paper, we propose and study a delayed SVEIR epidemic model with vaccination and saturation incidence. The existence and local stability of equilibria are addressed. By using Lyapunov functionals and the Lyapunov-LaSalle invariance principle, it is shown that if the basic reproduction number is less than or equal to one, the disease-free equilibrium is globally asymptotically stable and the disease will disappear; and if the basic reproduction number is greater than one, the endemic equilibrium is globally asymptotically stable and the disease will persist. Some numerical simulations are performed to illustrate the analytic results.Global exponential periodicity of complex-valued neural networks with discontinuous activation functions.https://zbmath.org/1449.342402021-01-08T12:24:00+00:00"Zou, Yao"https://zbmath.org/authors/?q=ai:zou.yao"Zeng, Chunna"https://zbmath.org/authors/?q=ai:zeng.chunna"Hu, Jin"https://zbmath.org/authors/?q=ai:hu.jinSummary: In this paper, we investigate a type of complex-valued neural networks with discontinuous activation functions. By using Filippov differential inclusion theory, Leray-Schauder alternative theorem and Lyapunov function, we obtain the sufficient conditions for the global exponential periodicity of the neural network. The simulation shows the effectiveness of the results.Operational matrices of Chebyshev cardinal functions and their application for solving delay differential equations arising in electrodynamics with error estimation.https://zbmath.org/1449.651412021-01-08T12:24:00+00:00"Heydari, M."https://zbmath.org/authors/?q=ai:heydari.mohammad-taghi|heydari.mehdi|heydari.masoud|heydari.m-m|heydari.mohammadhossein|heydari.mahdi|heydari.maysam|heydari.majeed|heydari.mohammad-hossien|heydari.maryam|heydari.mojgan"Loghmani, G. B."https://zbmath.org/authors/?q=ai:loghmani.ghasem-barid|loghmani.g-barid"Hosseini, S. M."https://zbmath.org/authors/?q=ai:hosseini.seyed-mahmood|hosseini.s-majid|hosseini.seyed-mohammad|hosseini.syed-mohammad-mahdi|hosseini.said-mohammad-mehdi|hosseini.seyedeh-marzieh|hosseini.seyed-morteza|hoseini.s-mohsen|hosseini.seyed-mahmoud|hosseini.seyed-mohammad-hassan|hosseini.seyedeh-masoumehSummary: In this paper, a new and effective direct method to determine the numerical solution of pantograph equation, pantograph equation with neutral term and Multiple-delay Volterra integral equation with large domain is proposed. The pantograph equation is a delay differential equation which arises in quite different fields of pure and applied mathematics, such as number theory, dynamical systems, probability, mechanics and electrodynamics. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration, product and delay of the Chebyshev cardinal functions are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a pantograph equation can be transformed to a system of algebraic equations. An efficient error estimation for the Chebyshev cardinal method is also introduced. Some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results.Existence of solutions for impulsive differential inclusions with upper and lower solutions in the reverse order.https://zbmath.org/1449.340582021-01-08T12:24:00+00:00"Luo, Yan"https://zbmath.org/authors/?q=ai:luo.yan"Xie, Wenzhe"https://zbmath.org/authors/?q=ai:xie.wenzheSummary: In this paper, we discuss the existence of solutions for nonlinear boundary problems of first-order impulsive differential inclusions. In the presence of a lower solution \(\alpha\) and an upper solution \(\beta\) in the reverse order \(\beta \leq \alpha\), we establish existence results by using Martelli's fixed point theorem.Existence of positive periodic solutions for a class of first-order functional differential equations.https://zbmath.org/1449.342392021-01-08T12:24:00+00:00"Zhu, Yan"https://zbmath.org/authors/?q=ai:zhu.yanSummary: In this paper, we use the global bifurcation theorem to study the existence of positive \(T\)-periodic solutions of the first-order functional differential equation
\[u'(t) - a(t) u(t) + \lambda g(t) f(u(t-\tau (t))) = 0,\, t \in \mathbb{R},\]
where \(\lambda > 0\) is a parameter, \(a \in C(\mathbb{R}, [0,\infty))\), \(g \in C(\mathbb{R}, [0, \infty))\), \(a\not\equiv{0}\), \(g\not\equiv{0}\), \(\tau \in C (\mathbb{R}, \mathbb{R})\), \(a\), \(g\), \(\tau \) are \(T\)-periodic functions, \(f \in C([0,\infty),[0,\infty))\). We construct the global structure of the set of positive \(T\)-periodic solutions of the equation and establish some existence results of the positive \(T\)-periodic solutions of this equation.Existence and uniqueness of positive solutions for a class of fractional impulsive differential equations with boundary value problems.https://zbmath.org/1449.340982021-01-08T12:24:00+00:00"Zheng, Fengxia"https://zbmath.org/authors/?q=ai:zheng.fengxia"Gu, Chuanyun"https://zbmath.org/authors/?q=ai:gu.chuanyunSummary: By using the fixed point theorem for mixed monotone operator, a new criterion for the existence and uniqueness of positive solution of boundary value problems of a class of fractional impulsive differential equations
\[\begin{cases}
{}^CD_{0^+}^q u(t) = f(t, u(t), u(t)), t \in J' = J\backslash \{t_1,t_2,\cdots, t_m\}, J = [0,1],\\
\Delta u(t_k) = I_k(u(t_k), u(t_k)), \Delta u'(t_k) = J_k(u(t_k), u(t_k)), k = 1, 2, \cdots, m,\\
au(0) - bu(1) = 0,\, au'(0) - bu'(1) = 0
\end{cases}\]
is established, where \(1 < q < 2\), \({}^CD_{0^+}^q\) is the Caputo fractional derivative.Zeros of abelian integral for a kind of Hamiltonian systems.https://zbmath.org/1449.341112021-01-08T12:24:00+00:00"Yang, Jihua"https://zbmath.org/authors/?q=ai:yang.jihua"Zhang, Erli"https://zbmath.org/authors/?q=ai:zhang.erliSummary: In this paper, we obtain an upper bound for the number of zeros of the Abelian integral for a class of Hamiltonian systems. The abelian integral has \(k + 2\) generators which satisfy two different Picard-Fuchs equations. Finally, we present two examples to illustrate an application of the theoretical result.Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane-Emden-Fowler type.https://zbmath.org/1449.651712021-01-08T12:24:00+00:00"Turkyilmazoglu, M."https://zbmath.org/authors/?q=ai:turkyilmazoglu.mustafaSummary: The particular motivation of this work is to develop a computational method to calculate exact and analytic approximate solutions to singular strongly nonlinear initial or boundary value problems of Lane-Emden-Fowler type which model many phenomena in mathematical physics and astrophysics. A powerful algorithm is proposed based on the series representation of the solution via suitable base functions. The utilization of such functions converts the solution of a given nonlinear differential equation to the solution of algebraic equations. Error analysis and convergence of the method is presented. Comparisons with the other methods reveal validity, applicability and great potential of the method. Several physical problems are treated to illustrative the good performance and high accuracy of the technique.A no-arbitrage theorem for uncertain stock model.https://zbmath.org/1449.911492021-01-08T12:24:00+00:00"Yao, Kai"https://zbmath.org/authors/?q=ai:yao.kaiSummary: Stock model is used to describe the evolution of stock price in the form of differential equations. In early years, the stock price was assumed to follow a stochastic differential equation driven by a Brownian motion, and some famous models such as Black-Scholes stock model and Black-Karasinski stock model were widely used. This paper assumes that the stock price follows an uncertain differential equation driven by Liu process rather than Brownian motion, and accepts Liu's stock model to simulate the uncertain market. Then this paper proves a no-arbitrage determinant theorem for Liu's stock model and presents a sufficient and necessary condition for no-arbitrage. Finally, some examples are given to illustrate the usefulness of the no-arbitrage determinant theorem.Measure functional differential equations with infinite delay: differentiability of solutions with respect to parameters.https://zbmath.org/1449.342142021-01-08T12:24:00+00:00"Li, Baolin"https://zbmath.org/authors/?q=ai:li.baolin"Xu, Zhiyan"https://zbmath.org/authors/?q=ai:xu.zhiyanSummary: In this paper, we establish the differentiability of solutions with respect to parameters for measure functional differential equations with infinite delay by using the differentiability of solutions with respect to parameters for generalized ordinary differential equations.Global asymptotic robust stability and partial robust stability for a Lotka-Volterra model with infinite delays.https://zbmath.org/1449.342522021-01-08T12:24:00+00:00"Zhong, Lingli"https://zbmath.org/authors/?q=ai:zhong.lingli"Li, Shuyong"https://zbmath.org/authors/?q=ai:li.shuyongSummary: This paper is devoted to the investigation of the robust stability and partial robust stability of a class of Lotka-Volterra models with infinite delays and uncertain parameters. By constructing Lyapunov functional, using Lyapunov-LaSalle type theorem and stability theory of dynamical systems on the interval, some sufficient conditions for determining the global asymptotic robust stability and partial robust stability of the system are obtained. Finally, a numerical example is given to verify the validity of the obtained results.Periodic solutions and stability of nonlinear differential system with delays.https://zbmath.org/1449.342342021-01-08T12:24:00+00:00"Huang, Minghui"https://zbmath.org/authors/?q=ai:huang.minghui"Zhao, Guorui"https://zbmath.org/authors/?q=ai:zhao.guorui"Jin, Chuhua"https://zbmath.org/authors/?q=ai:jin.chuhuaSummary: By using Krasnoselskii's fixed point theorem, the existence of periodic solutions for nonlinear neutral differential system with delays are given. Some sufficient conditions for the uniqueness of periodic solutions and stability of zero solutions are obtained by the using contraction mapping principle. The conclusions generalize corresponding results in the literature.Second-order multiple-point boundary value problems with upper and lower solutions in the reverse order.https://zbmath.org/1449.340602021-01-08T12:24:00+00:00"Li, Haiyan"https://zbmath.org/authors/?q=ai:li.haiyan"Wang, Min"https://zbmath.org/authors/?q=ai:wang.min.2Summary: In this paper, we discuss second-order multiple-point boundary value problems with upper and lower solutions in the reverse order. We get the existence of the maximal solution and the minimal solution by introducing an increasing operator and giving a monotone iterative sequence. The uniqueness is proved by using the contraction mapping principle.Positive ground-state solution to singular Emden-Fowler equation with Dirichlet boundary value condition.https://zbmath.org/1449.340882021-01-08T12:24:00+00:00"Wang, Jia"https://zbmath.org/authors/?q=ai:wang.jia"Gao, Guifeng"https://zbmath.org/authors/?q=ai:gao.guifeng"Wang, Xinke"https://zbmath.org/authors/?q=ai:wang.xinke"Mao, Anmin"https://zbmath.org/authors/?q=ai:mao.anminSummary: In this paper, we consider the singular Emden-Fowler equation with Dirichlet boundary value condition. By using the Nehari method, we obtain the existence of positive ground-state solution and our work improves and extends some existing results.Forwards and pullback behaviour of a non-autonomous predator-prey system with the Beddington-DeAngelis functional response.https://zbmath.org/1449.341562021-01-08T12:24:00+00:00"Shen, Yixin"https://zbmath.org/authors/?q=ai:shen.yixin"Pu, Zhilin"https://zbmath.org/authors/?q=ai:pu.zhilin"Hu, Huashu"https://zbmath.org/authors/?q=ai:hu.huashuSummary: For the non-autonomous predator-prey system with the Beddington-DeAngelis response, in this paper, we use the sub-super solution, logistic function and comparison principle to explore the asymptotic behaviour of the solution. The time asymptotic behaviour as \(t \to \infty\) is considered, including forward permanence and predator extinction. Then, we explore the asymptotic behaviour as \(s \to -\infty\), including the existence of the pullback attractor and pullback permanence.Synchronization analysis of complex dynamic networks with aperiodic quantized intermittent control.https://zbmath.org/1449.342602021-01-08T12:24:00+00:00"Wu, Dongmei"https://zbmath.org/authors/?q=ai:wu.dongmei"Feng, Jianwen"https://zbmath.org/authors/?q=ai:feng.jianwen"Wang, Jinyi"https://zbmath.org/authors/?q=ai:wang.jinyi"Zhao, Yi"https://zbmath.org/authors/?q=ai:zhao.yi.1Summary: The synchronization of complex dynamic networks with multiple time-varying delays has attracted wide attentions. Based on aperiodic quantized intermittent control strategy, we discuss the exponential synchronization problem of a class of coupled dynamic complex networks with time-varying delays. By constructing an appropriate time-dependent Lyapunov function and through strict theoretical analysis, we derive some sufficient conditions for exponential synchronization of the dynamic networks. The numerical simulation results demonstrate the effectiveness and validity of the theoretical results.Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints.https://zbmath.org/1449.490202021-01-08T12:24:00+00:00"Mahmudov, Elimhan N."https://zbmath.org/authors/?q=ai:mahmudov.elimhan-nSummary: The present paper studies a new class of problems of optimal control theory with differential inclusions described by fourth order Sturm-Liouville type differential operators (SLDOs). Then, there arises a rather complicated problem with simultaneous determination of the SLDOs with variable coefficients and a Mayer functional depending of high order derivatives of searched functions. The sufficient conditions, containing both the Euler-Lagrange and Hamiltonian type inclusions and ``transversality'' conditions are derived. Formulation of the transversality conditions at the endpoints \(t = 0\) and \(t = 1\) of the considered time interval plays a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions. The main idea of the proof of optimality conditions of Mayer problem for differential inclusions with fourth order SLDO is the use of locally-adjoint mappings. The method is demonstrated in detail as an example for the semilinear optimal control problem, for which the Weierstrass-Pontryagin maximum principle is obtained.Fractional generalised homotopy analysis method for solving nonlinear fractional differential equations.https://zbmath.org/1449.652932021-01-08T12:24:00+00:00"Saratha, S. R."https://zbmath.org/authors/?q=ai:saratha.s-r"Bagyalakshmi, M."https://zbmath.org/authors/?q=ai:bagyalakshmi.morachan"Sai Sundara Krishnan, G."https://zbmath.org/authors/?q=ai:saisundarakrishnan.gSummary: This paper presents a novel hybrid technique which is developed by incorporating the fractional derivatives in the generalised integral transform method. Homotopy analysis method is combined with fractional generalised integral transform method to solve the fractional order nonlinear differential equations. The performance of the proposed method is analysed by solving various categories of nonlinear fractional differential equations like Navier Stokes's model and Riccatti equations, etc. Unlike the other analytical methods, the hybrid method provides a better way to control the convergence region of the obtained series solution through an auxiliary parameter \(h\). Furthermore, as proposed in this paper, the `Fractional Generalised Homotopy Analysis Method' along with the several examples reveal that this method can be effectively used as a tool for solving various kinds of nonlinear fractional differential equations.Dynamical behavior of an epidemic model for a vector-borne disease with treatment.https://zbmath.org/1449.341382021-01-08T12:24:00+00:00"Guo, Shumin"https://zbmath.org/authors/?q=ai:guo.shumin"Ghosh, Mini"https://zbmath.org/authors/?q=ai:ghosh.miniSummary: In this paper, we investigate an epidemic model of a vector-borne disease with treatment. The reproduction number \({R_0}\) of the model is obtained. The equilibria and the threshold of the model are determined by \({R_0}\). By using Bendixson-Dulac theorem, it is shown that the unique positive equilibrium for the model is globally stable if \({R_0}\) is greater than 1. Theoretical results obtained here can help us to explore the method of controlling the spread of vector-borne disease. Finally, numerical simulations for the model are presented to illustrate our mathematical findings.On the generalization of nonlinear fractional differential equations with non-separated boundary conditions.https://zbmath.org/1449.340372021-01-08T12:24:00+00:00"Xing, Yanyuan"https://zbmath.org/authors/?q=ai:xing.yanyuan"Xiao, Huafeng"https://zbmath.org/authors/?q=ai:xiao.huafengSummary: In this paper, the existence and uniqueness results of nonlinear fractional differential equations with non-separated boundary conditions are investigated. The Banach's fixed point theorem and Leray-Schauder degree theory are applied to establish the results. Some examples are given to illustrate the main result. Relevant results are generalized and improved.The existence of extremal solutions for fractional \(p\)-Laplacian problems with the right-handed Riemann-Liouville fractional derivative.https://zbmath.org/1449.340382021-01-08T12:24:00+00:00"Xue, Tingting"https://zbmath.org/authors/?q=ai:xue.tingting"Fan, Xiaolin"https://zbmath.org/authors/?q=ai:fan.xiaolin"Xu, Jiabo"https://zbmath.org/authors/?q=ai:xu.jiaboSummary: In this paper, we study the solvability of fractional \(p\)-Laplacian problems involving the right-hand Riemann-Liouville derivative. By applying monotone iterative technique, lower and upper solutions method and the Banach fixed point theorem, we obtain sufficient conditions for the existence and uniqueness of extremal solutions, and extend the existing results. Finally, we provide an examples to illustrate the results.Countering violent extremism: a mathematical model.https://zbmath.org/1449.910922021-01-08T12:24:00+00:00"Santoprete, Manuele"https://zbmath.org/authors/?q=ai:santoprete.manueleSummary: The term radicalization refers to the process of developing extremist religious political or social beliefs and ideologies. Radicalization becomes a threat to national security when it leads to violence. Prevention and de-radicalization initiatives are part of a set of strategies used to combat violent extremism, which taken together are known as countering violent extremism (CVE). Prevention programs aim to stop the radicalization process before it starts. De-radicalization programs attempt to reform convicted extremists with the ultimate goal of social reintegration. We describe prevention and de-radicalization programs mathematically using a compartmental model. The prevention initiatives are modeled by including a vaccination compartment, while the de-radicalization process is modeled by including a treatment compartment. The model exhibits a threshold dynamics characterized by the basic reproduction number \(R_0\). When \(R_0 < 1\) the system has a unique equilibrium that is asymptotically stable. When \(R_0 > 1\) the system has another equilibrium called ``endemic equilibrium'', which is globally asymptotically stable. These results are established by using Lyapunov functions and LaSalle's invariance principle. Analyzing the basic reproduction number we determine that a combination of prevention and de-radicalization seems to provide the most effective intervention. We perform numerical simulations to confirm our theoretical results. These simulation also show that de-radicalization seems to be more effective to counter radicalization than prevention for our choice of parameters. For other choices of parameters the situation is reversed.Initial time difference quasilinearization method for fractional differential equations involving generalized Hilfer fractional derivative.https://zbmath.org/1449.340222021-01-08T12:24:00+00:00"Kucche, Kishor D."https://zbmath.org/authors/?q=ai:kucche.kishor-d"Mali, Ashwini D."https://zbmath.org/authors/?q=ai:mali.ashwini-dSummary: We present the quasilinearization method with initial time difference for nonlinear fractional differential equations (FDEs) involving generalized Hilfer fractional derivative under various conditions on the nonlinear function involved in the right hand side of the equation. An essential comparison result concerning lower and upper solutions is obtained for this generalized FDEs without demanding the Hölder continuity assumption.Invariant algebraic surfaces of a modified coupled dynamos model.https://zbmath.org/1449.341302021-01-08T12:24:00+00:00"Wu, Jiankun"https://zbmath.org/authors/?q=ai:wu.jiankun"Xie, Feng"https://zbmath.org/authors/?q=ai:xie.fengSummary: A coupled dynamos model considering two loss characteristics can be described as a three-dimensional nonlinear autonomous system, which exhibits very complicated dynamics. In this paper, invariant algebraic surfaces of this system are investigated from the view of integrability. Using the method of characteristic curves for solving linear partial differential equations, we obtain the parameter conditions when the system has invariant algebraic surfaces.Memory state feedback control and stability criteria for a class of neutral delay interconnected systems.https://zbmath.org/1449.342712021-01-08T12:24:00+00:00"Zhao, Feifei"https://zbmath.org/authors/?q=ai:zhao.feifei"Ji, Zhoupeng"https://zbmath.org/authors/?q=ai:ji.zhoupengSummary: The problem of memory feedback control is studied when the time-delay has little effect on the system for a class of neutral interconnected systems with time-delay. A memory feedback controller is designed and a new Lyapunov function is given to analyze the stability of the system by combining the free-weight matrix and the formula of Newton-Leibniz. A sufficient condition for stability of neutral delay interconnected systems is given by linear matrix inequality. A numerical simulation example is given to illustrate the feasibility of the proposed method.Meromorphic solutions of a class of auxiliary differential equation and its applications.https://zbmath.org/1449.343142021-01-08T12:24:00+00:00"Gu, Yongyi"https://zbmath.org/authors/?q=ai:gu.yongyi"Kong, Yinying"https://zbmath.org/authors/?q=ai:kong.yinyingSummary: This paper introduces a method to find exact solutions of nonlinear partial differential equations -- complex method, and derives meromorphic solutions for a class of algebraic differential equation by the mentioned method. The results are used to seek exact solutions of nonlinear differential equations. Exact solutions of the Vakhnenko-Parkes equation and Dodd-Bullough-Mikhailov equation are obtained.Positive periodic solutions for second-order singular differential equations with damping terms.https://zbmath.org/1449.340722021-01-08T12:24:00+00:00"Chen, Ruipeng"https://zbmath.org/authors/?q=ai:chen.ruipeng"Li, Xiaoya"https://zbmath.org/authors/?q=ai:li.xiaoyaSummary: This paper studies the existence of positive periodic solutions of
\[u'' + p(t)u' + q (t)u = f (t,u) + c (t),\]
where \(p\), \(q\), \(c \in L^1 (\mathbb{R}/T\mathbb{Z};\mathbb{R})\), \(f\) is a Carathéodory function and is singular when \(u = 0\). By means of the fixed point theory, several existence theorems are established for the above equation, and some recent results in the literature are generalized and improved.Qualitative analysis of an SIRI epidemic model with stochastic effects.https://zbmath.org/1449.341332021-01-08T12:24:00+00:00"Gao, Jianzhong"https://zbmath.org/authors/?q=ai:gao.jianzhong"Zhang, Tailei"https://zbmath.org/authors/?q=ai:zhang.taileiSummary: An SIRI bilinear epidemic model with stochastic effects is studied. The global existence, uniqueness and boundedness of its positive solution are proved by using stopping time theory and Lyapunov analysis method. It is also shown that the solution of the stochastic model oscillates around the corresponding deterministic disease-free equilibrium and endemic equilibrium points, and sufficient conditions for persistence in mean of the solution of the stochastic model and disease extinction are obtained. Finally, numerical simulations are carried out to illustrate of theoretical results.Existence of positive solutions for a class of periodic boundary value problems of nonlinear second-order systems.https://zbmath.org/1449.340812021-01-08T12:24:00+00:00"Ma, Mantang"https://zbmath.org/authors/?q=ai:ma.mantangSummary: We consider the existence of positive solutions for the periodic boundary value problems of nonlinear second-order systems \[\begin{cases}u'' + A (t)u = \Lambda G (t)F (u),\, 0 < t < 1, \\ u (0) = u (1), u' (0) = u' (1),\end{cases}\] where \(u = ({u_1}, \cdots, {u_n})^{\mathrm{T}}\), \(A (t) = {\mathrm{diag}}[{a_1} (t),\cdots,{a_n} (t)]\), \({a_i} (t)\) can change the sign in \([0,1] (i = 1,\cdots, n)\), \(G (t) = {\mathrm{diag}}[{g_1} (t),\cdots, {g_n} (t)]\), \(F (u) = ({f_1} (u),\cdots,{f_n} (u))^{\mathrm{T}}\), \(\Lambda = {\mathrm{diag}} ({\lambda_1}, \cdots, {\lambda_n})\), \({\lambda_i}\) is a positive parameter \( (i = 1, \cdots, n)\). Under the assumption that the nonlinear term \(F\) satisfies superlinear, sublinear and asymptotic growth condition, the existence of positive solutions of the problem is obtained by using the fixed-point theorem of cone expansion-compression. The conclusions in this paper generalize and improve related results.Bifurcations of coupled nonlinear oscillators with similar kinematics.https://zbmath.org/1449.370262021-01-08T12:24:00+00:00"Nikitina, N. V."https://zbmath.org/authors/?q=ai:nikitina.nelly-vladimirovna"Talimonova, O. Yu."https://zbmath.org/authors/?q=ai:talimonova.o-yuSummary: The application of the principle of skew symmetry for nonlinear systems that represent a bunch of nonlinear Van der Pol oscillators is analyzed. A bunch of oscillators can (depending on the parameters) form systems of coupled regular limiting cycles and coupled attractors with chaotic or conditionally periodic winding of the trajectory. At a slight change in the parameters of oscillators, the scale of two limiting cycles changes. A strong change in the parameters and the coupling coefficient leads to the appearance of limiting cycles with chaotic winding of the trajectory. When considering three connected limiting cycles, one can reduce them to two ones with a periodic winding and one limiting cycle with a conditionally periodic winding. To clarify the nature of the winding of the trajectories, a topological analysis of the trajectory should be done. In this case, the equations in variations are constructed, and the characteristic indicators of solutions are found.Numerical approach for solving the Riccati and logistic equations via QLM-rational Legendre collocation method.https://zbmath.org/1449.651652021-01-08T12:24:00+00:00"Khader, Mohamed M."https://zbmath.org/authors/?q=ai:khader.mohamed-m"Adel, M."https://zbmath.org/authors/?q=ai:adel.mohamed-hSummary: In the presented study, we are presenting the approximate solutions of two important equations, the Riccati and Logistic equations; the presented technique is based on the rational Legendre function. Since all the studied models are nonlinear, we convert these nonlinear equations to a sequence of linear ordinary differential equations (ODEs), then by applying the quasi-linearization method (QLM) on these resulting ODEs at each iteration, these ODEs will be converted to a simple linear system of algebraic equations which can be solved. Two numerical examples are presented and we compared between the approximate and the exact solutions.Linear feedback control of cardiac pacemaker chaotic model.https://zbmath.org/1449.930612021-01-08T12:24:00+00:00"Liu, Shuang"https://zbmath.org/authors/?q=ai:liu.shuang"Chen, Lu"https://zbmath.org/authors/?q=ai:chen.lu"Wang, Tao"https://zbmath.org/authors/?q=ai:wang.tao.4|wang.tao.8|wang.tao.1|wang.tao.2|wang.tao.6|wang.tao.7|wang.tao|wang.tao.3|wang.tao.5|wang.tao.9"Yue, Lijuan"https://zbmath.org/authors/?q=ai:yue.lijuanSummary: A mathematical chaotic model of pacemaker was improved in this paper. The basic dynamics analysis was given, numerical simulation and circuit simulation were carried out to prove the chaotic characteristics of the model. Using linear feedback theory, a simple linear feedback controller is designed to control the pacemaker chaos model. The numerical simulation and circuit simulation results show that the linear feedback controller can quickly control the model from the chaotic state to the periodic state. This makes the cardiac pacemaker recovered from pathology to the normal motion mechanism, so that the heart can restore sinus rhythm.Stochastic permanence of solution to stochastic non-autonomous logistic equation with jumps.https://zbmath.org/1449.600972021-01-08T12:24:00+00:00"Borysenko, O. D."https://zbmath.org/authors/?q=ai:borysenko.oleksandr-d"Borysenko, D. O."https://zbmath.org/authors/?q=ai:borysenko.d-oSummary: It is investigated a non-autonomous logistic differential equation with disturbance of coefficients by white noise, centered and non-centered Poisson noises. The coefficients of equation are locally Lipschitz continuous but do not satisfy the linear growth condition. This equation describes the dynamics of population in the Verhulst model which takes into account the logistic effect: an increase of the population size produces a decrease in fertility and an increase in mortality; since resources are limited, if the population size exceeds some threshold level, the habitat cannot support the growth. The property of stochastic permanence is desirable since it means the long time survival in a population dynamics. Sufficient conditions for the stochastic permanence of population in the considered model are obtained.On the reduction of a nonlinear Noetherian differential-algebraic boundary-value problem to a noncritical case.https://zbmath.org/1449.340632021-01-08T12:24:00+00:00"Chuĭko, S. M."https://zbmath.org/authors/?q=ai:chuiko.sergei-m"Nesmelova, O. V."https://zbmath.org/authors/?q=ai:nesmelova.o-vSummary: We construct necessary and sufficient conditions for the existence of solution and iterative scheme for the approximate solutions of nonlinear Noetherian differential-algebraic boundary value problem in critical case.On two special functions, generalizing the Mittag-Leffler type function, their properties and applications.https://zbmath.org/1449.330192021-01-08T12:24:00+00:00"Ogorodnikov, Evgeniĭ Nikolaevich"https://zbmath.org/authors/?q=ai:ogorodnikov.evgenii-nikolaevichSummary: Two special functions, concerning Mittag-Leffler type functions, are studied. The first is the modification of generalized Mittag-Leffler function, which was introduced by A. A. Kilbas and M. Saigo; the second is the special case of the first one. The relation of these functions with some elementary and special functions and their role in solving of Abel-Volterra integral equations is indicated. The formulas of the fractional integration and differentiation in sense of Riemann-Liouville and Kober are presented. The applications to Cauchy type problems for some linear fractional differential equations with Riemann-Liouville and Kober derivatives are noticed.On some class of functional-differential equations.https://zbmath.org/1449.342132021-01-08T12:24:00+00:00"Kyrov, Vladimir Aleksandrovich"https://zbmath.org/authors/?q=ai:kyrov.vladimir-aleksandrovichSummary: In this paper we consider special functional-differential equations arising in geometry for the metric functions. We prove a theorem on the form of the metric functions.Studies on an SIR model with bilinear incidence.https://zbmath.org/1449.920482021-01-08T12:24:00+00:00"Wang, Mengpin"https://zbmath.org/authors/?q=ai:wang.mengpin"Zou, Shaofen"https://zbmath.org/authors/?q=ai:zou.shuliangSummary: An SIR epidemic model with bilinear incidence was considered and studied. By virtue of differential inequalities and comparison principle, we exploit the SIR model under the basic reproductive number \({R_0} > 1\) to get the uniqueness and global stability of the endemic equilibrium.The symmetries of the \( (2 + 1)\)-dimensional WGC equation and the Volterra lattice equation.https://zbmath.org/1449.342432021-01-08T12:24:00+00:00"Li, Wenting"https://zbmath.org/authors/?q=ai:li.wenting"Liu, Yan"https://zbmath.org/authors/?q=ai:liu.yan.5|liu.yan.8|liu.yan|liu.yan.7|liu.yan.2|liu.yan.3|liu.yan.4|liu.yan.1|liu.yan.6"Jiang, Kun"https://zbmath.org/authors/?q=ai:jiang.kunSummary: Discrete Lie symmetry reduction procedure is a classical method to study the differential-difference equation. Discrete Lie symmetry reduction procedure is used to study the \( (2 + 1)\)-dimensional WGC equation and the Volterra lattice equation. The infinite dimensional Lie algebra and symmetry of these two equations are obtained. The \( (2 + 1)\)-dimensional WGC equation is a rational differential-difference equation, so it is necessary to consider the denominator constraint in the process of reduction. We can not directly apply the discrete Lie symmetry reduction procedure to the nonlinear discrete Volterra lattice equation. In order to solve this problem, we use a similarity transformation method to convert it into an equation which can be used to perform symmetric reduction.On the permanence of the positive absolutely continuous solutions of the generalized Mackey-Glass model.https://zbmath.org/1449.342892021-01-08T12:24:00+00:00"Kiskinov, Hristo"https://zbmath.org/authors/?q=ai:kiskinov.hristo"Zahariev, Andrey"https://zbmath.org/authors/?q=ai:zahariev.andrey-ivanov"Zlatev, Stoyan"https://zbmath.org/authors/?q=ai:zlatev.stoyan-iSummary: The aim of the present paper is to study one of possible generalizations of the Mackey-Glass model of respiratory dynamics. Existence of unique global absolutely continuous positive solutions of the Cauchy problem, their boundedness and permanence are proved. Moreover, an example is given which shows that the conditions introduced in this paper are sharp and cannot be weakened even for ordinary differential equations of this type.Scattering problem for Dirac system with nonlocal potentials.https://zbmath.org/1449.470252021-01-08T12:24:00+00:00"Cojuhari, P. A."https://zbmath.org/authors/?q=ai:cojuhari.petru-a"Nizhnik, L. P."https://zbmath.org/authors/?q=ai:nizhnik.leonid-pavlovichThe authors consider the problem \[ i\frac{d\psi_1 (x)}{dx}+v_1(x)\psi_+=\lambda \psi_1(x), \] \[ -i\frac{d\psi_2 (x)}{dx}+v_2(x)\psi_+=\lambda \psi_2(x), \] where \(v_1,v_2\in L_1(0,\infty)\cap L_2(0,\infty)\), \(\psi_+=\frac12 [\psi_1(0)+\psi_2(0)]\), with the boundary condition \[ \psi_1(0)-\psi_2(0)-i\int\limits_0^\infty [\psi_1(x)\overline{v_1(x)}+\psi_2(x)\overline{v_2(x)}]\,dx=0. \] An explicit expression for the scattering operator is described.
Reviewer: Anatoly N. Kochubei (Kyïv)Stability and Hopf bifurcation analysis in a mutualistic model with time-delay.https://zbmath.org/1449.342872021-01-08T12:24:00+00:00"He, Shun"https://zbmath.org/authors/?q=ai:he.shun"Li, Mei"https://zbmath.org/authors/?q=ai:li.meiSummary: This paper considers a mutualistic model with time-delay. The boundedness of solution is proved by comparison principle, and sufficient conditions for the global asymptotical stability of the positive equilibrium of the model are obtained by constructing Lyapunov function. Then by using the eigenvalue theory and taking the time delay as the parameter, the existence of Hopf bifurcation is studied. Finally, numerical simulations are given to illustrate the theory.On splitting and stability of linear stationary singularly perturbed differential equations.https://zbmath.org/1449.342032021-01-08T12:24:00+00:00"Osypova, O. V."https://zbmath.org/authors/?q=ai:osypova.o-v"Cherevko, I. M."https://zbmath.org/authors/?q=ai:cherevko.igor-mSummary: In this paper, we study linear stationary singularly perturbed systems of differential equations by the method of integral manifolds. An explicit form of non-degenerate replacement of variables is obtained, which splits the input system into two independent subsystems. The initial conditions are split and the principle of construction is established to study the stability of the zero solution. The possibility of using zero approximation of the integral manifold of slow variables for investigating the stability of the solution of the input singularly perturbed system is considered.Rationalized Haar wavelet bases to approximate the solution of the first Painlevé equations.https://zbmath.org/1449.651522021-01-08T12:24:00+00:00"Erfanian, Majid"https://zbmath.org/authors/?q=ai:erfanian.majid"Mansoori, Amin"https://zbmath.org/authors/?q=ai:mansoori.aminSummary: In this article, using the properties of the rationalized Haar (RH) wavelets and the matrix operator, a method is presented for calculating the numerical approximation of the first Painlevé equations solution. Also, an upper bound of the error is given and by applying the Banach fixed point theorem the convergence analysis of the method is stated. Furthermore, an algorithm to solve the first Painlevé equation is proposed. Finally, the reported results are compared with some other methods to show the effectiveness of the proposed approach.Existence and continuation of solutions of Hilfer fractional differential equations.https://zbmath.org/1449.340132021-01-08T12:24:00+00:00"Bhairat, Sandeep P."https://zbmath.org/authors/?q=ai:bhairat.sandeep-pSummary: In the present paper, we consider initial value problems for Hilfer fractional differential equations and for system of Hilfer fractional differential equations. By using equivalent integral equations and some fixed point theorems, we study the local existence of solutions. We extend these local existence results globally with the help of continuation theorems and generalized Gronwall inequality.The Newton-Kantorovich method in the theory of autonomous Noetherian boundary-value problems in the case of parametric resonance.https://zbmath.org/1449.340622021-01-08T12:24:00+00:00"Chuĭko, S. M."https://zbmath.org/authors/?q=ai:chuiko.sergei-m"Nesmelova, O. V."https://zbmath.org/authors/?q=ai:nesmelova.o-vSummary: We have found conditions for solvability and a convergent iterative scheme for solutions of a nonlinear autonomous Noetherian boundary-value problem in the case of parametric resonance. As an example of applying the scheme, some approximations to the solution of a periodic boundary-value problem for an autonomous equation of the Duffing type with a parametric perturbation are determined. To control the accuracy of the approximations, residuals in the original equation are applied.Partial states linearized synchronization of the single parameter Chen system.https://zbmath.org/1449.341852021-01-08T12:24:00+00:00"Li, Dekui"https://zbmath.org/authors/?q=ai:li.dekuiSummary: The partial states linearized synchronization problem of the single parameter Chen system is studied in this paper. By the feedback linearized method and the error analysis of the linearized system, it is found that if only the second state variable of the response system is controlled, all states of the single parameter Chen system can be synchronized. Finally, the numerical simulations show that the theoretical analysis is correct and the synchronization controller is effective.On the \((\alpha,\beta)\)-Scott-Blair anti-Zener arrangement.https://zbmath.org/1449.340192021-01-08T12:24:00+00:00"Hassouna, M."https://zbmath.org/authors/?q=ai:hassouna.m"Ouhadan, A."https://zbmath.org/authors/?q=ai:ouhadan.abdelaziz"El Kinani, E. H."https://zbmath.org/authors/?q=ai:el-kinani.el-hassanSummary: In this paper, we study some fractional rheological anti-Zener equations that are derived from the ordinary anti-Zener model. We analyze three situations : the theoretical anti-Zener, the \(\alpha\)-Scott-Blair anti-Zener and the \((\alpha,\beta)\)-Scott-Blair anti-Zener models. The Mellin transform technique and Fox-H functions are investigated to derive relaxation modulus and the creep compliance function corresponding to each model. The limit cases are discussed.Existence and uniqueness of solutions for Caputo-Hadamard type fractional differential equations.https://zbmath.org/1449.340322021-01-08T12:24:00+00:00"Shi, Linfei"https://zbmath.org/authors/?q=ai:shi.linfei"Li, Chengfu"https://zbmath.org/authors/?q=ai:li.chengfuSummary: In this paper, we study a class of Caputo-Hadamard fractional differential equations with boundary value problems. By using Banach fixed point theorem and the method of upper and lower solutions, the existence and uniqueness results of the solutions are obtained, which generalize some results about ordinary differential equations with boundary value problems. As an application, two examples are given to illustrate our main results.Nonlocal integral boundary value problem of Bagley-Torvik type fractional differential equations and inclusions.https://zbmath.org/1449.340142021-01-08T12:24:00+00:00"Chen, Lizhen"https://zbmath.org/authors/?q=ai:chen.lizhen"Ibrahim, Badawi Hamza Eibadawi"https://zbmath.org/authors/?q=ai:ibrahim.badawi-hamza-eibadawi"Li, Gang"https://zbmath.org/authors/?q=ai:li.gang.8Summary: In this article, we consider the Bagley-Torvik type fractional differential equation \[{}^cD^{v_1}l (t)-a{}^cD^{v_2}l (t) = g (t,l (t))\]
and the differential inclusion
\[{}^cD^{v_1}l (t)-a{}^cD^{v_2}l (t) \in G (t,l (t)), t \in (0,1)\]
subject to
\[l (0) = {l_0}, \text{ and }l (1) = \lambda'\int_0^\omega \frac{(\omega-s)^{\chi-1}l (s)}{\Gamma (\chi)}ds,\]
where \(1 < v_1 \leq 2\), \(1 \leq v_2 < v_1\), \(0 < \omega \leq 1\), \(\chi = v_1 - v_2 > 0\), \(a\), \(\lambda'\) are given constants. By using the Leray-Schauder degree theory and fixed point theorems, we prove the existence of solutions. Our results extend existence theorems for the classical Bagley-Torvik equation and some related models.Anti-periodic boundary value problem of a class of fractional impulsive differential equation.https://zbmath.org/1449.340362021-01-08T12:24:00+00:00"Xing, Yanyuan"https://zbmath.org/authors/?q=ai:xing.yanyuan"Liu, Fang"https://zbmath.org/authors/?q=ai:liu.fang|liu.fang.1Summary: The existence and uniqueness of solution for an anti-periodic boundary value problem of a class of nonlinear fractional impulsive differential equation are obtained by using the contraction mapping principle. An actual example is presented.