Recent zbMATH articles in MSC 34https://zbmath.org/atom/cc/342022-11-17T18:59:28.764376ZWerkzeugEmbedded Picard-Vessiot extensionshttps://zbmath.org/1496.031572022-11-17T18:59:28.764376Z"Brouette, Quentin"https://zbmath.org/authors/?q=ai:brouette.quentin"Cousins, Greg"https://zbmath.org/authors/?q=ai:cousins.greg"Pillay, Anand"https://zbmath.org/authors/?q=ai:pillay.anand"Point, Francoise"https://zbmath.org/authors/?q=ai:point.francoiseSummary: We prove that if \(T\) is a theory of large, bounded, fields of characteristic 0 with almost quantifier elimination, and \(T_D\) is the model companion of \(T\cup\{\)``\(\partial\) is a derivation''\(\}\), then for any model \((\mathcal U,\partial)\) of \(T_D\), differential subfield \(K\) of \(\mathcal U\) such that \(C_K\vDash T\), and linear differential equation \(\partial Y= AY\) over \(K\), there is a Picard-Vessiot extension \(L\) of \(K\) for the equation with \(K\leq L\leq\mathcal U\), i.e. \(L\) can be embedded in \(\mathcal U\) over \(K\), as a differential field. Moreover such \(L\) is unique to isomorphism over \(K\) as a differential field. Likewise for the analogue for strongly normal extensions for logarithmic differential equations in the sense of Kolchin.Quantitative problems on the size of \(G\)-operatorshttps://zbmath.org/1496.130392022-11-17T18:59:28.764376Z"Lepetit, G."https://zbmath.org/authors/?q=ai:lepetit.gabrielA \(G\)-function is a power series satisfying certain hypotheses. Examples of \(G\)-functions are certain hypergeometric series with rational parameters, polylogarithms and the function \(1/(1-z)\). \(G\)-operators, a class of differential operators containing the differential operators of minimal order annihilating Siegel's \(G\)-functions, satisfy the Galochkin condition of moderate growth, encoded by a \(p\)-adic quantity, the size.
Previous works of Chudnovsky, André and Dwork have provided inequalities between the size of a \(G\)-operator and certain computable constants depending among others on its solutions. The author recalls André's idea to attach a notion of size to differential modules and details his results on the behavior of the size relatively to the standard algebraic operations on the modules. This allows to prove a quantitative version of André's generalization of Chudnovsky's Theorem: for \(f(z)=\Sigma _{\alpha ,k,\ell}c_{\alpha ,k,\ell}\log ^k(z)f_{\alpha ,k,\ell}(z)\), where \(f_{\alpha ,k,\ell}\) are \(G\)-functions, one can determine an upper bound on the size of the minimal operator \(L\) over \(\bar{\mathbb{Q}}(z)\) of \(f(z)\) in terms of quantities depending on the \(f_{\alpha ,k,\ell}\), the rationals \(\alpha\) and the integers \(k\).
The author gives two applications of this result: an estimation of the size of a product of two \(G\)-operators depending on the size of each operator and a computation of a constant appearing in a certain Diophantine problem.
Reviewer: Vladimir P. Kostov (Nice)Distinguishability of the descriptor systems with regular pencilhttps://zbmath.org/1496.150072022-11-17T18:59:28.764376Z"Dastgeer, Zoubia"https://zbmath.org/authors/?q=ai:dastgeer.zoubia"Younus, Awais"https://zbmath.org/authors/?q=ai:younus.awais"Tunç, Cemil"https://zbmath.org/authors/?q=ai:tunc.cemilThe authors consider a linear hybrid descriptor system and focus on the property that the distinguishability of this type of systems is imperative.
The paper contains two results. The first result is related to the distinguishability of the descriptor system under study, for which some criteria and equivalent conditions related to polynomial input distinguishability, analytic and smooth input distinguishability are developed.
The second result concerns equivalent criteria for input distinguishability of descriptor systems with a regular pencil by using the Laplace transform and the Cayley-Hamilton theorem.
Reviewer: Ioannis Dassios (Dublin)Some new Gronwall-Bihari type inequalities associated with generalized fractional operators and applicationshttps://zbmath.org/1496.260272022-11-17T18:59:28.764376Z"Ayari, Amira"https://zbmath.org/authors/?q=ai:ayari.amira"Boukerrioua, Khaled"https://zbmath.org/authors/?q=ai:boukerrioua.khaledSummary: In this paper, we derive some generalizations of certain Gronwall-Bihari type inequality for generalized fractional operators unifying Riemann-Liouville and Hadamard fractional operators for functions in one variable, which provide explicit bounds on unknown functions. To show the feasibility of the obtained inequalities, two illustrative examples are also introduced.Neumann boundary condition for a nonlocal biharmonic equationhttps://zbmath.org/1496.310032022-11-17T18:59:28.764376Z"Turmetov, Batirkhan Khudaĭbergenovich"https://zbmath.org/authors/?q=ai:turmetov.batirkhan-khudaybergenovich"Karachik, Valeriĭ Valentinovich"https://zbmath.org/authors/?q=ai:karachik.valery-vSummary: The solvability conditions for a class of boundary value problems for a nonlocal biharmonic equation in the unit ball with the Neumann conditions on the boundary are studied. The nonlocality of the equation is generated by some orthogonal matrix. The presence and uniqueness of a solution to the proposed Neumann boundary condition is examined, and an integral representation of the solution to the Dirichlet problem in terms of the Green's function for the biharmonic equation in the unit ball is obtained.
First, some auxiliary statements are established: the Green's function of the Dirichlet problem for the biharmonic equation in the unit ball is given, the representation of the solution to the Dirichlet problem in terms of this Green's function is written, and the values of the integrals of the functions perturbed by the orthogonal matrix are found. Then a theorem for the solution to the auxiliary Dirichlet problem for a nonlocal biharmonic equation in the unit ball is proved. The solution to this problem is written using the Green's function of the Dirichlet problem for the regular biharmonic equation. An example of solving a simple problem for a nonlocal biharmonic equation is given. Next, we formulate a theorem on necessary and sufficient conditions for the solvability of the Neumann boundary condition for a nonlocal biharmonic equation. The main theorem is proved based on two lemmas, with the help of which it is possible to transform the solvability conditions of the Neumann boundary condition to a simpler form. The solution to the Neumann boundary condition is presented through the solution to the auxiliary Dirichlet problem.The hypergeometric function, the confluent hypergeometric function and WKB solutionshttps://zbmath.org/1496.330022022-11-17T18:59:28.764376Z"Aoki, Takashi"https://zbmath.org/authors/?q=ai:aoki.takashi"Takahashi, Toshinori"https://zbmath.org/authors/?q=ai:takahashi.toshinori"Tanda, Mika"https://zbmath.org/authors/?q=ai:tanda.mikaTake the Gauss hypergeometric equation and instead of parameters \(a,b,c\) let us write \(a=\alpha_0+\alpha \eta, b=\beta_0+\beta\eta, c=\gamma_0+\gamma\eta\) and instead of the unknown function \(w\) let us take \(w=x^{-c/2}(1-x)^{(-1/2)(a+b-c+1)}\psi\). Then the equation is written in the form
\[
(-\frac{d^2}{dx^2}+\eta^2Q)\psi=0,\quad Q=\sum_{j=0}^N \eta^{-j}Q_j(x).
\]
To such an equation the WKB analysis can be applied. It gives formal solutions that using the Borel summation can be transformed to analytic solutions. In the paper under review a relation between these solutions and the classical solutions of the Gauss equation is established.
Also analogous results for the equation for the function \(F_{1,1}\) are obtained.
Reviewer: Dmitry Artamonov (Moskva)Asymptotics of multiple orthogonal Hermite polynomials \(H_{n_1,n_2}(z,\alpha)\) determined by a third-order differential equationhttps://zbmath.org/1496.330082022-11-17T18:59:28.764376Z"Dobrokhotov, S. Yu."https://zbmath.org/authors/?q=ai:dobrokhotov.sergei-yu"Tsvetkova, A. V."https://zbmath.org/authors/?q=ai:tsvetkova.anna-vSummary: In the paper, we study the asymptotics of multiple orthogonal Hermite polynomials \(H_{n_1,n_2}(z,\alpha)\) that are determined by orthogonality relations with respect to two weights that are Gaussian exponents with shifted maxima. These polynomials can be defined using recurrence relations, and also, as shown by A. I. Aptekarev, A. Branquinho, and W. Van Assche, as certain solutions to a third-order differential equation. Starting from this differential equation, we obtain asymptotics of such polynomials as \(|n|=\sqrt{n_1^2+n_2^2} \rightarrow \infty\) in the form of the Airy function \(A_i\) and its derivative \(A_i'\) of a compound argument.Properties of \(\psi\)-Mittag-Leffler fractional integralshttps://zbmath.org/1496.330122022-11-17T18:59:28.764376Z"Oliveira, D. S."https://zbmath.org/authors/?q=ai:de-souza-oliveira.fabiano|oliveira.daniela-s|oliveira.david-senaSummary: This paper aims to investigate properties of fractional integrals with three-parameters Mittag-Leffler function kernel. We prove that the Cauchy problem and the Volterra integral equation are equivalent. We find a closed-form to the solution of the Cauchy problem using successive approximations method and \(\psi\)-Caputo fractional derivative.On some generalization of the hyperbolic and trigonometric functions and their applicationshttps://zbmath.org/1496.340012022-11-17T18:59:28.764376Z"Bondarenko, B. A."https://zbmath.org/authors/?q=ai:bondarenko.boris-a(no abstract)On linearizability via nonlocal transformations and first integrals for second-order ordinary differential equationshttps://zbmath.org/1496.340022022-11-17T18:59:28.764376Z"Sinelshchikov, Dmitry I."https://zbmath.org/authors/?q=ai:sinelshchikov.dmitry-iSummary: Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering the general case of a linearization problem via certain nonlocal transformations. In addition, we show that each equation from the linearizable family admits a transcendental first integral and study particular cases when this first integral is autonomous or rational. Thus, as a byproduct of solving this linearization problem we obtain a classification of second-order differential equations admitting a certain transcendental first integral. To demonstrate effectiveness of our approach, we consider several examples of autonomous and non-autonomous second order differential equations, including generalizations of the Duffing and Van der Pol oscillators, and construct their first integrals and general solutions. We also show that the corresponding first integrals can be used for finding periodic solutions, including limit cycles, of the considered equations.The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methodshttps://zbmath.org/1496.340032022-11-17T18:59:28.764376Z"Wang, Yanjie"https://zbmath.org/authors/?q=ai:wang.yanjie"Zhang, Beibei"https://zbmath.org/authors/?q=ai:zhang.beibei"Cao, Bo"https://zbmath.org/authors/?q=ai:cao.boSummary: The exact traveling wave solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities are studied using the dynamical system and the first integral methods. Taking different parameter conditions, we obtain periodic wave solutions, exact solitary wave solutions, kink wave solutions, and anti-kink wave solutions.Stability, boundedness and controllability of solutions of measure functional differential equationshttps://zbmath.org/1496.340042022-11-17T18:59:28.764376Z"Andrade da Silva, F."https://zbmath.org/authors/?q=ai:andrade-da-silva.f"Federson, M."https://zbmath.org/authors/?q=ai:federson.marcia"Toon, E."https://zbmath.org/authors/?q=ai:toon.eduardIn the present paper, converse Lyapunov results on uniform boundedness for the very general class of generalized differential equations are established. Relations between stability and boundedness of solutions are also obtained. Using Lyapunov techniques, asymptotic controllability is characterized as well. As the theory of measure functional differential equations is a particular case of this wide setting, corresponding theorems are derived for measure functional differential equations.
Reviewer: Bianca-Renata Satco (Suceava)Stable RBF-RA method for solving fuzzy fractional kinetic equationhttps://zbmath.org/1496.340052022-11-17T18:59:28.764376Z"Jafari, Hossein"https://zbmath.org/authors/?q=ai:jafari.hossein|jafari.hossein.1"Kazemi, Behshid Fakhr"https://zbmath.org/authors/?q=ai:kazemi.behshid-fakhrSummary: The direct method based on the flat radial basis functions (RBFs) for obtaining numerical solution of differential equations is highly ill-conditioned. Therefore, many studies have been dedicated to overcome this ill-conditioning by using different techniques. Here, the RBF algorithm based on vector-valued rational approximations is utilised to obtain the numerical solution of fuzzy fractional differential equations. This stable method can be applied with any sort of smooth RBF easily and accurately. To illustrate the accuracy and stability of the presented algorithm, we focus on solving the kinetic model with fuzzy fractional derivative.Existence of positive solutions for weighted fractional order differential equationshttps://zbmath.org/1496.340062022-11-17T18:59:28.764376Z"Abdo, Mohammed S."https://zbmath.org/authors/?q=ai:abdo.mohammed-salem"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet"Ali, Saeed M."https://zbmath.org/authors/?q=ai:ali.saeed-m"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamal"Jarad, Fahd"https://zbmath.org/authors/?q=ai:jarad.fahdSummary: In this paper, we deliberate two classes of initial value problems for nonlinear fractional differential equations under a version weighted generalized of Caputo fractional derivative given by \textit{F. Jarad} et al. [Fractals 28, No. 8, Article ID 2040011, 12 p. (2020; Zbl 1489.26006)]. We get a formula for the solution through the equivalent fractional integral equations to the proposed problems. The existence and uniqueness of positive solutions have been obtained by using lower and upper solutions. The acquired results are demonstrated by building the upper and lower control functions of the nonlinear term with the aid of Banach and Schauder fixed point theorems. The acquired results are demonstrated by pertinent numerical examples along with the Bashforth Moulton prediction correction scheme and Matlab.A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers-Ulam stabilityhttps://zbmath.org/1496.340072022-11-17T18:59:28.764376Z"Alam, Mehboob"https://zbmath.org/authors/?q=ai:alam.mehboob"Zada, Akbar"https://zbmath.org/authors/?q=ai:zada.akbar"Popa, Ioan-Lucian"https://zbmath.org/authors/?q=ai:popa.ioan-lucian"Kheiryan, Alireza"https://zbmath.org/authors/?q=ai:kheiryan.alireza"Rezapour, Shahram"https://zbmath.org/authors/?q=ai:rezapour.shahram"Kaabar, Mohammed K. A."https://zbmath.org/authors/?q=ai:kaabar.mohammedSummary: In this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results.Some properties of implicit impulsive coupled system via \(\varphi \)-Hilfer fractional operatorhttps://zbmath.org/1496.340082022-11-17T18:59:28.764376Z"Almalahi, Mohammed A."https://zbmath.org/authors/?q=ai:almalahi.mohammed-a"Panchal, Satish K."https://zbmath.org/authors/?q=ai:panchal.satish-kushabaSummary: The major goal of this work is investigating sufficient conditions for the existence and uniqueness of solutions for implicit impulsive coupled system of \(\varphi \)-Hilfer fractional differential equations (FDEs) with instantaneous impulses and terminal conditions. First, we derive equivalent fractional integral equations of the proposed system. Next, by employing some standard fixed point theorems such as Leray-Schauder alternative and Banach, we obtain the existence and uniqueness of solutions. Further, by mathematical analysis technique we investigate the Ulam-Hyers (UH) and generalized UH (GUH) stability of solutions. Finally, we provide a pertinent example to corroborate the results obtained.On a nonlocal implicit problem under Atangana-Baleanu-Caputo fractional derivativehttps://zbmath.org/1496.340092022-11-17T18:59:28.764376Z"Alnahdi, Abeer S."https://zbmath.org/authors/?q=ai:alnahdi.abeer-s"Jeelani, Mdi Begum"https://zbmath.org/authors/?q=ai:jeelani.mdi-begum"Abdo, Mohammed S."https://zbmath.org/authors/?q=ai:abdo.mohammed-salem"Ali, Saeed M."https://zbmath.org/authors/?q=ai:ali.saeed-m"Saleh, S."https://zbmath.org/authors/?q=ai:saleh.sagvan|saleh.s-q|saleh.s-a|saleh.sami|saleh.saleh-a|saleh.saad-j|saleh.sahar-mohamad|saleh.siti-hidayah-muhad|saleh.sina|saleh.samera-m|saleh.s-v|saleh.shanti-faridah|saleh.shokryaSummary: In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana-Baleanu-Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means of Schauder's and Banach's fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results.Existence and Ulam stability results for two-orders fractional differential equationhttps://zbmath.org/1496.340102022-11-17T18:59:28.764376Z"Atmania, R."https://zbmath.org/authors/?q=ai:atmania.rahima"Bouzitouna, S."https://zbmath.org/authors/?q=ai:bouzitouna.sSummary: In this paper, we deal with the existence of a unique solution and some Ulam's type stability concepts for an initial value problem of a class of two-orders fractional differential equations involving Caputo's fractional derivative. We investigate two types of Ulam stability: Ulam-Hyers stability and Ulam-Hyers-Rassias stability for the considered problem of two fractional orders. We use the Banach fixed point theorem and fractional calculus. Finally, we give an example to illustrate the results.Existence of solutions of discrete fractional problem coupled to mixed fractional boundary conditionshttps://zbmath.org/1496.340112022-11-17T18:59:28.764376Z"Bourguiba, Rim"https://zbmath.org/authors/?q=ai:bourguiba.rim"Cabada, Alberto"https://zbmath.org/authors/?q=ai:cabada.alberto"Kalthoum, Wanassi Om"https://zbmath.org/authors/?q=ai:kalthoum.wanassi-omSummary: In this paper, we introduce a two-point nonlinear boundary value problem for a finite fractional difference equation. An associated Green's function is constructed as a series of functions and some of its properties are obtained. Some existence results are deduced from fixed point theory and lower and upper solutions.Multiplicity solutions for a class of \(p\)-Laplacian fractional differential equations via variational methodshttps://zbmath.org/1496.340122022-11-17T18:59:28.764376Z"Chen, Yiru"https://zbmath.org/authors/?q=ai:chen.yiru"Gu, Haibo"https://zbmath.org/authors/?q=ai:gu.haiboSummary: While it is known that one can consider the existence of solutions to boundary-value problems for fractional differential equations with derivative terms, the situations for the multiplicity of weak solutions for the \(p\)-Laplacian fractional differential equations with derivative terms are less considered. In this article, we propose a new class of \(p\)-Laplacian fractional differential equations with the Caputo derivatives. The multiplicity of weak solutions is proved by the variational method and critical point theorem. At the end of the article, two examples are given to illustrate the validity and practicality of our main results.Fractional Fourier transform to stability analysis of fractional difffferential equations with Prabhakar derivativeshttps://zbmath.org/1496.340132022-11-17T18:59:28.764376Z"Deepa, S."https://zbmath.org/authors/?q=ai:deepa.s-n"Ganesh, A."https://zbmath.org/authors/?q=ai:ganesh.anumanthappa"Ibrahimov, V."https://zbmath.org/authors/?q=ai:ibrahimov.vagif-r"Santra, S. S."https://zbmath.org/authors/?q=ai:santra.shyam-sundar"Govindan, V."https://zbmath.org/authors/?q=ai:govindan.vediyappan"Khedher, K. M."https://zbmath.org/authors/?q=ai:khedher.khaled-mohamed"Noeiaghdam, S."https://zbmath.org/authors/?q=ai:noeiaghdam.samadSummary: In this paper, the authors introduce the Prabhakar derivative associated with the generalised Mittag-Leffler function. Some properties of the Prabhakar integrals, Prabhakar derivatives and some of their extensions, like fractional Fourier transform of Prabhakar integrals and fractional Fourier transform of Prabhakar derivatives are introduced. This note aims to study the Mittag-Leffler-Hyers-Ulam stability of the linear and nonlinear fractional differential equations with the Prabhakar derivative. Furthermore, we give a brief definition of the Mittag-Leffler-Hyers-Ulam problem and a method for solving fractional differential equations using the fractional Fourier transform. We show that the fractional differential equations are Mittag-Leffler-Hyers-Ulam stable in the sense of Prabhakar derivatives.A new reliable method and its convergence for nonlinear second-order fractional differential equationshttps://zbmath.org/1496.340142022-11-17T18:59:28.764376Z"Khalouta, Ali"https://zbmath.org/authors/?q=ai:khalouta.ali"Kadem, Abdelouahab"https://zbmath.org/authors/?q=ai:kadem.abdelouahabSummary: The main goal of this article is to propose a new reliable method to solve nonlinear second-order fractional differential equations in particular, nonlinear fractional Bratu-type equation. This method called the modified Taylor fractional series method (MTFSM). The fractional derivative is defined in the Liouville-Caputo sense. Simplicity, rapid convergence, and high accuracy are the advantages of this method. In addition, the MTFSM reduces the size of calculations by not requiring linearization, discretization, perturbation or any other restriction. Three numerical examples are exhibited to demonstrate the reliability and efficiency of the proposed method, and the solutions are considered as an infinite series that converge rapidly to the exact solutions. The results display that the MTFSM is very effective and accurate to solve this type of nonlinear fractional problems.Existence results for higher order fractional differential equations with integral boundary conditions via Kuratowski measure of noncompactneshttps://zbmath.org/1496.340152022-11-17T18:59:28.764376Z"Lachouri, Adel"https://zbmath.org/authors/?q=ai:lachouri.adel"Ardjouni, Abdelouaheb"https://zbmath.org/authors/?q=ai:ardjouni.abdelouaheb"Gouri, Nesrine"https://zbmath.org/authors/?q=ai:gouri.nesrine"Khelil, Kamel Ali"https://zbmath.org/authors/?q=ai:khelil.kamel-aliThe authors prove the existence of solutions for a higher order fractional differential equation involving a Caputo-Hadamard fractional derivative operator, supplemented with integral boundary conditions by applying the technique of Kuratowski measure of non-compactness combined with Mönch's fixed point theorem. An example illustrating the main result is presented.
Reviewer: Bashir Ahmad (Jeddah)Comment for ``Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel''https://zbmath.org/1496.340162022-11-17T18:59:28.764376Z"Li, Xiaoyan"https://zbmath.org/authors/?q=ai:li.xiaoyanSummary: In a published paper [\textit{A. Khan} et al., Chaos Solitons Fractals 127, 422--427 (2019; Zbl 1448.34046)], some miss prints were found. One is about the calculation of the solution and also about the expression of the solution using Green's function of the fractional differential equation studied in this paper; The others are for the properties for Green's function, these miss prints affected the deriving of the main results in Khan's paper, some corrections for Khan's paper should be needed. In this paper, we make some corrections and give the correct proof proceeding for the results, a new example is given to validate part of the proven results.The solution theory for the fractional hybrid \(q\)-difference equationshttps://zbmath.org/1496.340172022-11-17T18:59:28.764376Z"Ma, Kuikui"https://zbmath.org/authors/?q=ai:ma.kuikui"Gao, Lei"https://zbmath.org/authors/?q=ai:gao.lei(no abstract)Existence of solutions for tripled system of fractional differential equations involving cyclic permutation boundary conditionshttps://zbmath.org/1496.340182022-11-17T18:59:28.764376Z"Matar, Mohammed M."https://zbmath.org/authors/?q=ai:matar.mohammed-m"Amra, Iman Abo"https://zbmath.org/authors/?q=ai:amra.iman-abo"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-oSummary: In this paper, we introduce and study a tripled system of three associated fractional differential equations. Prior to proceeding to the main results, the proposed system is converted into an equivalent integral form by the help of fractional calculus. Our approach is based on using the addressed tripled system with cyclic permutation boundary conditions. The existence and uniqueness of solutions are investigated. We employ the Banach and Krasnoselskii fixed point theorems to prove our main results. Illustrative examples are presented to explain the theoretical results.On a coupled nonlinear fractional integro-differential equations with coupled non-local generalised fractional integral boundary conditionshttps://zbmath.org/1496.340192022-11-17T18:59:28.764376Z"Muthaiah, Subramanian"https://zbmath.org/authors/?q=ai:muthaiah.subramanianSummary: We investigate a coupled Liouville-Caputo fractional integro-differential equations (CLCFIDEs) with nonlinearities that depend on the lower order fractional derivatives of the unknown functions, and also fractional integrals of the unknown functions supplemented with the coupled non-local generalised Riemann-Liouville fractional integral (GRLFI) boundary conditions. The existence and uniqueness results have endorsed by Leray-Schauder nonlinear alternative, and Banach fixed point theorem respectively. Sufficient examples have also been supplemented to substantiate the proof, and we have discussed some variants of the given problem.Existence and uniqueness results for sequential \(\psi\)-Hilfer fractional differential equations with multi-point boundary conditionshttps://zbmath.org/1496.340202022-11-17T18:59:28.764376Z"Ntouyas, Sotiris K."https://zbmath.org/authors/?q=ai:ntouyas.sotiris-k"Vivek, Devaraj"https://zbmath.org/authors/?q=ai:vivek.devarajIn this paper, the authors establish existence and uniqueness results of solution for multi-point boundary value problems for sequential fractional differential equations involving $\psi$-Hilfer fractional derivative. The existence and uniqueness of a solution are obtained by the Banach contraction mapping principle. The nonlinear alternative of Leray-Schauder is applied to obtain the existence of at least one solution. Finally, they give some examples to illustrate their main results.
Reviewer: Thanin Sitthiwirattham (Bangkok)A new approach on the approximate controllability of fractional differential evolution equations of order \(1<r<2\) in Hilbert spaceshttps://zbmath.org/1496.340212022-11-17T18:59:28.764376Z"Raja, M. Mohan"https://zbmath.org/authors/?q=ai:raja.m-mohan"Vijayakumar, V."https://zbmath.org/authors/?q=ai:vijayakumar.velusamy"Udhayakumar, R."https://zbmath.org/authors/?q=ai:udhayakumar.r"Zhou, Yong"https://zbmath.org/authors/?q=ai:zhou.yongSummary: This manuscript is mainly focusing on approximate controllability for fractional differential evolution equations of order \(1<r<2\) in Hilbert spaces. We consider a class of control systems governed by the fractional differential evolution equations. By using the results on fractional calculus, cosine and sine functions of operators, and Schauder's fixed point theorem, a new set of sufficient conditions are formulated which guarantees the approximate controllability of fractional differential evolution systems. The results are established under the assumption that the associated linear system is approximately controllable. Then, we develop our conclusions to the ideas of nonlocal conditions. Lastly, we present theoretical and practical applications to support the validity of the study.Sufficient conditions for the existence of oscillatory solutions to nonlinear second order differential equationshttps://zbmath.org/1496.340222022-11-17T18:59:28.764376Z"Sethi, Abhay Kumar"https://zbmath.org/authors/?q=ai:sethi.abhay-kumar"Ghaderi, Mehran"https://zbmath.org/authors/?q=ai:ghaderi.mehran"Rezapour, Shahram"https://zbmath.org/authors/?q=ai:rezapour.shahram"Kaabar, Mohammed K. A."https://zbmath.org/authors/?q=ai:kaabar.mohammed"Inc, Mostafa"https://zbmath.org/authors/?q=ai:inc.mostafa"Masiha, Hashem Parvaneh"https://zbmath.org/authors/?q=ai:masiha.hashem-parvanehSummary: Some electrical events consist of mathematical models and due to the essence of electric current, their related differential equations have naturally an interesting property where all solutions are oscillatory. Many researchers have studied the necessary conditions for oscillatory solutions in the literature. But in this work, some sufficient conditions are investigated using Riccati transformation for nonlinear second order differential equations in which all solutions are oscillatory. Two illustrative examples are provided to validate our theoretical results. In addition, simulation results and numerical experiments are conducted to validate our obtained results.On the novel existence results of solutions for fractional Langevin equation associating with nonlinear fractional ordershttps://zbmath.org/1496.340232022-11-17T18:59:28.764376Z"Sintunavarat, Wutiphol"https://zbmath.org/authors/?q=ai:sintunavarat.wutiphol"Turab, Ali"https://zbmath.org/authors/?q=ai:turab.aliSummary: The Langevin equation is a core premise of the Brownian motion, which describes the development of essential processes in continuously changing situations. As a generalization of the classical one, the fractional Langevin equation offers a fractional Gaussian mechanism with two indices as parametrization, which is more flexible to model fractal systems. This paper aims to deals with a nonlinear fractional Langevin equation that involves two fractional orders with nonlocal integral boundary conditions. Our goal is to find the results related to the existence of solutions for the proposed Langevin equation by using the appropriate fixed point methods. An example is also presented to illustrate the importance of our result.Basic theory of differential equations with fractional substantial derivative in Banach spaceshttps://zbmath.org/1496.340242022-11-17T18:59:28.764376Z"Wang, Yejuan"https://zbmath.org/authors/?q=ai:wang.yejuan"Wang, Yaxiong"https://zbmath.org/authors/?q=ai:wang.yaxiong.1Summary: The theory of local existence, extremal solutions and global existence of solutions are established for an abstract fractional differential equation with fractional substantial derivative in a Banach space by assuming that the nonlinear term is weakly continuous in some sense. We then apply the theory results to a first order lattice system with fractional substantial derivative, the existence of a compact global attractor is established. In addition, the global attractor is a singleton is also proved under extra Lipschitz conditions.Tangent nonlinear equation in context of fractal fractional operators with nonsingular kernelhttps://zbmath.org/1496.340252022-11-17T18:59:28.764376Z"Zafar, Zain Ul Abadin"https://zbmath.org/authors/?q=ai:zafar.zain-ul-abadin"Sene, Ndolane"https://zbmath.org/authors/?q=ai:sene.ndolane"Rezazadeh, Hadi"https://zbmath.org/authors/?q=ai:rezazadeh.hadi"Esfandian, Nafiseh"https://zbmath.org/authors/?q=ai:esfandian.nafisehSummary: In this manuscript, we investigate the approximate solutions to the tangent nonlinear packaging equation in the context of fractional calculus. It is an important equation because shock and vibrations are unavoidable circumstances for the packaged goods during transport from production plants to the consumer. We consider the fractal fractional Caputo operator and Atangana-Baleanu fractal fractional operator with nonsingular kernel to obtain the numerical consequences. Both fractal fractional techniques are equally good, but the Atangana-Baleanu Caputo method has an edge over Caputo method. For illustrations and clarity of our main results, we provided the numerical simulations of the approximate solutions and their physical interpretations. This paper contributes to the new applications of fractional calculus in packaging systems.Implicit nonlinear fractional differential equations of variable orderhttps://zbmath.org/1496.340262022-11-17T18:59:28.764376Z"Benkerrouche, Amar"https://zbmath.org/authors/?q=ai:benkerrouche.amar"Souid, Mohammed Said"https://zbmath.org/authors/?q=ai:souid.mohammed-said"Sitthithakerngkiet, Kanokwan"https://zbmath.org/authors/?q=ai:sitthithakerngkiet.kanokwan"Hakem, Ali"https://zbmath.org/authors/?q=ai:hakem.aliSummary: In this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.An adaptation of the modified decomposition method in solving nonlinear initial-boundary value problems for ODEshttps://zbmath.org/1496.340272022-11-17T18:59:28.764376Z"Bougoffa, Lazhar"https://zbmath.org/authors/?q=ai:bougoffa.lazhar"Rach, Randolph C."https://zbmath.org/authors/?q=ai:rach.randolph-cSummary: This paper considers an interesting variation of the modified decomposition method, which permits determination of the solution of nonlinear initial-boundary value problems for second-order ODEs appearing in physics such as the \textit{Thomas-Fermi}, \textit{Bratu} and \textit{Troesch equations}.The abstract Cauchy problem in locally convex spaceshttps://zbmath.org/1496.340282022-11-17T18:59:28.764376Z"Kruse, Karsten"https://zbmath.org/authors/?q=ai:kruse.karstenSummary: We derive necessary and sufficient criteria for the uniqueness and existence of solutions of the abstract Cauchy problem in locally convex Hausdorff spaces. Our approach is based on a suitable notion of an asymptotic Laplace transform and extends results of Langenbruch beyond the class of Fréchet spaces.Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearitieshttps://zbmath.org/1496.340292022-11-17T18:59:28.764376Z"Kuang, Juhong"https://zbmath.org/authors/?q=ai:kuang.juhong"Chen, Weiyi"https://zbmath.org/authors/?q=ai:chen.weiyi(no abstract)Embedded solitons in second-harmonic-generating latticeshttps://zbmath.org/1496.340302022-11-17T18:59:28.764376Z"Susanto, Hadi"https://zbmath.org/authors/?q=ai:susanto.hadi"Malomed, Boris A."https://zbmath.org/authors/?q=ai:malomed.boris-aSummary: Embedded solitons are exceptional modes in nonlinear-wave systems with the propagation constant falling in the system's propagation band. An especially challenging topic is seeking for such modes in nonlinear dynamical lattices (discrete systems). We address this problem for a system of coupled discrete equations modeling the light propagation in an array of tunnel-coupled waveguides with a combination of intrinsic quadratic (second-harmonic-generating) and cubic nonlinearities. Solutions for discrete embedded solitons (DESs) are constructed by means of two analytical approximations, adjusted, severally, for broad and narrow DESs, and in a systematic form with the help of numerical calculations. DESs of several types, including ones with identical and opposite signs of their fundamental-frequency and second-harmonic components, are produced. In the most relevant case of narrow DESs, the analytical approximation produces very accurate results, in comparison with the numerical findings. In this case, the DES branch extends from the propagation band into a semi-infinite gap as a family of regular discrete solitons. The study of stability of DESs demonstrates that, in addition to ones featuring the well-known property of semi-stability, there are linearly stable DESs which are genuinely robust modes.Extensions of Gronwall-Bellman type integral inequalities with two independent variableshttps://zbmath.org/1496.340312022-11-17T18:59:28.764376Z"Xie, Yihuai"https://zbmath.org/authors/?q=ai:xie.yihuai"Li, Yueyang"https://zbmath.org/authors/?q=ai:li.yueyang"Liu, Zhenhai"https://zbmath.org/authors/?q=ai:liu.zhenhaiSummary: In this paper, we establish several kinds of integral inequalities in two independent variables, which improve well-known versions of Gronwall-Bellman inequalities and extend them to fractional integral form. By using these inequalities, we can provide explicit bounds on unknown functions. The integral inequalities play an important role in the qualitative theory of differential and integral equations and partial differential equations.Local solvability and stability of the inverse problem for the non-self-adjoint Sturm-Liouville operatorhttps://zbmath.org/1496.340322022-11-17T18:59:28.764376Z"Bondarenko, Natalia P."https://zbmath.org/authors/?q=ai:bondarenko.natalia-pSummary: We consider the non-self-adjoint Sturm-Liouville operator on a finite interval. The inverse spectral problem is studied, which consists in recovering this operator from its eigenvalues and generalized weight numbers. We prove local solvability and stability of this inverse problem, relying on the method of spectral mappings. Possible splitting of multiple eigenvalues is taken into account.Uniqueness theorems for the impulsive Dirac operator with discontinuityhttps://zbmath.org/1496.340332022-11-17T18:59:28.764376Z"Zhang, Ran"https://zbmath.org/authors/?q=ai:zhang.ran.1|zhang.ran.2"Yang, Chuan-Fu"https://zbmath.org/authors/?q=ai:yang.chuanfuThe paper deals with an impulsive Dirac operator with discontinuity
\[ly:=-By^{\prime}+Q(x)y=\lambda\rho y,\qquad x\in(0,\pi),\] with the boundary conditions \[U(y):=y_{1}(0)\cos\alpha+y_{2}(0)\sin\alpha=0,\,\,V(y):=y_{1}(\pi)\cos\beta+y_{2}(\pi)\sin\beta=0\] and the jump conditions \[y_{1}(b+0)=a_{1}y_{1}(b-0),y_{2}(b+0)=a_{1}^{-1}y_{2}(b-0)+a_{2}y_{1}(b-0).\] Here \(\alpha\in(-\pi/2,\pi/2]\) and \(\beta\in(-\pi/2,\pi/2).\)
For studying the inverse problem for \(l\), the author introduced the new supplementary data to prove the uniqueness theorems. It is shown that the potential on the whole interval can be uniquely determined by these given data, which are the analogues of Borg, Marchenko and McLaughlin-Rundell theorems. The results in this paper can be viewed as the generalizing in [\textit{T. N. Harutyunyan}, Lobachevskii J. Math. 40, No. 10, 1489--1497 (2019; Zbl 1483.34028)].
Reviewer: Zhaoying Wei (Xi'an)On the solution set of a second-order integro-differential inclusionhttps://zbmath.org/1496.340342022-11-17T18:59:28.764376Z"Cernea, Aurelian"https://zbmath.org/authors/?q=ai:cernea.aurelianSummary: We consider a nonconvex and nonclosed second-order integro-differential inclusion and we prove the arcwise connectedness of the set of its mild solutions.Necessary and sufficient conditions for the nonincrease of scalar functions along solutions to constrained differential inclusionshttps://zbmath.org/1496.340352022-11-17T18:59:28.764376Z"Maghenem, Mohamed"https://zbmath.org/authors/?q=ai:maghenem.mohamed-adlene"Melis, Alessandro"https://zbmath.org/authors/?q=ai:melis.alessandro"Sanfelice, Ricardo G."https://zbmath.org/authors/?q=ai:sanfelice.ricardo-gThe paper provides necessary and sufficient conditions such that a given function \(B(.):{\mathbb{R}}^n\to \mathbb{R}\) is nonincreasing along the solutions of the following constrained differential inclusion
\[ x'\in F(x),\quad x(t)\in C \quad \forall \, t\in \mathrm{int}(\mathrm{dom} (x(t))), \]
where \(F(.):{\mathbb{R}}^n\to\mathcal{P}({\mathbb{R}}^n)\) is an upper semicontinuous set-valued map with compact convex values and \(C\subset {\mathbb{R}}^n\) is an arbitrary set.
Under certain technical hypotheses on the set \(C\) and on the function \(B(.)\) and under certain regularity hypotheses on the set-valued map \(F(.)\) such kind of results are obtained in three cases concerning the regularity of the function \(B(.)\). Namely, \(B(.)\) is asumed to be lower semicontinuous, locally Lipschitz and continuously differentiable.
Reviewer: Aurelian Cernea (Bucureşti)On the solution of boundary value problems for loaded ordinary differential equationshttps://zbmath.org/1496.340362022-11-17T18:59:28.764376Z"Providas, E."https://zbmath.org/authors/?q=ai:providas.efthimios|providas.efthinios"Parasidis, I. N."https://zbmath.org/authors/?q=ai:parasidis.ivan-nesterovich|parasidis.ioannis-nestoriosSummary: This chapter is devoted to the solution of the so-called \textit{loaded ordinary differential equations} which arise in applications in sciences and engineering. We propose a direct operator method for examining existence and uniqueness and constructing the solution in closed form to a class of boundary value problems for loaded \(n\)th-order ordinary differential equations with multipoint and integral boundary conditions.
For the entire collection see [Zbl 1483.00042].Secondary bifurcations in semilinear ordinary differential equationshttps://zbmath.org/1496.340372022-11-17T18:59:28.764376Z"Kan, Toru"https://zbmath.org/authors/?q=ai:kan.toruSummary: We consider the Neumann problem for the equation \(u_{xx}+\lambda f(u)=0\) in the punctured interval \((-1,1)\setminus \{0\}\), where \(\lambda >0\) is a bifurcation parameter and \(f(u)=u-u^3\). At \(x=0\), we impose the conditions \(u(-0)+au_x(-0)=u(+0)-au_x(+0)\) and \(u_x(-0)=u_x(+0)\) for a constant \(a > 0\) (the symbols \(+0\) and \(-0\) stand for one-sided limits). The problem appears as a limiting equation for a semilinear elliptic equation in a higher dimensional domain shrinking to the interval \((-1,1)\). First we prove that odd solutions and even solutions form families of branches \(\{\mathcal{C}^o_k\}_{k\in\mathbb{N}}\) and \(\{\mathcal{C}^e_k\}_{k\in\mathbb{N}}\), respectively. Both \(\mathcal{C}^o_k\) and \(\mathcal{C}^e_k\) bifurcate from the trivial solution \(u=0\). We then show that \(\mathcal{C}^e_k\) contains no other bifurcation point, while \(\mathcal{C}^o_k\) contains two points where secondary bifurcations occur. Finally we determine the Morse index of solutions on the branches. General conditions on \(f(u)\) for the same assertions to hold are also given.The dual eigenvalue problems of the conformable fractional Sturm-Liouville problemshttps://zbmath.org/1496.340382022-11-17T18:59:28.764376Z"Cheng, Yan-Hsiou"https://zbmath.org/authors/?q=ai:cheng.yan-hsiouSummary: In this paper, we are concerned with the eigenvalue gap and eigenvalue ratio of the Dirichlet conformable fractional Sturm-Liouville problems. We show that this kind of differential equation satisfies the Sturm-Liouville property by the Prüfer substitution. That is, the \(n\)th eigenfunction has \(n-1\) zero in \((0,\pi)\) for \(n\in \mathbb{N} \). Then, using the homotopy argument, we find the minimum of the first eigenvalue gap under the class of single-well potential functions and the first eigenvalue ratio under the class of single-barrier density functions. The result of the eigenvalue gap is different from the classical Sturm-Liouville problem.Inverse transmission eigenvalue problems for the Schrödinger operator with the Robin boundary conditionhttps://zbmath.org/1496.340392022-11-17T18:59:28.764376Z"Ma, Li-Jie"https://zbmath.org/authors/?q=ai:ma.lijie"Xu, Xiao-Chuan"https://zbmath.org/authors/?q=ai:xu.xiaochuanThe paper deals with the following transmission eigenvalue problem \(H(q,h)\)
\begin{align*} -\varphi^{\prime\prime}+q(x)\varphi=k^{2}\varphi,\qquad x\in(0,1),\\
-\varphi_{0}^{\prime\prime}=k^{2}\varphi_{0},\qquad x\in(0,1),\\
\varphi^{\prime}(0)+h\varphi(0)=0,\\
\varphi_{0}^{\prime}(0)+h_{0}\varphi_{0}(0)=0,\\
\varphi(1)=\varphi_{0}(1),\,\,\varphi^{\prime}(1)=\varphi_{0}^{\prime}(1). \end{align*}
Here \(k\) is the spectral parameter, \(q\) is the complex-valued potential \(q\in L_{2}(0,1)\), \(h_{0},h\in \mathbf{C}.\)
By considering the distinctness of the parameters \(h_{0},h\), the authors show that the potential \(q(x)\) and the parameter \(h\) can be uniquely determined by all transmission eigenvalues and a certain constant on the condition that \(h_{0}\) is known. This partially solves the open question raised by [\textit{T. Aktosun} and \textit{V. G. Papanicolaou}, Inverse Probl. 30, No. 7, Article ID 075001, 23 p. (2014; Zbl 1305.34145)]. Moreover, the authors prove the local solvability and stability of recovering \(h\) and \(q(x)\) in a particular set from the spectrum and the non-zero value \(q(1)\).
Reviewer: Zhaoying Wei (Xi'an)Boundary shape function method for nonlinear BVP, automatically satisfying prescribed multipoint boundary conditionshttps://zbmath.org/1496.340402022-11-17T18:59:28.764376Z"Liu, Chein-Shan"https://zbmath.org/authors/?q=ai:liu.chein-shan"Chang, Chih-Wen"https://zbmath.org/authors/?q=ai:chang.chih-wenSummary: It is difficult to exactly and automatically satisfy nonseparable multipoint boundary conditions by numerical methods. With this in mind, we develop a novel algorithm to find solution for a second-order nonlinear boundary value problem (BVP), which automatically satisfies the multipoint boundary conditions prescribed. A novel concept of boundary shape function (BSF) is introduced, whose existence is proven, and it can satisfy the multipoint boundary conditions a priori. In the BSF, there exists a free function, from which we can develop an iterative algorithm by letting the BSF be the solution of the BVP and the free function be another variable. Hence, the multipoint nonlinear BVP is properly transformed to an initial value problem for the new variable, whose initial conditions are given arbitrarily. The BSF method (BSFM) can find very accurate solution through a few iterations.Existence of solutions for functional boundary value problems at resonance on the half-linehttps://zbmath.org/1496.340412022-11-17T18:59:28.764376Z"Sun, Bingzhi"https://zbmath.org/authors/?q=ai:sun.bingzhi"Jiang, Weihua"https://zbmath.org/authors/?q=ai:jiang.weihuaSummary: By defining the Banach spaces endowed with the appropriate norm, constructing a suitable projection scheme, and using the coincidence degree theory due to Mawhin, we study the existence of solutions for functional boundary value problems at resonance on the half-line with \(\operatorname{dim}\operatorname{Ker}L = 1\). And an example is given to show that our result here is valid.Using an integrating factor to transform a second order BVP to a fixed point problemhttps://zbmath.org/1496.340422022-11-17T18:59:28.764376Z"Avery, Richard I."https://zbmath.org/authors/?q=ai:avery.richard-i"Anderson, Douglas R."https://zbmath.org/authors/?q=ai:anderson.douglas-robert"Henderson, Johnny"https://zbmath.org/authors/?q=ai:henderson.johnnySummary: Using an integrating factor, a second order boundary value problem is transformed into a fixed point problem. We provide growth conditions for the existence of a fixed point to the associated operator for this transformation and conclude that the index of the operator applying the standard Green's function approach is zero; this does not guarantee the existence of a solution, demonstrating the value and potential for this new transformation.
For the entire collection see [Zbl 1483.00042].On resonant mixed Caputo fractional differential equationshttps://zbmath.org/1496.340432022-11-17T18:59:28.764376Z"Guezane-Lakoud, Assia"https://zbmath.org/authors/?q=ai:guezane-lakoud.assia"Kılıçman, Adem"https://zbmath.org/authors/?q=ai:kilicman.ademSummary: The purpose of this study is to discuss the existence of solutions for a boundary value problem at resonance generated by a nonlinear differential equation involving both right and left Caputo fractional derivatives. The proofs of the existence of solutions are mainly based on Mawhin's coincidence degree theory. We provide an example to illustrate the main result.Existence results for first derivative dependent \(\varphi \)-Laplacian boundary value problemshttps://zbmath.org/1496.340442022-11-17T18:59:28.764376Z"Talib, Imran"https://zbmath.org/authors/?q=ai:talib.imran"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabetSummary: Our main concern in this article is to investigate the existence of solution for the boundary-value problem
\[
\begin{aligned}
& (\phi \bigl(x^\prime(t)\bigr)^\prime=g_1 \bigl(t,x(t),x^\prime(t)\bigr),\quad \forall t\in [0,1], \\
& \Upsilon_1\bigl(x(0),x(1),x^\prime(0)\bigr)=0, \\
& \Upsilon_2\bigl(x(0),x(1),x^\prime(1)\bigr)=0,
\end{aligned}
\] where \(g_1:[0,1]\times \mathbb{R}^2\rightarrow \mathbb{R}\) is an \(L^1\)-Carathéodory function, \( \Upsilon_i:\mathbb{R}^3\rightarrow \mathbb{R}\) are continuous functions, \(i=1,2\), and \(\phi :(-a,a)\rightarrow \mathbb{R}\) is an increasing homeomorphism such that \(\phi (0)=0\), for \(0< a< \infty \). We obtain the solvability results by imposing some new conditions on the boundary functions. The new conditions allow us to ensure the existence of at least one solution in the sector defined by well ordered functions. These ordered functions do not require one to check the definitions of lower and upper solutions. Moreover, the monotonicity assumptions on the arguments of boundary functions are not required in our case. An application is considered to ensure the applicability of our results.On the existence of solutions to boundary value problems for interval-valued differential equations under gH-differentiabilityhttps://zbmath.org/1496.340452022-11-17T18:59:28.764376Z"Wang, Hongzhou"https://zbmath.org/authors/?q=ai:wang.hongzhou"Rodríguez-López, Rosana"https://zbmath.org/authors/?q=ai:rodriguez-lopez.rosanaSummary: In this paper, we study first order interval-valued differential equations with the boundary value condition \(x(0) = \alpha x(T)\). By constructing proper operators, some sufficient conditions for the existence of solutions are provided for \(\alpha\in\mathbb{R}\backslash\{0, 1\}\), corresponding to non-periodic boundary value problems. At the end of this paper, some examples are shown to illustrate the theorems proved.Shooting method in the application of boundary value problems for differential equations with sign-changing weight functionhttps://zbmath.org/1496.340462022-11-17T18:59:28.764376Z"Yue, Xu"https://zbmath.org/authors/?q=ai:yue.xu"Xiaoling, Han"https://zbmath.org/authors/?q=ai:xiaoling.han(no abstract)Weak and strong singularities problems to Liénard equationhttps://zbmath.org/1496.340472022-11-17T18:59:28.764376Z"Xin, Yun"https://zbmath.org/authors/?q=ai:xin.yun"Hu, Guixin"https://zbmath.org/authors/?q=ai:hu.guixinSummary: This paper is devoted to an investigation of the existence of a positive periodic solution for the following singular Liénard equation:
\[
x^{\prime\prime}+f\bigl(x(t)\bigr)x^\prime(t)+a(t)x= \frac{b(t)}{x^{\alpha}}+e(t),
\] where the external force \(e(t)\) may change sign, \( \alpha\) is a constant and \(\alpha >0\). The novelty of the present article is that for the first time we show that weak and strong singularities enables the achievement of a new existence criterion of positive periodic solution through an application of the Manásevich-Mawhin continuation theorem. Recent results in the literature are generalized and significantly improved, and we give the existence interval of periodic solution of this equation. At last, two examples and numerical solution (phase portraits and time portraits of periodic solutions of the example) are given to show applications of the theorem.Existence and Ulam-Hyers stability of positive solutions for a nonlinear model for the antarctic circumpolar currenthttps://zbmath.org/1496.340482022-11-17T18:59:28.764376Z"Fečkan, Michal"https://zbmath.org/authors/?q=ai:feckan.michal"Li, Qixiang"https://zbmath.org/authors/?q=ai:li.qixiang"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrongAuthors' abstract: In this paper, we study the existence of positive solutions for the nonlinear model of the antarctic circumpolar current and analyze their Ulam-Hyers stability. By introducing some conditions on the ocean nonlinear vorticity function depending on other functions and initial values, we establish sufficient conditions to guarantee the existence, multiplicity, location and construction of positive solutions via the approach of fixed point theorem in cones, upper and lower solutions, and monotone method, respectively. Finally, we show the existence and uniqueness and Ulam-Hyers stability of positive solutions when the ocean nonlinear vorticity function has uniformly Lipschitz continuity.
Reviewer: Hanying Feng (Shijiazhuang)Radial regular and rupture solutions for a PDE problem with gradient term and two parametershttps://zbmath.org/1496.340492022-11-17T18:59:28.764376Z"Ghergu, Marius"https://zbmath.org/authors/?q=ai:ghergu.marius"Miyamoto, Yasuhito"https://zbmath.org/authors/?q=ai:miyamoto.yasuhitoThis paper is concerned with radial solutions to the problem:
\[-\Delta U=\frac{\lambda +\delta|\nabla U|^2}{1-U},\,U>0\text{ in }B,\text{ with }U=0\text{ on }\partial B,\] where \(B \subset \mathbb{R}^N\) \((N \ge 2)\) denotes an open unit ball and \(\lambda, \delta > 0\) are real numbers. The authors study regular and rupture radial solutions to the given problem satisfying \( 0 <U< 1\) in \(B\) and \( U(0) = 1\), respectively. It has been shown that there exist (a) infinitely many regular solutions for suitable values of \(\lambda\) and \(\delta\) (Theorem 1.2(ii)); (b) exactly one rupture solution for \(0 < \delta < N/2\) (Theorem 1.2 (i)); and (c) infinitely many rupture solutions for \(\delta \ge N/2\) with small positive values of \(\lambda\) (Theorem 1.6) to the problem at hand. Some preliminary results are also presented to establish the main theorems.
Reviewer: Bashir Ahmad (Jeddah)Existence-uniqueness of positive solutions to nonlinear impulsive fractional differential systems and optimal controlhttps://zbmath.org/1496.340502022-11-17T18:59:28.764376Z"Song, Shu"https://zbmath.org/authors/?q=ai:song.shu"Zhang, Lingling"https://zbmath.org/authors/?q=ai:zhang.lingling"Zhou, Bibo"https://zbmath.org/authors/?q=ai:zhou.bibo"Zhang, Nan"https://zbmath.org/authors/?q=ai:zhang.nan.1Summary: In this thesis, we investigate a kind of impulsive fractional order differential systems involving control terms. By using a class of \(\varphi \)-concave-convex mixed monotone operator fixed point theorem, we obtain a theorem on the existence and uniqueness of positive solutions for the impulsive fractional differential equation, and the optimal control problem of positive solutions is also studied. As applications, an example is offered to illustrate our main results.Multiple positive solutions for one dimensional third order \(p\)-Laplacian equations with integral boundary conditionshttps://zbmath.org/1496.340512022-11-17T18:59:28.764376Z"Yang, You-yuan"https://zbmath.org/authors/?q=ai:yang.youyuan"Wang, Qi-ru"https://zbmath.org/authors/?q=ai:wang.qiruThis paper is devoted to the following third order equation coupled to integral boundary conditions: \[ \left\{\begin{array}{l} \left(\Phi_{p}\left(u^{\prime \prime}\right)\right)^{\prime}+h(t) f(t, u(t))=0, \quad t \in(0,1), \\
u(0)-\alpha \,u^{\prime}(0)=\int_{0}^{1} g_{1}(s) u(s) d s, \\
u(1)+\beta \,u^{\prime}(1)=\int_{0}^{1} g_{2}(s) u(s) d s, \quad u^{\prime \prime}(0)=0. \end{array}\right. \] Here \(\alpha, \beta \geq 0\), \(p>1\), \(\Phi_{p}(u)=|u|^{p-2} u\) is called the one-dimensional \(p\)-Laplacian operator and \( \Phi_{p}^{-1}(u)=\Phi_{q}(u)=|u|^{q-2} u\) with \(\frac{1}{p}+\frac{1}{q}=1.\)
Under suitable assumptions on the function \(f\), the authors deduce the existence of at least three positive solutions of the considered problem. The results are obtained from Avery-Peterson fixed point theorem. The used arguments consist on the construction of the Green's function related to the linear problem
\[
u''(t)+y(t)=0, \; t \in(0,1), \quad u(0)-\alpha\, u^{\prime}(0)=0, \; u(1)+\beta\, u^{\prime}(1)=0,
\]
and recursive methods for kernel functions.
Reviewer: Alberto Cabada (Santiago de Compostela)Existence-uniqueness and monotone iteration of positive solutions to nonlinear tempered fractional differential equation with \(p\)-Laplacian operatorhttps://zbmath.org/1496.340522022-11-17T18:59:28.764376Z"Zhou, Bibo"https://zbmath.org/authors/?q=ai:zhou.bibo"Zhang, Lingling"https://zbmath.org/authors/?q=ai:zhang.lingling"Xing, Gaofeng"https://zbmath.org/authors/?q=ai:xing.gaofeng"Zhang, Nan"https://zbmath.org/authors/?q=ai:zhang.nan.1Summary: In this paper, without requiring the complete continuity of integral operators and the existence of upper-lower solutions, by means of the sum-type mixed monotone operator fixed point theorem based on the cone \(P_h\), we investigate a kind of \(p\)-Laplacian differential equation Riemann-Stieltjes integral boundary value problem involving a tempered fractional derivative. Not only the existence and uniqueness of positive solutions are obtained, but also we can construct successively sequences for approximating the unique positive solution. As an application of our fundamental aims, we offer a realistic example to illustrate the effectiveness and practicability of the main results.The existence of solutions for Sturm-Liouville differential equation with random impulses and boundary value problemshttps://zbmath.org/1496.340532022-11-17T18:59:28.764376Z"Li, Zihan"https://zbmath.org/authors/?q=ai:li.zihan"Shu, Xiao-Bao"https://zbmath.org/authors/?q=ai:shu.xiaobao"Miao, Tengyuan"https://zbmath.org/authors/?q=ai:miao.tengyuanSummary: In this article, we consider the existence of solutions to the Sturm-Liouville differential equation with random impulses and boundary value problems. We first study the Green function of the Sturm-Liouville differential equation with random impulses. Then, we get the equivalent integral equation of the random impulsive differential equation. Based on this integral equation, we use Dhage's fixed point theorem to prove the existence of solutions to the equation, and the theorem is extended to the general second order nonlinear random impulsive differential equations. Then we use the upper and lower solution method to give a monotonic iterative sequence of the generalized random impulsive Sturm-Liouville differential equations and prove that it is convergent. Finally, we give two concrete examples to verify the correctness of the results.On degenerate boundary conditions and finiteness of the spectrum of boundary value problemshttps://zbmath.org/1496.340542022-11-17T18:59:28.764376Z"Sadovnichii, Victor A."https://zbmath.org/authors/?q=ai:sadovnichii.viktor-antonovich"Sultanaev, Yaudat T."https://zbmath.org/authors/?q=ai:sultanaev.yaudat-talgatovich"Akhtyamov, Azamat M."https://zbmath.org/authors/?q=ai:akhtyamov.azamat-moukhtarovichSummary: It is shown that for the asymmetric diffusion operator the case when the characteristic determinant is identically equal to zero is impossible and the only possible degenerate boundary conditions are the Cauchy conditions. In the case of a symmetric diffusion operator, the characteristic determinant is identically equal to zero if and only if the boundary conditions are false-periodic boundary conditions and is identically equal to a constant other than zero if and only if its boundary conditions are generalized Cauchy conditions. All degenerate boundary conditions for a spectral problem with a third-order differential equation \(y^{\prime \prime \prime}(x) = \lambda y(x)\) are described. The general form of degenerate boundary conditions for the fourth-order differentiation operator \(D^4\) is found. Twelve classes of boundary value eigenvalue problems are described for the operator \(D^4\), the spectrum of which fills the entire complex plane. It is known that spectral problems whose spectrum fills the entire complex plane exist for differential equations of any even order. John Locker posed the following problem (eleventh problem): Are there similar problems for odd-order differential equations? A positive answer is given to this question. It is proved that spectral problems, the spectrum of which fills the entire complex plane, exist for differential equations of any odd order. Thus, the problem of John Locker is resolved. John Locker posed a problem (tenth problem): Can a spectral boundary value problem have a finite spectrum? Boundary value problems with a polynomial occurrence of a spectral parameter in a differential equation are considered. It is shown that the corresponding boundary value problem can have a predetermined finite spectrum in the case when the roots of the characteristic equation are multiple. If the roots of the characteristic equation are not multiple, then there can be no finite spectrum. Thus, John Locker's tenth problem is resolved.
For the entire collection see [Zbl 1483.00042].On strong singular fractional version of the Sturm-Liouville equationhttps://zbmath.org/1496.340552022-11-17T18:59:28.764376Z"Shabibi, Mehdi"https://zbmath.org/authors/?q=ai:shabibi.mehdi"Zada, Akbar"https://zbmath.org/authors/?q=ai:zada.akbar"Masiha, Hashem Parvaneh"https://zbmath.org/authors/?q=ai:masiha.hashem-parvaneh"Rezapour, Shahram"https://zbmath.org/authors/?q=ai:rezapour.shahramSummary: The Sturm-Liouville equation is among the significant differential equations having many applications, and a lot of researchers have studied it. Up to now, different versions of this equation have been reviewed, but one of its most attractive versions is its strong singular version. In this work, we investigate the existence of solutions for the strong singular version of the fractional Sturm-Liouville differential equation with multi-points integral boundary conditions. Also, the continuity depending on coefficients of the initial condition of the equation is examined. An example is proposed to demonstrate our main result.Some notes on conformable fractional Sturm-Liouville problemshttps://zbmath.org/1496.340562022-11-17T18:59:28.764376Z"Wang, Wei-Chuan"https://zbmath.org/authors/?q=ai:wang.wei-chuanSummary: The conformable fractional eigenvalue problem
\[
-D_x^{\alpha}D_x^{\alpha}y+q(x)y=\lambda \rho (x)y
\] is considered. We employ an easy and efficient method to derive its eigenvalue asymptotic expansion. On the basis of this result, we also investigate Ambarzumyan problems related to this eigenvalue problem as an application.Dependence of eigenvalues of Sturm-Liouville problems with eigenparameter-dependent boundary conditions and interface conditionshttps://zbmath.org/1496.340572022-11-17T18:59:28.764376Z"Zhang, Hai-Yan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Ao, Ji-jun"https://zbmath.org/authors/?q=ai:ao.jijun"Li, Meng-lei"https://zbmath.org/authors/?q=ai:li.meng-leiIn this paper, the authors study continuous dependence and differential expressions of eigenvalues for the second-order Sturm-Liouville equation \[ -\left ( p(x) y'(x) \right )' + q(x)y =\lambda w(x)y, \quad x \in J = [a,c) \cup (c, b], \tag{1}\] with the boundary conditions \begin{align*} (\alpha_1' \lambda - \alpha_1) y(a) - (\alpha_2' \lambda - \alpha_2) p y'(a) =0, \\
(\beta_1' \lambda + \beta_1) y(b) - (\beta_2' \lambda + \beta_2) p y'(b) =0, \tag{2}\end{align*} and the interface conditions \begin{align*} y(c+) - \alpha_3 y(c-) - \beta_3 p y'(c-) = 0, \\
p y'(c+) - \alpha_4 y(c-) - \beta_4 p y'(c-) = 0. \tag{3}\end{align*}
The authors prove that the eigenvalues for the equation depend not only continuously but also smoothly on the parameters of the equation: the coefficients, the boundary conditions, the interface conditions, as well as the endpoints. Finally, the authors list the derivative formulas of eigenvalues with respect to all the parameters.
Reviewer: Gang Meng (Beijing)Indefinite integrals from Wronskians and related linear second-order differential equationshttps://zbmath.org/1496.340582022-11-17T18:59:28.764376Z"Conway, John T."https://zbmath.org/authors/?q=ai:conway.john-thomasSummary: Many indefinite integrals are derived for Bessel functions and associated Legendre functions from particular transformations of their differential equations which are closely linked to Wronskians. A large portion of the results for Bessel functions is known, but all the results for associated Legendre functions appear to be new. The method can be applied to many other special functions. All results have been checked by differentiation using Mathematica.Existence of positive solutions for singular Dirichlet boundary value problems with impulse and derivative dependencehttps://zbmath.org/1496.340592022-11-17T18:59:28.764376Z"Jin, Fengfei"https://zbmath.org/authors/?q=ai:jin.fengfei"Yan, Baoqiang"https://zbmath.org/authors/?q=ai:yan.baoqiangSummary: In this paper, we present a theorem for some impulsive boundary problems with derivative dependence by the upper and lower solutions method. Using the theorem obtained, we consider the existence of positive solutions of some class of singular impulsive boundary problems.Two solutions to Kirchhoff-type fourth-order implusive elastic beam equationshttps://zbmath.org/1496.340602022-11-17T18:59:28.764376Z"Liu, Jian"https://zbmath.org/authors/?q=ai:liu.jian.1"Yu, Wenguang"https://zbmath.org/authors/?q=ai:yu.wenguangSummary: In this paper, the existence of two solutions for superlinear fourth-order impulsive elastic beam equations is obtained. We get two theorems via variational methods and corresponding two-critical-point theorems. Combining with the Newton-iterative method, an example is presented to illustrate the value of the obtained theorems.Correctness conditions for high-order differential equations with unbounded coefficientshttps://zbmath.org/1496.340612022-11-17T18:59:28.764376Z"Ospanov, Kordan N."https://zbmath.org/authors/?q=ai:ospanov.kordan-nauryzkhanovichSummary: We give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.Non-existence, existence, and uniqueness of limit cycles for a generalization of the van der Pol-Duffing and the Rayleigh-Duffing oscillatorshttps://zbmath.org/1496.340622022-11-17T18:59:28.764376Z"Cândido, Murilo R."https://zbmath.org/authors/?q=ai:candido.murilo-r"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Valls, Claudia"https://zbmath.org/authors/?q=ai:valls.claudiaConsider the planar polynomial system
\[
\frac{{dx}}{{dt}} = y, \quad \frac{{dy}}{{dt}} = -x -x^3-\varepsilon y(b+x^2+ay^2),\tag{1}
\]
where \( a, b, \varepsilon \) are real numbers satisfying \( a>0, \varepsilon>0 \). The authors prove that (1) has no limit cycle for \(b \ge 0 \) and a unique stable limit cycle for \(b<0\). Additionally they present the correponding global phase portraits in the Poincarè disk.
Reviewer: Klaus R. Schneider (Berlin)Global dynamics of hybrid van der Pol-Rayleigh oscillatorshttps://zbmath.org/1496.340632022-11-17T18:59:28.764376Z"Chen, Hebai"https://zbmath.org/authors/?q=ai:chen.hebai"Tang, Yilei"https://zbmath.org/authors/?q=ai:tang.yilei"Xiao, Dongmei"https://zbmath.org/authors/?q=ai:xiao.dongmeiConsider the planar polynomial system
\[
\frac{{dx}}{{dt}} = y, \quad \frac{{dy}}{{dt}} = -x -y( \alpha + x^2 +\beta y^2),\tag{1}
\]
where \( \alpha\) and \(\beta \) are real parameters. System (1) contains the van der Pol term \(x^2y\) and the Rayleigh term \( \beta y^3\); therefore, it is called the hybrid van der Pol-Rayleigh system. The authors study the global phase portrait of System (1) in dependence on the parameters \( \alpha \) and \( \beta \). For this goal they determine the bifurcation curves in the parameter plane which define 7 simply connected regions, where each region corresponds to a structurally stable phase portrait of System (1) in the Poincaré disk. The studies show that the hybrid van der Pol-Rayleigh system can have more than one limit cycle. Results about the location of these limit cycles are presented.
Reviewer: Klaus R. Schneider (Berlin)Periodic orbits bifurcating from a Hopf equilibrium of 2-dimensional polynomial Kolmogorov systems of arbitrary degreehttps://zbmath.org/1496.340642022-11-17T18:59:28.764376Z"Djedid, Djamila"https://zbmath.org/authors/?q=ai:djedid.djamila"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Makhlouf, Amar"https://zbmath.org/authors/?q=ai:makhlouf.amarSummary: A Hopf equilibrium of a differential system in \(\mathbb{R}^2\) is an equilibrium point whose linear part has eigenvalues \(\pm\omega i\) with \(\omega\neq 0\), where \(i=\sqrt{-1}\). We provide necessary and sufficient conditions for the existence of a limit cycle bifurcating from a Hopf equilibrium of 2-dimensional polynomial Kolmogorov systems of arbitrary degree. We provide an estimation of the bifurcating small limit cycle and also characterize the stability of this limit cycle.The center conditions for a perturbed cubic center via the fourth-order Melnikov functionhttps://zbmath.org/1496.340652022-11-17T18:59:28.764376Z"Asheghi, Rasoul"https://zbmath.org/authors/?q=ai:asheghi.rasoulSummary: In this paper, we first consider a cubic integrable system under general quadratic perturbations. We then study the Melnikov functions of the perturbed system up to the fourth order. Our studies show that the first four Melnikov functions are sufficient to obtain the center conditions for the perturbed system.On generation of a limit cycle from a separatrix loop of a sewn saddle-nodehttps://zbmath.org/1496.340662022-11-17T18:59:28.764376Z"Roĭtenberg, Vladimir Shleĭmovich"https://zbmath.org/authors/?q=ai:roitenberg.vladimir-shleimovichSummary: The article considers dynamical systems on the plane, defined by continuous piecewise smooth vector fields. Such systems are used as mathematical models of real processes with switching. An important task is to find the conditions for the generation of periodic trajectories when the parameters change. The paper describes the bifurcation of the birth of a periodic trajectory from the loop of the separatrix of a sewn saddle-node -- an analogue of the classical bifurcation of the separatrix loop of a saddle-node of a smooth dynamical system. Consider a one-parameter family \(\{ X_\varepsilon \}\) of continuous piecewise-smooth vector fields on the plane. Let \(z^0\) be a point on the switching line. Let's choose the local coordinates \(x,y\) in which \(z^0\) has zero coordinates, and the switching line is given by the equation \(y = 0\). Let the vector field \(X_0\) in a semi-neighborhood \(y \ge 0 (y \le 0)\) coincide with a smooth vector field \(X_0^+ (X_0^- )\), for which the point \(z^0\) is a stable rough node (rough saddle), and the proper subspaces of the matrix of the linear part of the field in \(z^0\) do not coincide with the straight line \(y = 0\). The singular point \(z^0\) is called a sewn saddle-node. There is a single trajectory \(L_0\) that is \(\alpha \)-limit to \(z^0 \) -- the outgoing separatrix of the point \(z^0 \). It is assumed that \(L_0\) is also \(\omega \)-limit to \(z^0\), and enters \(z^0\) in the leading direction of the node of the field \(X_0^+ \). For generic family, when the parameter \(\varepsilon\) changes, the sewn saddle-node either splits into a rough node and a rough saddle, or disappears. In the paper it is proved that in the latter case the only periodic trajectory of the field \(X_\varepsilon\) is generated from the contour \(L_0 \cup \{ z^0 \} \) -- a stable limit cycle.Lyapunov functions, Krasnosel'skii canonical domains, and the existence of Poisson bounded solutionshttps://zbmath.org/1496.340672022-11-17T18:59:28.764376Z"Lapin, K. S."https://zbmath.org/authors/?q=ai:lapin.kirill-sSummary: The notions of Poisson boundedness and Poisson partial boundedness of solutions of systems are introduced. Based on the Lyapunov function method and Krasnosel'skii's method of canonical domains, a sufficient condition for the existence of Poisson bounded solutions is obtained.Enhancing synchrony in asymmetrically weighted multiplex networkshttps://zbmath.org/1496.340682022-11-17T18:59:28.764376Z"Anwar, Md Sayeed"https://zbmath.org/authors/?q=ai:anwar.md-sayeed"Kundu, Srilena"https://zbmath.org/authors/?q=ai:kundu.srilena"Ghosh, Dibakar"https://zbmath.org/authors/?q=ai:ghosh.dibakarSummary: We uncover the transition scenarios for two different kinds of synchronization appearing in a multiplex network, namely intralayer and interlayer synchronization, considering different weighting mechanisms among the coupled oscillators. Particularly, we choose an unweighted Erdös-Rényi interaction topology and then the weights are associated to the edges depending on the ratio of the adjacent nodal degrees. While considering intralayer connectivity, whether stronger nodal influences are received from a high degree node or a low degree node, the networks are classified accordingly as hubs-attracting or hubs-repelling, respectively. The obtained synchronization phenomena considering these two networks are compared with that of the unweighted network. Our explorations reveal that the intralayer synchronization is greatly enhanced in case of hubs-attracting network whereas it is de-enhanced for hubs-repelling network compared to the unweighted network. The obtained numerical results are also verified by analytically deriving necessary stability conditions using master stability function approach. We also explain the synchrony phenomena through spectral analysis of the corresponding Laplacian matrices and find that higher heterogeneity among the input intensities of the nodes gives lesser synchronizability and vice-versa. Moreover, the sustainability of interlayer synchronization is also scrutinized against progressive removal of interlayer links following either increasing or decreasing degree sequence of the nodes.Dynamical robustness in a heterogeneous network of globally coupled nonlinear oscillatorshttps://zbmath.org/1496.340692022-11-17T18:59:28.764376Z"Gowthaman, I."https://zbmath.org/authors/?q=ai:gowthaman.i"Singh, Uday"https://zbmath.org/authors/?q=ai:singh.uday-pratap|singh.uday-narayan|singh.uday-partap|singh.uday-prasad"Chandrasekar, V. K."https://zbmath.org/authors/?q=ai:chandrasekar.v-k"Senthilkumar, D. V."https://zbmath.org/authors/?q=ai:senthilkumar.dharmapuri-vijayanSummary: Deterioration or failure of even a fraction of the microscopic constituents of a large class of networks leads to the loss of the macroscopic activity of the network as a whole. We deduce the evolution equation of two macroscopic order parameters from a globally coupled network of heterogeneous oscillators following the self-consistent field approach under the strong coupling limit. The macroscopic order parameter is used to classify the stable nontrivial steady state and the macroscopic oscillatory state of the network. We examine the dynamical robustness of the network by including a limiting factor that limits the degree of diffusion and a self-feedback factor in the network in addition to the heterogeneity of the network. The heterogeneity is introduced using the parameter specifying the distance from the Hopf bifurcation, which is drawn from a random statistical distribution. We also deduce the critical stability curves from the evolution equation of the macroscopic order parameters demarcating the stable nontrivial steady state and the macroscopic oscillatory state in the system parameter space. We show that a large heterogeneity and a large self-feedback factor facilitates the onset of the stable macroscopic oscillatory state by destabilizing the aging transition state, whereas limiting the degree of the diffusion favors the sustained macroscopic oscillation of the heterogeneous network.Emergence of extreme events in coupled systems with time-dependent interactionshttps://zbmath.org/1496.340702022-11-17T18:59:28.764376Z"Kumarasamy, Suresh"https://zbmath.org/authors/?q=ai:kumarasamy.suresh"Srinivasan, Sabarathinam"https://zbmath.org/authors/?q=ai:srinivasan.sabarathinam"Gogoi, Pragjyotish Bhuyan"https://zbmath.org/authors/?q=ai:gogoi.pragjyotish-bhuyan"Prasad, Awadhesh"https://zbmath.org/authors/?q=ai:prasad.awadheshDynamics of two or more nonlinear interacting systems being a subject of this paper is a very interesting topic. Usually a general solution to equations of motion of such a system cannot be found in symbolic form and so numerical methods are applied. In this work, the system of two Stuart-Landau (SL) oscillators in the presence of time-dependent interactions is studied and Mathematica software is used to find numerical solutions to the equations of motion for some chosen values of the system parameters. The authors consider two cases of time-varying coupling strengths. In the first case parameter determining the interactions is a periodic function of time and so the interaction depends on time explicitly. In the second case parameter determining the interactions depends on the distance or coordinates which are functions of time and so such a case is considered as ``implicit time dependence''. Note that in the first case interaction of two SL oscillators depends on the parameter $d=1+\gamma$ that is a function of time because $\gamma=f\cos [[(\Omega t)]]$. As the parameter \(d\) appears in the denominator of the term describing this interaction becomes very large for $f=0.999$ when $\cos [[(\Omega t)]]=-1$ or $\gamma=-0.999$. So it is quite natural that large amplitude oscillations may occur that are considered as extreme events, and this results are confirmed by the corresponding numerical solutions of the equations of motion. At the same time the authors introduce the ``quasi-stable equilibrium points'' a sense of which is not clear. It seems such equilibrium points are determined as solutions of the equations which are obtained if the right-hand sides of equations (3) are equated to zero. But these equations contain function of time $\gamma=f\cos [[(\Omega t)]]$ and so the corresponding ``equilibrium points'' are functions of time, as well. The question arises what is the sense of such solutions and what does it mean stability of such ``equilibrium points''? The authors equate the Jacobian matrix (5) to zero and state that ``solving this equation, we get the stability of the equilibrium points in the systems''. This statement is very strange because the Jacobian matrix is a function of gamma and so it is a function of time. So it would be good to give a definition of stability which is used to define the corresponding criteriums of stability. This can help to understand the results and to reproduce all the calculations if a reader will be interested in their checking. Summarizing, one can state that the problem is very interesting but the results obtained are very questionable and must be checked carefully.
Reviewer: Alexander Prokopenya (Warszawa)Explosive and semi-explosive death in coupled oscillatorshttps://zbmath.org/1496.340712022-11-17T18:59:28.764376Z"Sun, Zhongkui"https://zbmath.org/authors/?q=ai:sun.zhongkui"Liu, Shutong"https://zbmath.org/authors/?q=ai:liu.shutong"Zhao, Nannan"https://zbmath.org/authors/?q=ai:zhao.nannanSummary: In this paper, we propose a new kind of transition process from oscillation to death state that is different from the traditional explosive death called semi-explosive death. In this process, two kinds of death arise in different ways. We investigate the occurrence of semi-explosive death transition for the first time on globally conjugate-coupled Van der Pol (VdP) oscillators with asymmetry factor. Semi-explosive death is an irreversible transition of half first order and half second order. That is, this process not only happens with an abrupt, irreversible transition that is a common feature of first order phase transition, but also includes a continuous second order change. Moreover, the forward and the backward second order transition points for this transition have been obtained theoretically, which is in complete agreement with the numerical results. Finally, the details of the transition mechanisms between semi-explosive death and explosive death along with dependence of asymmetry factor and damping coefficient are also discussed theoretically and numerically.Bifurcation of symmetric solutions for the sublinear Moore-Nehari differential equationhttps://zbmath.org/1496.340722022-11-17T18:59:28.764376Z"Kajikiya, Ryuji"https://zbmath.org/authors/?q=ai:kajikiya.ryujiIn this paper, the author considers the bifurcation of symmetric nodal solutions for the sublinear Moore-Nehari differential equation \[u^{\prime\prime}+h\left( x,\lambda\right) \left\vert u\right\vert ^{p-1}u=0,~u\left( -1\right) =u\left( 1\right) =0,\tag{1.1} \] where \(0<p<1,\) \(h\left( x,\lambda\right) =0\) for \(\left\vert x\right\vert <\lambda\) and \(h\left( x,\lambda\right) =1\) for \(\lambda\leq\left\vert x\right\vert \leq1\) and \(\lambda\in\left( 0,1\right) \) is a bifurcation parameter. Denote the set of all solutions \(\left( \lambda,u\right) \) for (1.1) by \(S\) and define \(S_{n}:=\left\{ \left( \lambda,u_{n}\left( x,\lambda\right) \right) :~0<\lambda<1\right\} .\) Then the author gives the following main theorem:
Theorem. Let \(0<p<1\) and let \(n\) be a non-negative integer. If \(n\) is even, then \(S_{n}\) does not bifurcate. Let \(n\) be an odd integer. Then there exists a unique number \(\lambda_{\ast}\left( n\right) \in\left( 0,1\right) \) such that \(S_{n}\) bifurcates at \(\lambda=\lambda_{\ast}\left( n\right) \) only and it does not have any other bifurcation point. We denote the set of all solutions for (1.1) by \(S\) and put \(\lambda_{\ast}:=\lambda_{\ast}\left( n\right) \) and \(u_{\ast}:=u_{n}\left( x,\lambda_{\ast}\left( n\right) \right) \). Then there exists a subset \(C_{n}\) of \(S\) satisfying the conditions below:
(i) \(C_{n}\) bifurcates from \(S_{n}\) at \((\lambda_{\ast},u_{\ast})\).
(ii) \(C_{n}\) is closed, connected and bounded in \((0,1)\times C_{0} ^{1}[-1,1].\)
(iii) \(C_{n}\) \(\cup\) \(S_{n}\) is a connected component of \(S\) having \((\lambda_{\ast},u_{\ast})\).
(iv) \(C_{n}\) \(\cap\) \(S_{n}=\{(\lambda_{\ast},u_{\ast})\}.\)
(v) Put \(n=2m-1\) with \(m\geq1\). Let \(C_{m,m-1}\left( C_{m-1,m}\right) \) denote the set of \((\lambda,u)\in C_{2m-1}\) such that \(u\) is an \((m,m-1)\)-solution (\((m-1,m)\)-solution, respectively). Then it holds that \(\ C_{2m-1}=C_{m,m-1}\cup C_{m-1,m}\cup\{(\lambda_{\ast},u_{\ast})\}\). Therefore, each point in \(C_{2m-1}\backslash\) \(\{(\lambda_{\ast},u_{\ast})\}\) is either an \((m,m-1)\)-solution or an \((m-1,m)\)-solution of (1.1).
(vi) If \((\lambda,u)\in C_{2m-1}\), then \((\lambda,-u(-x))\in C_{2m-1}\). Hence, \((\lambda,u)\) is in \(C_{m,m-1}\) if and only if \((\lambda,-u(-x))\) lies in \(C_{m-1,m}\).
(vii) For any \(\lambda\in(\lambda_{\ast},1)\), there exist points \(u,v\) such that \((\lambda,u)\in C_{m,m-1}\) and \((\lambda,v)\in\) \(C_{m-1,m}\).
(viii) For any \((\lambda,u)\in\) \(C_{n}\), \(u^{\prime}\left( 0\right) \) is positive.
(ix) \(\left\Vert u\right\Vert _{C^{1}}\rightarrow0\) as \(\lambda\rightarrow1\) with \((\lambda,u)\in S\). In particular, this assertion is valid for \((\lambda,u)\in\) \(C_{n}\) also.
The author gives some preliminary lemmas and properties for the proof and then proves the above theorem.
Reviewer: Fatma Hıra (Atakum)A missing generic local fold-fold bifurcation in planar Filippov systemshttps://zbmath.org/1496.340732022-11-17T18:59:28.764376Z"Siller, Tiago E."https://zbmath.org/authors/?q=ai:siller.tiago-eIn this paper, the author present a new generic local bifurcation of the visible-invisible two-fold type in a class of planar Filippov systems. The normal form of this type of bifurcation is given by \begin{eqnarray*} \left( \begin{array}{c} \dot{x} \\
\dot{y} \end{array} \right) = Z(x, y)= \left
\{ \begin{array}{ll} \left( \begin{array}{c} 1 \\
\lambda_1 x \end{array} \right) & \mbox{if } y> 0, \\
\left( \begin{array}{c} -1 \\
-\lambda_2 (x-\alpha) \end{array} \right) & \mbox{if } y< 0, \end{array} \right. \end{eqnarray*}
where \(\lambda_1>0\) and \(\lambda_2>0\) with the constraints: \[ \frac{1}{2}<\frac{\lambda_1}{\lambda_2}<1. \] This system has two tangency points at \((0, 0)\) and \((\alpha, 0)\) that bound a crossing region and a pseudonode for \(\alpha>0\). These three points collide and the crossing region contracts until it disappears at \(\alpha=0\). Then the pseudonode and the crossing region reappear when \(\alpha<0\). This case of bifurcation phenomena has not been reported in previous works.
Reviewer: Zhengdong Du (Chengdu)Limit cycles appearing from piecewise smooth perturbations to a reversible nonlinear centerhttps://zbmath.org/1496.340742022-11-17T18:59:28.764376Z"Sun, Dan"https://zbmath.org/authors/?q=ai:sun.dan.1"Peng, Linping"https://zbmath.org/authors/?q=ai:peng.linpingIn this paper, the authors investigate the maximum number of limit cycles of a class of planar piecewise smooth polynomial differential systems of the following form \begin{eqnarray*} \left( \begin{array}{c} \dot{x} \\
\dot{y} \end{array} \right) = \left( \begin{array}{c} -y+xy(x^2+y^2)^n\\
x+y^2(x^2+y^2)^n \end{array} \right)+\varepsilon \left \{ \begin{array}{ll} \left( \begin{array}{c} p_1(x,y) \\
q_1(x,y) \end{array} \right) & \mbox{if } y> 0, \\
\left( \begin{array}{c} p_2(x,y) \\
q_2(x,y) \end{array} \right) & \mbox{if } y< 0, \end{array} \right. \end{eqnarray*} where \(p_k(x,y)\) and \(q_k(x,y)\) (\(k=1, 2\)) are homogeneous polynomials of degree \(2n+2\), \(n\) is a positive integer, \(\varepsilon\) is a small parameter. When \(\varepsilon=0\), the unperturbed system has a period annulus which starts at the center \((0, 0)\) and terminates with the separatrix passing the infinite degenerate singularity on the equator. By using the averaging theory for discontinuous systems and the complex method based on the Argument Principle, the authors obtain the lower and the upper bounds for the maximum number of limit cycles bifurcating from the period annulus and the center \((0, 0)\) for \(|\varepsilon|\) sufficiently small.
Reviewer: Zhengdong Du (Chengdu)On Landesman-Lazer conditions and the fundamental theorem of algebrahttps://zbmath.org/1496.340752022-11-17T18:59:28.764376Z"Amster, Pablo"https://zbmath.org/authors/?q=ai:amster.pabloIn this paper, the author deals with the differential system
\[
u'(t)+g(u(t))=p(t), \tag{1}
\]
where \(g:\mathbb{R}^2 \rightarrow \mathbb{R}^2\) is bounded and \(p\) is continuous and \(T\)-periodic. Two results for the existence of at least one \(T\)-periodic solution for system (1) are obtained when \(g\) satisfies Landesman-Lazer type conditions. The connection of the second result with the fundamental theorem of algebra is stated.
Furthermore, the author treats the following delay systems
\[
u'(t)=g(u(t))+p(t,u(t),u(t-\tau)), \tag{2}
\]
where \(\tau>0\) and \(p\) is bounded, continuous and \(T\)-periodic in the first coordinate. Under similar conditions, two theorems for the existence of at least one \(T\)-periodic solution for system (2) are proved.
Reviewer: Chun Li (Chongqing)On the existence of periodic solutions to one class of systems of nonlinear differential equationshttps://zbmath.org/1496.340762022-11-17T18:59:28.764376Z"Demidenko, G. V."https://zbmath.org/authors/?q=ai:demidenko.gennadii-vThe author investigates the existence of periodic solutions for a class of nonlinear ordinary differential equations. The exponential dichotomy of the linear part is also considered. Stability criteria for the periodic state are also proposed.
Reviewer: Gani T. Stamov (Sliven)Subharmonic solutions in reversible non-autonomous differential equationshttps://zbmath.org/1496.340772022-11-17T18:59:28.764376Z"Eze, Izuchukwu"https://zbmath.org/authors/?q=ai:eze.izuchukwu"García-Azpeitia, Carlos"https://zbmath.org/authors/?q=ai:garcia-azpeitia.carlos"Krawcewicz, Wieslaw"https://zbmath.org/authors/?q=ai:krawcewicz.wieslaw-z"Lv, Yanli"https://zbmath.org/authors/?q=ai:lv.yanliLet \(p>0\) be a fixed number. The authors are interested in subharmonic solutions of the system
\[
\ddot{u}(t) = f(t,u(t)),\ u(t)\in V
\]
where \(f(t,u)\) is a continuous map, \(p\)-periodic with respect to the temporal variable. More precisely, let \(V := \mathbb{R}^k\) and let \(p = 2 \pi\) without loss of generality. Assume that \(f: \mathbb{R}\times V \rightarrow V\) is a continuous function satisfying the following symmetry conditions:
\begin{enumerate}
\item[(\(S_1\))] For all \(t \in \mathbb{R}\) and \(x \in V\) we have \(f(t+2\pi,x) = f(t,x)\) (\textit{dihedral symmetry});
\item[(\(S_2\))] For all \(t \in \mathbb{R}\) and \(x \in V\) we have \(f(-t,x) = f(t,x)\) (\textit{time-reversibility});
\item[(\(S_3\))] For all \(t \in \mathbb{R}\) and \(x \in V\) we have \(f(t,-x) = -f(t,x)\) (\textit{antipodal \(\mathbb{Z}_2\)-symmetry}).
\end{enumerate}
The symmetric properties of the system of study allow reformulation of the problem of existence of the subharmonic \(2\pi m\)-periodic solutions as a question about the operator equation \(\mathcal{F}(u)=0\) with \(D_m\times \mathbb{Z}_2\)-symmetries in the functional space \(\mathcal{E} := C_{2\pi m}^2(\mathbb{R};V)\). The authors introduce an additional symmetry to the system of study before proving several results on the existence and multiplicity of subharmonic solutions. Namely, let \(\Gamma\) be a finite group then
\begin{enumerate}
\item[(\(S_4\))] For all \(t \in \mathbb{R}\), \(x \in V\), and \(\sigma \in \Gamma\), we have \(f(t,\sigma x) = \sigma f(t,x)\) (\textit{\(\Gamma\)-equivariant}).
\end{enumerate}
The last condition allows for the restatement of the original problem as the \(G\)-equivariant operator equation with respect to the full group
\[
G := \Gamma \times D_m \times \mathbb{Z}_2.
\]
If the isotropy group \(G_u\) of a solution \(u\) satisfies \(\{ e \} \times \mathbb{Z}_m \times\{ 1 \} \nleq G_u\), then \(u\) is a subharmonic solution.
The authors prove several novel results in the paper. Most notably Theorems 2.6 and 2.10. The main technical tool is Brower \(\textbf{G}\)-equivariant degree theory. Given a group \(G\) corresponding \(\textbf{G}\)-equivariant Brower degree is computed using the computer algebra system GAP. In addition, the authors discuss the bifurcation problem of subharmonic solutions in the case of a system depending on an extra parameter \(\alpha\). The paper is clear and easy to follow.
Reviewer: Predrag Punosevac (Pittsburgh)Existence of positive \(S\)-asymptotically periodic solutions of the fractional evolution equations in ordered Banach spaceshttps://zbmath.org/1496.340782022-11-17T18:59:28.764376Z"Li, Qiang"https://zbmath.org/authors/?q=ai:li.qiang.3|li.qiang.1|li.qiang.4|li.qiang.2"Liu, Lishan"https://zbmath.org/authors/?q=ai:liu.lishan|liu.lishan.1"Wei, Mei"https://zbmath.org/authors/?q=ai:wei.meiIn this article, there is considered the following initial value problem for abstract fractional evolution equations
\[\begin{split} & ^C D^q_t u(t) + A u(t)= F(t,u(t)), \quad t \geq 0 , \\
& u(0)=u_0, \end{split}\tag{1} \] where \((E,||.||)\) is an ordered Banach space, whose positive cone \(K = \{ x \in E : x \geq \theta \}\) is a normal cone with normal constant \(N\), \(\theta\) is the zero element of \(E\), \(^C D^q_t\) is the Caputo fractional derivative of order \(q \in (0,1)\) with lower terminal zero, \(A : D(A) \subset E \to E\) is a closed linear (not necessarily bounded) operator, \(A\) generates a \(C_0\)-semigroup \(T(t) (t \geq 0)\) in \(E\) and \(F : \mathbb{R}^+ \times E \to E\) is a given continuous function.
The main results of this article are obtained sufficient conditions for the existence of mild positive solutions \(u(t)\) of (1), which are \(S\)-asymptotically \(\omega\)-periodic, i.e. \(u(t)\) is continuous, bounded and there exists an asymptotic period \(\omega >0\) of \(u\), such that \(\lim_{t \to \infty} || u(t+\omega)-u(t)||=0\).
The results are obtained without assuming the existence of upper and lower solutions, by using monotone iterative method and a fixed point theorem. It can be noted that Lipschitz condition is no longer needed, which makes the results more applicable.
An example illustrating the obtained results is given too.
Reviewer: Hristo S. Kiskinov (Plovdiv)Dynamics of a new multistable 4D hyperchaotic Lorenz system and its applicationshttps://zbmath.org/1496.340792022-11-17T18:59:28.764376Z"Leutcho, Gervais Dolvis"https://zbmath.org/authors/?q=ai:leutcho.gervais-dolvis"Wang, Huihai"https://zbmath.org/authors/?q=ai:wang.huihai"Fozin, Theophile Fonzin"https://zbmath.org/authors/?q=ai:fozin.theophile-fonzin"Sun, Kehui"https://zbmath.org/authors/?q=ai:sun.kehui"Njitacke, Zeric Tabekoueng"https://zbmath.org/authors/?q=ai:njitacke.zeric-tabekoueng"Kengne, Jacques"https://zbmath.org/authors/?q=ai:kengne.jacquesBased on the three-dimensional Lorenz system, the authors constructs a new four-dimensional hyperchaotic system
\begin{align*} & \dot{x}=a\left( y-x \right), \\
& \dot{y}=bx-y-xz+u, \\
& \dot{z}=-cz+{{x}^{2}}, \\
& \dot{u}=-dx\left| x \right|+u, \\
\end{align*}
where \(a,b,c,d\) are positive constants. The authors study the dynamics of the new system. The results show that the system has three unstable equilibrium points and exhibits complex non-linear phenomena such as symmetry breaking, torus, chaos, hyperchaos, and heterogeneous multistability. In addition, a set of coexisting attractors has been found, consisting of both periodic and chaotic attractors. The implementation of the new hyperchaotic system based on a digital signal processor has been made. The proposed hyperchaotic system is used to construct an image encryption algorithm. Standard security analysis tools show that the proposed algorithm is effective.
Reviewer: Eduard Musafirov (Grodno)Impulsive conformable fractional stochastic differential equations with Poisson jumpshttps://zbmath.org/1496.340802022-11-17T18:59:28.764376Z"Ahmed, Hamdy M."https://zbmath.org/authors/?q=ai:ahmed.hamdy-mSummary: In this article, periodic averaging method for impulsive conformable fractional stochastic differential equations with Poisson jumps are discussed. By using stochastic analysis, fractional calculus, Doob's martingale inequality and Cauchy-Schwarz inequality, we show that the solution of the conformable fractional impulsive stochastic differential equations with Poisson jumps converges to the corresponding averaged conformable fractional stochastic differential equations with Poisson jumps and without impulses.On asymptotically ideal invariant equivalence of double sequenceshttps://zbmath.org/1496.340812022-11-17T18:59:28.764376Z"Dündar, Erdinç"https://zbmath.org/authors/?q=ai:dundar.erdinc"Ulusu, Uğur"https://zbmath.org/authors/?q=ai:ulusu.ugur"Nuray, Fatih"https://zbmath.org/authors/?q=ai:nuray.fatihSummary: In this study, the concepts of asymptotically \(\mathcal{I}^{\sigma}_2\)-equivalent, asymptotically invariant equivalent, strongly asymptotically invariant equivalent and \(p\)-strongly asymptotically invariant equivalent for double sequences are defined. Also, we investigate relationships among these new type equivalence concepts.Existence of an invariant foliation near a locally integral surface of neutral typehttps://zbmath.org/1496.340822022-11-17T18:59:28.764376Z"Il'in, Yu."https://zbmath.org/authors/?q=ai:ilin.yurii-anatolevich|ilin.yuri-aThe author investigates a class of nonlinear differential systems without linear terms on the right-hand side in a neighbourhood of the equilibrium point. Criteria for the existence of a foliation into surfaces of stable type in some neighborhood of a neutral surface are established. The main results are obtained by using logarithmic norms of the Jacobi matrices of the right-hand sides.
Reviewer: Gani T. Stamov (Sliven)Solutions of fractional Verhulst model by modified analytical and numerical approacheshttps://zbmath.org/1496.340832022-11-17T18:59:28.764376Z"Hasan, Shatha"https://zbmath.org/authors/?q=ai:hasan.shatha"Hadid, Samir"https://zbmath.org/authors/?q=ai:hadid.samir-b"Al-Smadi, Mohammed"https://zbmath.org/authors/?q=ai:al-smadi.mohammed-h"Arqub, Omar Abu"https://zbmath.org/authors/?q=ai:arqub.omar-abu"Momani, Shaher"https://zbmath.org/authors/?q=ai:momani.shaher-mSummary: In this chapter, we are interested in the fractional release of the Verhulst model according to Caputo's sense, which is popular in applying environmental, biological, chemical and social studies describing the population growth model. Such a model, which is sometimes called logistic growth model related to systems in which the rate of change depends on their previous memory. In the light of this, three advanced numerical and analytical algorithms are presented to obtain approximate solutions for different classes of logistical growth problems, including reproducing kernel algorithm, fractional residual series algorithm and successive substitutions algorithm. The first technique relies on the reproducing property that characterises a specific function of building a complete orthogonal system at desired Hilbert spaces. The RPS technique relies on residual error function and generalised Taylor series to reduce residual errors and generate a converging power series, while the last technique converts the fractional logistic model to Volterra integral equation based on Riemann-Liouville integral operator. To demonstrate consistency with the theoretical framework, some realistic applications are tested to show the accuracy and efficiency of the proposed schemes. Numerical results are displayed in tables and figures for different fractional orders to illustrate the effect of the fractional parameter on population growth behaviour. The results confirm that the proposed schemes are very convenient, effective and do not require long-term calculations.
For the entire collection see [Zbl 1464.65006].In-phase and anti-phase spikes synchronization within mixed bursters of the pre-Bözinger complexhttps://zbmath.org/1496.340842022-11-17T18:59:28.764376Z"Liu, Moutian"https://zbmath.org/authors/?q=ai:liu.moutian"Duan, Lixia"https://zbmath.org/authors/?q=ai:duan.lixiaSummary: In this paper, the transition from anti-phase spike synchronization to in-phase spike synchronization within mixed bursters is investigated in a two-coupled pre-Bözinger complex (pre-BötC) network. In this two-coupled neuronal network, the communication between two pre-BötC networks is based on electrical and synaptic coupling. The results show that the electrical coupling accelerates in-phase spike synchronization within mixed bursters, but synaptic coupling postpones this kind of synchronization. Synaptic coupling promotes anti-phase spike synchronization when electrical coupling is weak. At the same time, the in-phase spike synchronization within dendritic bursters occurs earlier than that within somatic bursters. Asymmetric periodic somatic bursters appear in the transition state from anti-phase spikes to in-phase spikes. We also use fast/slow decomposition and bifurcation analysis to clarify the dynamic mechanism for the two types of synchronization.Dynamics of a nonlinear differential advertising model with single parameter sales promotion strategyhttps://zbmath.org/1496.340852022-11-17T18:59:28.764376Z"Ma, Junhai"https://zbmath.org/authors/?q=ai:ma.junhai"Jiang, Hui"https://zbmath.org/authors/?q=ai:jiang.huiSummary: Advertising and sales promotion are two important specific marketing communications tools. In this paper, nonlinear differential equation and single parameter sales promotion strategy are introduced into an advertising model and investigated quantitatively. The existence and stability of period-\(nT\) (\(n = 1, 2, 4, 8\)) solutions are investigated. Interestingly, both period doubling bifurcation and inverse flip bifurcation occur at different parameter values in the same advertising model. The results show that the system enters into chaos from stable state through flip bifurcation and enters into stable state from chaos through inverse flip bifurcation. An effective control strategy, which suppresses flip bifurcation and promotes inverse flip bifurcation, is proposed to eliminate chaos. These results have some significant theoretical and practical value in related markets.Analysis of a class of Lotka-Volterra systemshttps://zbmath.org/1496.340862022-11-17T18:59:28.764376Z"Moza, G."https://zbmath.org/authors/?q=ai:moza.gheorghe"Constantinescu, D."https://zbmath.org/authors/?q=ai:constantinescu.constanta-dana"Efrem, R."https://zbmath.org/authors/?q=ai:efrem.raluca"Bucur, L."https://zbmath.org/authors/?q=ai:bucur.laurentiu|bucur.liliana-maria"Constantinescu, R."https://zbmath.org/authors/?q=ai:constantinescu.radu-dThe authors study the dynamics of the two-dimensional Lotka-Volterra system
\begin{align*} & \dot{x}=x{{P}_{1}}(\mu ,x,y), \\
& \dot{y}=y{{P}_{2}}(\mu ,x,y), \\
\end{align*}
where \(x\ge 0\), \(y\ge 0\), \({{P}_{i}}(\mu ,x,y)={{\mu }_{i}}+{{p}_{i1}}x+{{p}_{i2}}y+{{p}_{i3}}xy+{{p}_{i4}}{{x}^{2}}+{{p}_{i5}}{{y}^{2}}\), \({{p}_{ij}}={{p}_{ij}}(\mu )\) are smooth functions of variable \(({{\mu }_{1}},{{\mu }_{2}})\in {{\mathbb{R}}^{2}}\) such that \({{p}_{12}}(0){{p}_{21}}(0)\ne 0\), \(\left| {{\mu }_{1}} \right|\) and \(\left| {{\mu }_{2}} \right|\) are small. The local behavior of the model is studied in two different degenerate cases. Sixteen different bifurcation diagrams with forty different regions are presented, describing the behavior of the model in these cases.
Reviewer: Eduard Musafirov (Grodno)Analytic solution of the SEIR epidemic model via asymptotic approximanthttps://zbmath.org/1496.340872022-11-17T18:59:28.764376Z"Weinstein, Steven J."https://zbmath.org/authors/?q=ai:weinstein.steven-j"Holland, Morgan S."https://zbmath.org/authors/?q=ai:holland.morgan-s"Rogers, Kelly E."https://zbmath.org/authors/?q=ai:rogers.kelly-e"Barlow, Nathaniel S."https://zbmath.org/authors/?q=ai:barlow.nathaniel-sSummary: An analytic solution is obtained to the SEIR Epidemic Model. The solution is created by constructing a single second-order nonlinear differential equation in \(\ln S\) and analytically continuing its divergent power series solution such that it matches the correct long-time exponential damping of the epidemic model. This is achieved through an asymptotic approximant [\textit{N. S. Barlow} et al., Classical Quantum Gravity 34, No. 13, Article ID 135017, 16 p. (2017; Zbl 1367.83019)] in the form of a modified symmetric Padé approximant that incorporates this damping. The utility of the analytical form is demonstrated through its application to the COVID-19 pandemic.Bifurcation analysis of an ecological model with nonlinear state-dependent feedback control by Poincaré map defined in phase sethttps://zbmath.org/1496.340882022-11-17T18:59:28.764376Z"Zhang, Qianqian"https://zbmath.org/authors/?q=ai:zhang.qianqian"Tang, Sanyi"https://zbmath.org/authors/?q=ai:tang.sanyiIn this paper, a predator-prey model with state-dependent pulse interventions, including spraying insecticides and releasing natural enemies, in which the implementation of control measures depends on whether the weighted value of pest density and its change rate reaches the action threshold is proposed to develop qualitative analysis techniques. First, the threshold condition for the existence and stability of the boundary periodic solutions is discussed and then define a one-parameter family of discrete (Poincaré) maps. By analyzing the properties of those discrete maps, it is concluded that if there is no interior equilibrium for the system without control measures and the releasing amount of natural enemies is greater than zero, then there is at least one positive periodic solution. While if the system without control measures has an internal equilibrium and only the chemical tactic is applied, then there could be an unstable positive periodic solution near the boundary periodic solution. As a result, backward bifurcation and bi-stability occur. The analytical techniques developed here could be applied to analyze more generalized models and other fields, including infectious disease control.
This paper aims to investigate the periodic solutions and the bifurcations associated with the control strategy for the proposed ecological model (\textit{Lotka-Volterra predator-prey model with state-dependent pulse and the signal that triggers pest control action is a convex combination of size and growth speed of the pest population}) (1.2). In Section 2, the relevant definitions and notations of planar impulsive semi-dynamic system and the dynamical behavior of the ODE model, Model (1.2) without control measures, are summarized. In Section 3, the dynamics of impulsive Model (1.2) are discussed when the ODE model has no positive equilibrium, including the exact ranges of impulsive set and phase set and the existence and stability of the boundary periodic solution. In Section 3.2, a Poincaré map for the proposed system is established, and by studying the properties of that map, the existence of an order-1 internal periodic solution is obtained. Furthermore, in Section 4, when the ODE model has a positive equilibrium, the bifurcation of a one-parameter family of discrete maps is analyzed with respect to the key parameters near the trivial fixed point. And the existence and stability of a positive fixed point is obtained, which is the positive periodic solution of the proposed impulsive System (1.2). To sum up the whole paper, the detailed conclusions and discussions are given in the last section.
Reviewer: Abdullah Özbekler (Ankara)Remote synchronization in a small star-like network of spin-torquehttps://zbmath.org/1496.340892022-11-17T18:59:28.764376Z"Kuptsov, Pavel Vladimirovich"https://zbmath.org/authors/?q=ai:kuptsov.pavel-vladimirovich"Kruglov, Vyacheslav Pavlovich"https://zbmath.org/authors/?q=ai:kruglov.vyacheslav-pavlovichSummary: We consider a mathematical model of field coupled spin-torque oscillators network. The model is described by a set of the Landau-Lifshitz-Gilbert-Slonczewski magnetization equations, which are coupled via an additional term to the effective field. For this model a generic form of the Jacobi matrix is explicitly derived. The network of four oscillators coupled as a star is considered: the central oscillator is coupled with three others and they do not have direct couplings with each other. For this network the Lyapunov's exponent chart is computed and the area in the parameter space is found where the network demonstrates remote synchronization. In this regime the peripheral oscillators are synchronized with each other but not synchronized with the central one.Extension of synchronizability analysis based on vital factors: extending validity to multilayer fully coupled networkshttps://zbmath.org/1496.340902022-11-17T18:59:28.764376Z"Wang, Li"https://zbmath.org/authors/?q=ai:wang.li.4"Jia, Xiaoyu"https://zbmath.org/authors/?q=ai:jia.xiaoyu"Pan, Xiuyu"https://zbmath.org/authors/?q=ai:pan.xiuyu"Xia, Chengyi"https://zbmath.org/authors/?q=ai:xia.chengyiSummary: In this paper, we study the synchronizability of multilayer fully coupled networks and their simplest equivalent networks and then discuss the vital factors affecting the synchronizability. First, we use the master stability method to theoretically deduce the synchronizability of a multilayer fully coupled network and its simplest equivalent network. It is proven that a multilayer fully coupled network has the same synchronizability as its corresponding simplest equivalent network. Second, we conduct experiments to verify the derivation and observe the process of synchronization. Finally, we analyze the factors affecting the synchronizability of multilayer fully coupled networks. The five influencing factors are identified as the intralayer coupling strength, the interlayer coupling strength, the number of nodes in each network layer, the number of vital nodes and the number of layers in the network. The current findings allow us to gain a deeper understanding of the synchronization behavior and properties of real-world networks.Individual nonuniform dichotomy and admissibility for linear skew-products semiflows over a semiflowhttps://zbmath.org/1496.340912022-11-17T18:59:28.764376Z"Onofrei, Oana Romina"https://zbmath.org/authors/?q=ai:onofrei.oana-romina"Preda, Petre"https://zbmath.org/authors/?q=ai:preda.petreSummary: The aim of this paper is to give a new characterization of the admissibility of the pair \((L^1; L^\infty)\) to the case of linear skew-product semiflows over semiflows, which satisfy the following conditions: the cocycle \( \pi = (\Phi \sigma)\) has no exponential growth and \(K\) the constant from the ``boundedness'' theorem it depends on \(\theta \in \Theta\), by using the ``input-output'' technique.Existence and stability of periodic oscillations of a smooth and discontinuous oscillatorhttps://zbmath.org/1496.340922022-11-17T18:59:28.764376Z"Liang, Zaitao"https://zbmath.org/authors/?q=ai:liang.zaitao"Yang, Yanjuan"https://zbmath.org/authors/?q=ai:yang.yanjuanSummary: In this paper, we study the existence, multiplicity and Lyapunov stability of periodic oscillations of a SD oscillator which exhibits both discontinuous and smooth dynamics depending on the value of the smoothness parameter \(\alpha\). Both linear stability and nonlinear stability results are obtained. The proof is based on some stability criteria of second order differential equations combined with the quantitative information obtained by the method of upper and lower solutions. Moreover, some numerical simulations are provided to illustrate the results.Turnpike properties of solutions of a differential inclusion with a Lyapunov function. IIhttps://zbmath.org/1496.340932022-11-17T18:59:28.764376Z"Zaslavski, Alexander J."https://zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: We study the turnpike phenomenon for approximate solutions of optimal problems governed by a differential inclusion with a Lyapunov function. This differential inclusion generates a dynamical system which has a prototype in mathematical economics. In our previous research we obtained turnpike results for a collection of approximate optimal trajectories with a fixed initial point. In the present paper, under a certain assumption, we extend these results for all approximate optimal trajectories.
For Part I see [ibid. 7, No. 3, 1085--1102 (2022; Zbl 1490.34061)].Asymptotic expansions for a degenerate canard explosionhttps://zbmath.org/1496.340942022-11-17T18:59:28.764376Z"Qin, Bo-Wei"https://zbmath.org/authors/?q=ai:qin.bo-wei"Chung, Kwok-Wai"https://zbmath.org/authors/?q=ai:chung.kwok-wai"Algaba, Antonio"https://zbmath.org/authors/?q=ai:algaba.antonio"Rodríguez-Luis, Alejandro J."https://zbmath.org/authors/?q=ai:rodriguez-luis.alejandro-jThe problem of a turning point (including degenerate turning point) is a key issue in singular perturbation theory. This article develops a new way, namely, the asymptotic expansion approach of suitable order, to detect canard explosion generated by the higher-order (degenerate) turning point in a two-dimensional singularly perturbated system. By this method, the canard curve can also be calculated with respect to the perturbation parameter to arbitrary order. It is shown that this asymptotic expansion-based method is quite effective to analyze the canard explosion phenomena generated by the degenerate turning point.
Reviewer: Jianhe Shen (Fuzhou)Generalized solutions of differential equations with the derivatives of functionals in Banach spaceshttps://zbmath.org/1496.340952022-11-17T18:59:28.764376Z"Falaleev, M. V."https://zbmath.org/authors/?q=ai:falaleev.mikhail-valentinovich"Grazhdantseva, E. Y."https://zbmath.org/authors/?q=ai:grazhdantseva.elena-yurevnaIn the paper under review, the authors investigate the initial value problem for a differential equation with the derivatives of the functionals in Banach spaces. The solution is sought in the space of generalized functions with the support bounded on the left and its has the form of convolution of the fundamental solution of the differential operator with the right-hand side of the equation. The authors also establish some sufficient conditions for the solvability of the initial-value problem under consideration in the classes of finitely differentiable functions.
Reviewer: Marko Kostić (Novi Sad)Variation of constants formula and exponential dichotomy for nonautonomous non-densely defined Cauchy problemshttps://zbmath.org/1496.340962022-11-17T18:59:28.764376Z"Magal, Pierre"https://zbmath.org/authors/?q=ai:magal.pierre"Seydi, Ousmane"https://zbmath.org/authors/?q=ai:seydi.ousmaneLet \(X\) be a Banach space and \(A:D(A)\to X\) be a linear operator with possibly non-dense domain. Denote \(\overline{D(A)}=X_0\). Let \(\{B(t)\}_{t\in\mathbb{R}}\subset \mathcal L(X_0,X)\) be a locally bounded and strongly continuous family of linear operators.
Assume that \(\exists \omega\in\mathbb{R}\) and \(M\geq1\) s.t. \((\omega,+\infty)\subset\rho(A)\),
\[
\|(\lambda I-A)^{-k}\|_{\mathcal{L}(X_0,X)}\leq M(\lambda-\omega)^{-k}\quad\forall \lambda>\omega,k\geq1
\]
and \(\lim_{\lambda\to\infty}(\lambda-A)^{-1}x=0\;\forall x\in X\). Suppose that for each \(\tau>0\) and \(f\in C([0,\tau],X)\) the equation \(u'_f=Au_f+f\) has a unique mild solution \(u_f\in C([0,\tau],X_0)\) with \(u_f(0)=0\). Suppose also that \(\sup_{t\in[-n,n]}\|B(t)\|_{\mathcal{L}(X_0,X)}<+\infty\) for all integer \(n\geq1\).
In \(X\) consider the differential non-homogeneous equation
\[
\frac{du(t)}{dt}=(A+B(t))u(t)+f(t),\quad t\geq t_0,\quad u(t_0)=x_0\in X_0.
\]
Then for each \(t_0\), \(x_0\in X_0\) and \(f\in C([t_0,+\infty],X)\) the equation has a unique mild solution
\[
u(t)=U_B(t,t_0)x_0+\lim_{\lambda\to+\infty} \int_{t_0}^tU_{B}(t,s)\lambda(\lambda I-A)^{-1}f(s)ds.
\]
Here \(U_B(t,s)\) is an evolution family for the related homogeneous equation.
If in addition \(\sup_{\mathbb{R}}\|B(t)\|_{\mathcal{L}(X_0,X)}<+\infty\), then the evolution \(U_B\) has an exponantial dichotomy. A related representation for the solution \(u\) is obtained.
Applications to PDEs with non-local conditions are given.
Reviewer: Nikita V. Artamonov (Moskva)Controllability of fractional evolution systems of Sobolev type via resolvent operatorshttps://zbmath.org/1496.340972022-11-17T18:59:28.764376Z"Yang, He"https://zbmath.org/authors/?q=ai:yang.he"Zhao, Yanjie"https://zbmath.org/authors/?q=ai:zhao.yanjieSummary: In this paper, we consider the nonlocal controllability of \(\alpha\in (1,2)\)-order fractional evolution systems of Sobolev type in abstract spaces. By utilizing fixed point theorems and the theory of resolvent operators we establish some sufficient conditions for the nonlocal controllability of Sobolev-type fractional evolution systems.Finite-time synchronization of nonlinear fractional chaotic systems with stochastic actuator faultshttps://zbmath.org/1496.340982022-11-17T18:59:28.764376Z"Sweetha, S."https://zbmath.org/authors/?q=ai:sweetha.s"Sakthivel, R."https://zbmath.org/authors/?q=ai:sakthivel.rathinasamy"Harshavarthini, S."https://zbmath.org/authors/?q=ai:harshavarthini.sSummary: This paper states with the objective of investigating the synchronization problem of nonlinear delayed fractional-order chaotic systems in conjunction with quantization, actuator faults, randomly occurring parametric uncertainties and exogenous disturbances. Moreover, the actuator faults are randomly occurring at any instant of time. The resultant random variables obeying Bernoulli distribution are introduced to account stochastic behavior. In spite of ensuring the robust performance, the finite-time synchronization of the addressed system is achieved and satisfies passive disturbance attenuation level by developing robust quantized stochastic reliable control protocol. As a consequence, the fast synchronization of the considered system is ensured in a finite time period. Owing to this perspective, the desired controller gain matrices can be obtained by solving developed linear matrix inequality. Further, the effectiveness of the theoretical result developed in this paper is validated via numerical simulation.Quasi-synchronization of heterogenous fractional-order dynamical networks with time-varying delay via distributed impulsive controlhttps://zbmath.org/1496.340992022-11-17T18:59:28.764376Z"Wang, Fei"https://zbmath.org/authors/?q=ai:wang.fei.1"Zheng, Zhaowen"https://zbmath.org/authors/?q=ai:zheng.zhaowen"Yang, Yongqing"https://zbmath.org/authors/?q=ai:yang.yongqingSummary: This paper investigates the quasi-synchronization problem of a heterogeneous dynamical network. All nodes have fractional order dynamical behavior with time-varying delay. The distributed impulsive control strategy is applied to drive all the nodes to approximately synchronize with the target orbit within a nonzero error bound. A new comparison principle of impulsive fractional order functional differential equation has been built at first. Then, based on the Lyapunov stability theory, some basic theories of fractional order functional differential equation, and the definition of an average impulsive interval, some quasi-synchronization criteria are derived with explicit expressions of the error bound. Both synchronizing impulses and desynchronizing impulses are discussed in this paper. Finally, two numerical examples are presented to illustrate the validity of the theoretical analysis.Qualitative and quantitative analysis of functional-differential equations of Goodwin typehttps://zbmath.org/1496.341002022-11-17T18:59:28.764376Z"Khidirov, B. N."https://zbmath.org/authors/?q=ai:khidirov.b-n"Khidirova, M. B."https://zbmath.org/authors/?q=ai:khidirova.m-b"Shakarov, A. R."https://zbmath.org/authors/?q=ai:shakarov.a-r(no abstract)Topological structure of the solution sets for a nonlinear delay evolutionhttps://zbmath.org/1496.341012022-11-17T18:59:28.764376Z"Wang, Rong-Nian"https://zbmath.org/authors/?q=ai:wang.rongnian"Ma, Zhong-Xin"https://zbmath.org/authors/?q=ai:ma.zhong-xin"Miranville, Alain"https://zbmath.org/authors/?q=ai:miranville.alain-mIn this paper, the authors study the following nonlinear delay evolution equation with multivalued perturbation on a noncompact interval \[\begin{cases} u'(t)\in -A(t)u(t)+f(t), ~~~ t\in \mathbb{R}^+,\\
f(t)\in F(t,u_t), ~~~ t\in \mathbb{R}^+,\\
u(t)=\phi(t), ~~~ t\in [-r,0], \end{cases} \] where \(A(t), t\in \mathbb{R}^+\) is a family of possibly unbounded operators on an infinite dimensional real Banach space \(\mathbb{X},\) \(F: \mathbb{R}^+\times C([-r,0], \mbox{conv}\mathbb{D})\to 2^{\mathbb{X}}\setminus\emptyset\) has convex closed values, \(F(t,\cdot)\) is upper hemicontinuous for a.e. \(t\in \mathbb{R}^+,\) \(\mathbb{D}\) is a nonempty closed subset of \(\mathbb{X},\) \(\mbox{conv}\mathbb{D}\) is the convex hull of \(\mathbb{D}\) and \(u_t(\cdot)\in C([-r,0],\mathbb{D})\) is defined by \(u_t(s)=u(t+s), s\in [-r,0].\) It is proved that the solution map, having nonempty and compact values, is an \(R_{\delta}\)-map, which maps any connected set into a connected set. Several examples illustrating the applicability of the obtained abstract results are also presented.
Reviewer: Sotiris K. Ntouyas (Ioannina)Nonlinear equations of fourth-order with \(p\)-Laplacian like operators: oscillation, methods and applicationshttps://zbmath.org/1496.341022022-11-17T18:59:28.764376Z"Bazighifan, Omar"https://zbmath.org/authors/?q=ai:bazighifan.omar"Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandraThe paper is concerned with fourth order differential equations of the types found in fluid dynamics, electromagnetism and quantum theory. The main aim is the study of oscillatory behaviour of solutions of these equations when they are driven by a \(p\)-Laplace differential operator. The main results of the paper are two theorems that provide results that, under given hypotheses, the fourth order equation under consideration is oscillatory. As the authors point out, these results extend the existing known results. The paper concludes with an example.
Reviewer: Neville Ford (Chester)Rigorous verification of Hopf bifurcations in functional differential equations of mixed typehttps://zbmath.org/1496.341032022-11-17T18:59:28.764376Z"Church, Kevin E. M."https://zbmath.org/authors/?q=ai:church.kevin-e-m"Lessard, Jean-Philippe"https://zbmath.org/authors/?q=ai:lessard.jean-philippeThe paper is concerned with the development of a numerical method to prove the existence of Hopf bifurcations in simple functional differential equations of mixed type, otherwise known as advance-delay equations, or sometimes as forward-backward equations. The question of interest is to consider the properties of the eigenvalues, and use is made of the Newton-Kantorovich theorem. The authors prove the existence of Hopf bifurcations in the Lasota-Wazewska-Czyzewska model and the existence of periodic traveling waves in the Fisher equation with nonlocal reaction. The overall objective of the work is to `develop numerical methods which can lead to computer-assisted proofs of existence of different type of dynamical objects arising in the study of differential equations.' Consequently, there is a section that discusses computer-assisted proofs of some of the theorems presented and links are provided to the relevant code.
Reviewer: Neville Ford (Chester)Periodic solutions to certain classes of third order delay differential equationshttps://zbmath.org/1496.341042022-11-17T18:59:28.764376Z"Olayemi, S. A."https://zbmath.org/authors/?q=ai:olayemi.s-a"Ogundare, B. S."https://zbmath.org/authors/?q=ai:ogundare.babatunde-sundaySummary: In this paper, existence of unique periodic solutions for two classes of third order delay differential equations are considered. The two equations are expressed in their integral equivalents with which suitable Green's function is constructed and its associated properties stated. The main tool adopted to establish the existence of unique periodic solutions to the classes of delay differential equations is the Krasnoselskii's fixed point theorem due to its applicability to the combination of a compact and contraction mappings which occur in dealing with perturbed differential operators.Bifurcation in car-following models with time delays and driver and mechanic sensitivitieshttps://zbmath.org/1496.341052022-11-17T18:59:28.764376Z"Padial, Juan Francisco"https://zbmath.org/authors/?q=ai:padial.juan-francisco"Casal, Alfonso"https://zbmath.org/authors/?q=ai:casal.alfonso-cSummary: In this work, we study a model of traffic flow along a one-way, one lane, road or street, the so-called car-following problem. We first present a historical evolution of models of this type corresponding to a successive improvement of requirements, to explain some real traffic phenomena. For both mathematical reasons and a better explanation of some of those phenomena, we consider more convenient and accurate requirements which lead to a better non-linear model with reaction delays, from several sources. The model can be written as an ordinary nonlinear delay differential equation. It has equilibrium solutions, which correspond to steady traffic. The mentioned reaction delays introduce perturbation terms in the equation, leading to of instabilities of equilibria and changes of the structure of the solutions. For some of the values of the delays, they may become oscillatory. We make a number of simulations to show these changes for different values of delays. We also show that, for certain values of the delays the above mentioned change of structure (representing regimes of real traffic) corresponds to a Hopf bifurcation.Discrete traveling waves in a relay system of Mackey-Glass equations with two delayshttps://zbmath.org/1496.341062022-11-17T18:59:28.764376Z"Preobrazhenskaya, M. M."https://zbmath.org/authors/?q=ai:preobrazhenskaya.m-mConsider a ring circuit of \(m\) identical Mackey-Glass generators of the form \[\dot{u}_{j}=-\beta u_{j}+\frac{\alpha\left(u_{j}(t-1)+u_{j-1}(t-\tau)\right)}{1+\left(u_{j}(t-1)+u_{j-1}(t-\tau)\right)^{\gamma}}, \quad u_{0} \equiv u_{m}, \quad j=1, \ldots, m\tag{1} \] with positive parameters \(\alpha,\beta,\gamma,\tau\). Letting \(\gamma \to \infty\) one obtains the limiting equation \[\dot{u}_{j}=-\beta u_{j}+\alpha\left(u_{j}(t-1)+u_{j-1}(t-\tau)\right) F\left(u_{j}(t-1)+u_{j-1}(t-\tau)\right), \quad u_{0} \equiv u_{m}, \quad j=1, \ldots, m \tag{2}\] with \[ F(u) := \lim _{\gamma \rightarrow+\infty} \frac{1}{1+u^{\gamma}}= \begin{cases}1, & 0<u<1, \\
1 / 2, & u=1, \\
0, & u>1.\end{cases} \] The motivation to study system (2) is twofold: on the one hand, \(\gamma\gg 1\) is a realistic assumption in applications, and on the other hand, equation (2) can be regarded as a relay circuit analogue of system (1).
The main result of the paper shows that for each positive integer \(m\geq 2\), there exists a range of values of the parameters such that for fixed \(\alpha, \beta,\tau\) and \(m\), there exists a discrete traveling wave solution of (2), that is, there is a \(\Delta>0\) for which system (2) has a periodic solution of the form \(u_j(t) = u_\ast (t + j\Delta)\).
The key observation is that discrete traveling wave solutions correspond to periodic solutions of the scalar version (i.e.\ \(m=1\)) of equation (2).
The proof is constructive: the exact formula for \(u_\ast\) is obtained.
Reviewer: Ábel Garab (Klagenfurt)Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive typehttps://zbmath.org/1496.341072022-11-17T18:59:28.764376Z"Zhu, Yu"https://zbmath.org/authors/?q=ai:zhu.yuSummary: In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type
\[
\bigl(x(t)-cx(t-\sigma)\bigr)''+f\bigl(x(t) \bigr)x'(t)-\varphi(t)x^{\mu}(t)+ \frac{\alpha(t)}{x^{\gamma}(t)}=e(t),
\] where \(f:(0,+\infty)\rightarrow \mathbb{R}\), \(\varphi(t)>0\) and \(\alpha(t)>0\) are continuous functions with \(T\)-periodicity in the \(t\) variable, \(c, \gamma\) are constants with \(|c|<1, \gamma\geq1\). Many authors obtained the existence of periodic solutions under the condition \(0<\mu\leq1\), and we extend the result to \(\mu>1\) by using Mawhin's continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem.Globally exponential stability of piecewise pseudo almost periodic solutions for neutral differential equations with impulses and delayshttps://zbmath.org/1496.341082022-11-17T18:59:28.764376Z"He, Jianxin"https://zbmath.org/authors/?q=ai:he.jianxin"Kong, Fanchao"https://zbmath.org/authors/?q=ai:kong.fanchao"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-jose"Qiu, Hongjun"https://zbmath.org/authors/?q=ai:qiu.hongjunImpulsive differential equations are very important class of differential equations whose dynamics is very rich. In this work, authors consider a delayed impulsive neutral differential equations. The coefficients are assumed to be bounded. The main objective is to establish the existence of piecewise pseudo almost periodic solution. The techniques used are contraction mapping principle and generalized Gronwall-Bellmain inequality. Moreover, the stability of such solution is also shown. The stability is globally exponential. At the end, an example with numerical illustration is provided by the authors.
Reviewer: Syed Abbas (Mandi)Further results on delay-dependent stability for neutral singular systems via state decomposition methodhttps://zbmath.org/1496.341092022-11-17T18:59:28.764376Z"Chen, Wenbin"https://zbmath.org/authors/?q=ai:chen.wenbin"Gao, Fang"https://zbmath.org/authors/?q=ai:gao.fang"She, Jinhua"https://zbmath.org/authors/?q=ai:she.jinhua"Xia, Weifeng"https://zbmath.org/authors/?q=ai:xia.weifengSummary: This paper studies the delay-dependent stability for neutral singular systems. In the light of state decomposition method, a novel augmented Lyapunov-Krasovskii functional including less decision variables is developed. Then by means of zero-value equations technology, some sufficient stability conditions in the form of linear matrix inequalities are acquired, which guarantees the non-impulsiveness, regularity and stability for the proposed neutral singular systems. The obtained stability criterion takes the sizes of both the discrete- and neutral- delays into account. They are less conservative than those presented by previous analytical approaches. Numerical examples are given to show the feasibility of our method and the interrelation between the discrete- and neutral-delays.A phase model with large time delayed couplinghttps://zbmath.org/1496.341102022-11-17T18:59:28.764376Z"Al-Darabsah, Isam"https://zbmath.org/authors/?q=ai:al-darabsah.isam"Campbell, Sue Ann"https://zbmath.org/authors/?q=ai:campbell.sue-annSummary: We consider two identical oscillators with weak, time delayed coupling. We start with a general system of delay differential equations then reduce it to a phase model. With the assumption of large time delay, the resulting phase model has an explicit delay and phase shift in the argument of the phases and connection function, respectively. Using the phase model, we prove that for any type of oscillators and any coupling, the in-phase and anti-phase phase-locked solutions always exist and give conditions for their stability. We show that for small delay these solutions are unique, but with large enough delay multiple solutions of each type with different frequencies may occur. We give conditions for the existence and stability of other types of phase-locked solutions. We discuss the various bifurcations that can occur in the phase model as the time delay is varied. The results of the phase model analysis are applied to Morris-Lecar oscillators with diffusive coupling and compared with numerical studies of the full system of delay differential equations. We also consider the case of small time delay and compare the results with the existing ones in the literature.A discussion on the approximate controllability of Hilfer fractional neutral stochastic integro-differential systemshttps://zbmath.org/1496.341112022-11-17T18:59:28.764376Z"Dineshkumar, C."https://zbmath.org/authors/?q=ai:dineshkumar.c"Udhayakumar, R."https://zbmath.org/authors/?q=ai:udhayakumar.r"Vijayakumar, V."https://zbmath.org/authors/?q=ai:vijayakumar.velusamy"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: This manuscript is mainly focusing on the approximate controllability of Hilfer fractional neutral stochastic integro-differential equations. The principal results of this article are proved based on the theoretical concepts related to the fractional calculus and Schauder's fixed-point theorem. Initially, we discuss the approximate controllability of the fractional evolution system. Then, we extend our results to the concept of nonlocal conditions. Finally, we provide theoretical and practical applications to assist in the effectiveness of the discussion.Theory and applications of equivariant normal forms and Hopf bifurcation for semilinear FDEs in Banach spaceshttps://zbmath.org/1496.341122022-11-17T18:59:28.764376Z"Guo, Shangjiang"https://zbmath.org/authors/?q=ai:guo.shangjiangThe paper extends existing methods for the analysis of autonomous delay differential equations on the existence of invariant manifolds to semilinear functional differential equations. The author summarises the paper in a succinct and comprehensive way: `We show that in the neighborhood of trivial solutions, variables can be chosen so that the form of the reduced vector field relies not only on the information of the linearized system at the critical point but also on the inherent symmetry. We observe that the normal forms give critical information about dynamical properties, such as generic local branching spatiotemporal patterns of equilibria and periodic solutions. As an important application of equivariant normal forms, we not only establish equivariant Hopf bifurcation theorem for semilinear FDEs in general Banach spaces, but also in a natural way derive criteria for the existence, stability, and bifurcation direction of branches of bifurcating periodic solutions. We employ these general results to obtain the existence of infinite many small-amplitude wave solutions for a delayed Ginzburg-Landau equation on a two-dimensional disk with the homogeneous Dirichlet boundary condition.' Sections of the paper on decomposition of the phase space, equivariant normal form, and Hopf bifurcation with symmetry provide detail and useful discussion.
Reviewer: Neville Ford (Chester)Existence and uniqueness of solutions for abstract integro-differential equations with state-dependent delay and applicationshttps://zbmath.org/1496.341132022-11-17T18:59:28.764376Z"Hernandez, Eduardo"https://zbmath.org/authors/?q=ai:hernandez.eduardo-m"Rolnik, Vanessa"https://zbmath.org/authors/?q=ai:rolnik.vanessa"Ferrari, Thauana M."https://zbmath.org/authors/?q=ai:ferrari.thauana-mIn this paper, the authors study the existence and uniqueness of solutions for a general class of abstract ordinary integro-differential equation with state dependent delay. The results are obtained by using a fixed point theorem. Some examples arising in the population dynamics and in the Solow's theory of economic growth are presented.
Reviewer: Krishnan Balachandran (Coimbatore)Stepanov-like pseudo almost periodic solutions of class \(r\) in \(\alpha \)-norm under the light of measure theoryhttps://zbmath.org/1496.341142022-11-17T18:59:28.764376Z"Zabsonre, Issa"https://zbmath.org/authors/?q=ai:zabsonre.issa"Nsangou, Abdel Hamid Gamal"https://zbmath.org/authors/?q=ai:nsangou.abdel-hamid-gamal"Kpoumiè, Moussa El-Khalil"https://zbmath.org/authors/?q=ai:kpoumie.moussa-el-khalil"Mboutngam, Salifou"https://zbmath.org/authors/?q=ai:mboutngam.salifouSummary: The aim of this work is to present some interesting results on weighted ergodic functions. We also study the existence and uniqueness of \((\mu,\nu)\)-weighted Stepanov-like pseudo almost periodic solutions class \(r\) for some partial differential equations in a Banach space when the delay is distributed using the spectral decomposition of the phase space developed by Adimy and his co-authors.Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operatorhttps://zbmath.org/1496.341152022-11-17T18:59:28.764376Z"Zhang, Xuping"https://zbmath.org/authors/?q=ai:zhang.xuping"Sun, Pan"https://zbmath.org/authors/?q=ai:sun.pan(no abstract)A study on controllability of impulsive fractional evolution equations via resolvent operatorshttps://zbmath.org/1496.341162022-11-17T18:59:28.764376Z"Gou, Haide"https://zbmath.org/authors/?q=ai:gou.haide"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: In this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the \((\alpha ,\beta)\)-resolvent operator, we concern with the term \(u^\prime(\cdot)\) and finding a control \(v\) such that the mild solution satisfies \(u(b)=u_b\) and \(u^\prime b)=u^\prime_b\). Finally, we present an application to support the validity study.Mathematical analysis of nonlinear integral boundary value problem of proportional delay implicit fractional differential equations with impulsive conditionshttps://zbmath.org/1496.341172022-11-17T18:59:28.764376Z"Ali, Arshad"https://zbmath.org/authors/?q=ai:ali.arshad"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamal"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet"Mahariq, Ibrahim"https://zbmath.org/authors/?q=ai:mahariq.ibrahim"Rashdan, Mostafa"https://zbmath.org/authors/?q=ai:rashdan.mostafaSummary: The current study is devoted to deriving some results about existence and stability analysis for a nonlinear problem of implicit fractional differential equations (FODEs) with impulsive and integral boundary conditions. The concerned problem involves proportional type delay term. By using Schaefer's fixed point theorem and Banach's contraction principle, the required conditions are developed. Also, different kinds of Ulam stability results are derived by using nonlinear analysis. Providing a pertinent example, we demonstrate our main results.Qualitative analysis of nonlinear coupled pantograph differential equations of fractional order with integral boundary conditionshttps://zbmath.org/1496.341182022-11-17T18:59:28.764376Z"Alrabaiah, Hussam"https://zbmath.org/authors/?q=ai:alrabaiah.hussam"Ahmad, Israr"https://zbmath.org/authors/?q=ai:ahmad.israr"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamal"Rahman, Ghaus Ur"https://zbmath.org/authors/?q=ai:rahman.ghaus-urSummary: In this research article, we develop a qualitative analysis to a class of nonlinear coupled system of fractional delay differential equations (FDDEs). Under the integral boundary conditions, existence and uniqueness for the solution of this system are carried out. With the help of Leray-Schauder and Banach fixed point theorem, we establish indispensable results. Also, some results affiliated to Ulam-Hyers (UH) stability for the system under investigation are presented. To validate the results, illustrative examples are given at the end of the manuscript.Solvability of nonlinear functional differential equations of fractional order in reflexive Banach spacehttps://zbmath.org/1496.341192022-11-17T18:59:28.764376Z"Hashem, H. H. G."https://zbmath.org/authors/?q=ai:hashem.hind-h-g"El-Sayed, A. M. A."https://zbmath.org/authors/?q=ai:el-sayed.ahmed-mohamed-ahmed"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-p"Ahmad, Bashir"https://zbmath.org/authors/?q=ai:ahmad.bashir.2In this paper, the authors discuss the solvability of nonlinear functional differential equations of fractional order in reflexive Banach spaces. There are two methods given in this paper. One is the coupled system approach, the other is the functional equation approach. By O'Regan's fixed point theorem in Banach spaces, the authors obtain weak and pseudo solutions for some initial value problems, which generalizes some related results in this topic.
Reviewer: Zhenbin Fan (Jiangsu)A new approach on approximate controllability of fractional evolution inclusions of order \(1<r<2\) with infinite delayhttps://zbmath.org/1496.341202022-11-17T18:59:28.764376Z"Raja, M. Mohan"https://zbmath.org/authors/?q=ai:raja.m-mohan"Vijayakumar, V."https://zbmath.org/authors/?q=ai:vijayakumar.velusamy"Udhayakumar, R."https://zbmath.org/authors/?q=ai:udhayakumar.rSummary: This manuscript is mainly focusing on the approximate controllability of fractional differential evolution inclusions of order \(1<r<2\) with infinite delay. We study our primary outcomes by using the theoretical concepts about fractional calculus, cosine, and sine function of operators and Dhage's fixed point theorem. Initially, we prove the approximate controllability for the fractional evolution system. The results are established under the assumption that the associated linear system is approximately controllable. Then, we develop our conclusions to the ideas of nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study.On a new structure of the pantograph inclusion problem in the Caputo conformable settinghttps://zbmath.org/1496.341212022-11-17T18:59:28.764376Z"Thabet, Sabri T. M."https://zbmath.org/authors/?q=ai:thabet.sabri-t-m"Etemad, Sina"https://zbmath.org/authors/?q=ai:etemad.sina"Rezapour, Shahram"https://zbmath.org/authors/?q=ai:rezapour.shahramSummary: In this work, we reformulate and investigate the well-known pantograph differential equation by applying newly-defined conformable operators in both Caputo and Riemann-Liouville settings simultaneously for the first time. In fact, we derive the required existence criteria of solutions corresponding to the inclusion version of the three-point Caputo conformable pantograph BVP subject to Riemann-Liouville conformable integral conditions. To achieve this aim, we establish our main results in some cases including the lower semi-continuous, the upper semi-continuous and the Lipschitz set-valued maps. Eventually, the last part of the present research is devoted to proposing two numerical simulative examples to confirm the consistency of our findings.Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delayshttps://zbmath.org/1496.341222022-11-17T18:59:28.764376Z"Jin, Shoubo"https://zbmath.org/authors/?q=ai:jin.shouboSummary: The Picard iteration method is used to study the existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays. Several sufficient conditions are specified to ensure that the equation has a unique solution. First, the stochastic Volterra-Levin equation is transformed into an integral equation. Then, to obtain the solution of the integral equation, the successive approximation sequences are constructed, and the existence and uniqueness of solutions for the stochastic Volterra-Levin equation are derived by the convergence of the sequences. Finally, two examples are given to demonstrate the validity of the theoretical results.Asymptotic behavior of a predator-prey system with delayshttps://zbmath.org/1496.341232022-11-17T18:59:28.764376Z"El-Owaidy, H. M."https://zbmath.org/authors/?q=ai:el-owaidy.hassan-mostafa"Ismail, M."https://zbmath.org/authors/?q=ai:ismail.m-i|ismail.mohamed-m|ismail.mohammad|ismail.m-a-h|ismail.mohd-tahir|ismail.mourad-el-houssieny|ismail.mahamod|ismail.mohammad-vaseem|ismail.mohammed|ismail.mehmet-s|ismail.mardhiyah|ismail.moshira-a|ismail.mohd-vaseem|ismail.m-ghazie|ismail.mat-rofa-bin|ismail.mourad|ismail.mohd-azmi|ismail.mohammad-s|ismail.munira|ismail.mohamed-a|ismail.mahmoud-h|ismail.m-n|ismail.muhammad-faizal(no abstract)Stability and bifurcation analyses of p53 gene regulatory network with time delayhttps://zbmath.org/1496.341242022-11-17T18:59:28.764376Z"Hou, Jianmin"https://zbmath.org/authors/?q=ai:hou.jianmin"Liu, Quansheng"https://zbmath.org/authors/?q=ai:liu.quansheng|liu.quansheng.1"Yang, Hongwei"https://zbmath.org/authors/?q=ai:yang.hongwei"Wang, Lixin"https://zbmath.org/authors/?q=ai:wang.lixin"Bi, Yuanhong"https://zbmath.org/authors/?q=ai:bi.yuanhongSummary: In this paper, based on a p53 gene regulatory network regulated by Programmed Cell Death 5(PDCD5), a time delay in transcription and translation of Mdm2 gene expression is introduced into the network, the effects of the time delay on oscillation dynamics of p53 are investigated through stability and bifurcation analyses. The local stability of the positive equilibrium in the network is proved through analyzing the characteristic values of the corresponding linearized systems, which give the conditions on undergoing Hopf bifurcation without and with time delay, respectively. The theoretical results are verified through numerical simulations of time series, characteristic values and potential landscapes. Furthermore, combined effect of time delay and several typical parameters in the network on oscillation dynamics of p53 are explored through two-parameter bifurcation diagrams. The results show p53 reaches a lower stable steady state under smaller PDCD5 level, the production rates of p53 and Mdm2 while reaches a higher stable steady state under these larger ones. But the case is the opposite for the degradation rate of p53. Specially, p53 oscillates at a smaller Mdm2 degradation rate, but a larger one makes p53 reach a low stable steady state. Besides, moderate time delay can make the steady state switch from stable to unstable and induce p53 oscillation for moderate value of these parameters. Theses results reveal that time delay has a significant impact on p53 oscillation and may provide a useful insight into developing anti-cancer therapy.Qualitative analysis of equations of the regulatory mechanisms of a multicellular organismhttps://zbmath.org/1496.341252022-11-17T18:59:28.764376Z"Khidirov, B. N."https://zbmath.org/authors/?q=ai:khidirov.b-n(no abstract)Qualitative spectral analysis of singular \(q\)-Sturm-Liouville operatorshttps://zbmath.org/1496.341262022-11-17T18:59:28.764376Z"Allahverdiev, Bilender P."https://zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaoglu"Tuna, Hüseyin"https://zbmath.org/authors/?q=ai:tuna.huseyinSummary: This paper considers properties of the spectrum of \(q\)-Sturm-Liouville operator derived from the \(q\)-Sturm-Liouville expression \[Ly := -\frac{1}{q} D_{q^{-1}} \left(p(x)D_q y(x)\right) + r(x) y(x), \quad 0 < x < a \le \infty.\] We prove that the regular symmetric \(q\)-Sturm-Liouville operator is semi-bounded from below. Using splittings technique, we will give some conditions for the self-adjoint operator associated with the singular \(q\)-Sturm-Liouville expression to have a discrete spectrum. We also investigate the continuous spectrum of this operator.Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension onehttps://zbmath.org/1496.341272022-11-17T18:59:28.764376Z"Galkowski, Jeffrey"https://zbmath.org/authors/?q=ai:galkowski.jeffreyThe abstract of the paper itself includes the most accurate and complete explanation about the content of the article under review. Here we quote it in full:
``In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let \(P\colon L^2(\mathbb{R})\to L^2(\mathbb{R})\) have the form \(P:=-\frac{d^2}{dx^2}+W\), where \(W\) is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, \(\mathbf{1}_{(-\infty,\lambda^2]}(P)\) has a full asymptotic expansion in powers of \(\lambda\). In particular, our class of potentials \(W\) is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, the class of potentials includes certain potentials with dense pure point spectrum. The proof combines the gauge transform methods of Parnovski-Shterenberg and Sobolev with Melrose's scattering calculus.''
Reviewer: Erdogan Sen (Tekirdağ)On a regularisation of a nonlinear differential equation related to the non-homogeneous Airy equationhttps://zbmath.org/1496.341282022-11-17T18:59:28.764376Z"Filipuk, Galina"https://zbmath.org/authors/?q=ai:filipuk.galina-v"Kecker, Thomas"https://zbmath.org/authors/?q=ai:kecker.thomas"Zullo, Federico"https://zbmath.org/authors/?q=ai:zullo.federicoIn nonlinear ordinary differential equations (ODE), there may exist points where the existence theorem of Cauchy does not apply. Then, a classical procedure (blow-up) is to resolve all such exceptional points by considering a (hopefully finite) sequence of equivalent systems defined in different coordinates, chosen so as to preserving the structure of singularities.
The present paper is a pedagogical introduction to this technique, on a toy nonlinear second order ODE (2.1) whose general solution is known since it is built from a linear ODE (inhomogeneous Airy) by elimination of the constant term.
However, the authors fail to get out of a usual trap in this procedure, which is the choice of a two-dimensional system equivalent to the second order ODE. This choice is not unique, and their choice (2.7) leads, according to the authors, to an infinite sequence of blow-ups. Since the toy ODE is linearizable, the sequence should be finite, and the authors should consider a choice different from (2.7).
For the entire collection see [Zbl 1481.26002].
Reviewer: Robert Conte (Gif-sur-Yvette)Differential polynomials generated by solutions of second order non-homogeneous linear differential equationshttps://zbmath.org/1496.341292022-11-17T18:59:28.764376Z"Belaïdi, Benharrat"https://zbmath.org/authors/?q=ai:belaidi.benharratSummary: This paper is devoted to studying the growth and the oscillation of solutions of the second order non-homogeneous linear differential equation
\[
f^{\prime\prime}+Ae^{a_1 z} f^\prime+B(z) e^{a_2 z}f = F(z) e^{a_1 z},
\]
where \(A\), \(a_1\), \(a_2\) are complex numbers, \(B (z)\) (\(\not\equiv 0\)) and \(F (z)\) (\(\not\equiv 0\)) are entire functions with order less than one. Moreover, we investigate the growth and the oscillation of some differential polynomials generated by solutions of the above equation.Statistical convergence in paranorm sense on time scaleshttps://zbmath.org/1496.341302022-11-17T18:59:28.764376Z"Gulsen, Tuba"https://zbmath.org/authors/?q=ai:gulsen.tuba"Koyunbakan, Hikmet"https://zbmath.org/authors/?q=ai:koyunbakan.hikmet"Yılmaz, Emrah"https://zbmath.org/authors/?q=ai:yilmaz.emrah-sercan"Altın, Yavuz"https://zbmath.org/authors/?q=ai:altin.yavuzSummary: In this study, we define statistical convergence and \(\lambda\)-statistical convergence in paranorm sense on anarbitrary time scale equipped with paranorm. Furthermore, we study on strongly \(\lambda_p\)-summability on timescales in paranorm sense. Eventually, some inclusion theorems are proved.Existence and uniqueness of solutions for a nonlinear coupled system of fractional differential equations on time scaleshttps://zbmath.org/1496.341312022-11-17T18:59:28.764376Z"Rao, Sabbavarapu Nageswara"https://zbmath.org/authors/?q=ai:rao.sabbavarapu-nageswaraSummary: In this paper, we establish the criteria for the existence and uniqueness of solutions of a two-point BVP for a system of nonlinear fractional differential equations on time scales.
\[
\begin{aligned}
\Delta_{a^{\star}}^{\alpha_1-1}x(t)&=f_1(t, x(t), y(t)),\quad t\in J:=[a,b]\cap \mathbb{T},\\
\Delta_{a^{\star}}^{\alpha_2-1}y(t)&=f_2(t, x(t), y(t)),\quad t\in J:=[a,b]\cap \mathbb{T},
\end{aligned}
\]
subject to the boundary conditions
\[
\begin{aligned}
x(a)=0,&\quad x^{\Delta}(b)=0,\quad x^{\Delta \Delta }(b)=0,\\
y(a)=0,&\quad y^{\Delta}(b)=0,\quad y^{\Delta \Delta }(b)=0.
\end{aligned}
\]
where \(\mathbb{T}\) is any time scale (nonempty closed subsets of the reals), \(2<\alpha_i<3\) and \(f_i\in C_{rd}([a,b]\times \mathbb{R}\times \mathbb{R}, \mathbb{R})\) and \(\Delta_{a^{\star }}^{\alpha_i-1}\) denotes the delta fractional derivative on time scales \(\mathbb{T}\) of order \(\alpha_i-1\) for \(i=1, 2\). By using the Banach contraction principle. Finally, an example is given to illustrate the main result.Slowly varying oscillations and waves. From basics to modernityhttps://zbmath.org/1496.350032022-11-17T18:59:28.764376Z"Ostrovsky, Lev"https://zbmath.org/authors/?q=ai:ostrovsky.lev-a|ostrovskii.lev-bPublisher's description: The beauty of the theoretical science is that quite different physical, biological, etc. phenomena can often be described as similar mathematical objects, by similar differential (or other) equations. In the 20th century, the notion of ``theory of oscillations'' and later ``theory of waves'' as unifying concepts, meaning the application of similar methods and equations to quite different physical problems, came into being. In the variety of applications (quite possibly in most of them), the oscillatory process is characterized by a slow (as compared with the characteristic period) variation of its parameters, such as the amplitude and frequency. The same is true for the wave processes.
This book describes a variety of problems associated with oscillations and waves with slowly varying parameters. Among them the nonlinear and parametric resonances, self-synchronization, attenuated and amplified solitons, self-focusing and self-modulation, and reaction-diffusion systems. For oscillators, the physical examples include the van der Pol oscillator and a pendulum, models of a laser. For waves, examples are taken from oceanography, nonlinear optics, acoustics, and biophysics. The last chapter of the book describes more formal asymptotic perturbation schemes for the classes of oscillators and waves considered in all preceding chapters.Singular perturbation and initial layer for the abstract Moore-Gibson-Thompson equationhttps://zbmath.org/1496.350332022-11-17T18:59:28.764376Z"Alvarez, Edgardo"https://zbmath.org/authors/?q=ai:alvarez.edgardo"Lizama, Carlos"https://zbmath.org/authors/?q=ai:lizama.carlosSummary: We investigate the singular limit of a third-order abstract equation in time, in relation to the complete second-order Cauchy problem on Banach spaces, where the principal operator is the generator of a strongly continuous cosine family. Assuming that an initial datum is ill prepared, the initial layer problem is studied. We show convergence, which is uniform on compact sets that stay away from zero, as long as initial data are sufficiently smooth. Our method employs suitable results from the theory of general resolvent families of operators. The abstract formulation of the third-order in time equation is inspired by the Moore-Gibson-Thompson equation, which is the linearization of a model that currently finds applications in the propagation of ultrasound waves, displacement of certain viscoelastic materials, flexible structural systems that possess internal damping and the theory of thermoelasticity.Statistical solution and Liouville type theorem for coupled Schrödinger-Boussinesq equations on infinite latticeshttps://zbmath.org/1496.351252022-11-17T18:59:28.764376Z"Li, Congcong"https://zbmath.org/authors/?q=ai:li.congcong"Li, Chunqiu"https://zbmath.org/authors/?q=ai:li.chunqiu"Wang, Jintao"https://zbmath.org/authors/?q=ai:wang.jintaoSummary: In this article, we are concerned with statistical solutions for the nonautonomous coupled Schrödinger-Boussinesq equations on infinite lattices. Firstly, we verify the existence of a pullback-\(\mathcal{D}\) attractor and establish the existence of a unique family of invariant Borel probability measures carried by the pullback-\(\mathcal{D}\) attractor for this lattice system. Then, it will be shown that the family of invariant Borel probability measures is a statistical solution and satisfies a Liouville type theorem. Finally, we illustrate that the invariant property of the statistical solution is indeed a particular case of the Liouville type theorem.On the relationship between the solutions of an abstract Euler-Poisson-Darboux equation and fractional powers of the operator coefficient in the equationhttps://zbmath.org/1496.351562022-11-17T18:59:28.764376Z"Glushak, A. V."https://zbmath.org/authors/?q=ai:glushak.a-vSummary: We consider an incomplete initial value problem for an abstract singular Euler-Poisson-Darboux equation. It is established that, under weakened requirements on the operator coefficient in the equation, fractional powers of this operator coefficient should be used to construct solutions. It is also shown that a fractional power of the operator coefficient relates the Dirichlet condition and the weighted Neumann condition in the case of a boundary value problem for the Euler-Poisson-Darboux equation.Global weak solutions to the Euler-Vlasov equations with finite energyhttps://zbmath.org/1496.352902022-11-17T18:59:28.764376Z"Cao, Wentao"https://zbmath.org/authors/?q=ai:cao.wentaoSummary: This paper concentrates on the global existence of weak solutions in \(L^p\) with finite energy to a type of one-dimensional compressible Euler-Vlasov equations, which models the interaction between the isentropic gas and dispersed particles. Approximate solutions are constructed by adding artificial viscosity. Then the uniform \(L^p\) estimates of the approximate solutions with respect to the artificial viscosity are established through some subtle analysis on level sets of density and relative velocity. The convergence of approximate solutions to the desired weak solutions is guaranteed by the \(L^p\) compensated compactness framework.Jacobi spectral discretization for nonlinear fractional generalized seventh-order KdV equations with convergence analysishttps://zbmath.org/1496.353432022-11-17T18:59:28.764376Z"Hafez, R. M."https://zbmath.org/authors/?q=ai:hafez.ramy-mahmoud"Youssri, Y. H."https://zbmath.org/authors/?q=ai:youssri.y-h(no abstract)An approximate method based on Bernstein polynomials for solving fractional PDEs with proportional delayshttps://zbmath.org/1496.354312022-11-17T18:59:28.764376Z"Ketabdari, A."https://zbmath.org/authors/?q=ai:ketabdari.ali"Farahi, M. H."https://zbmath.org/authors/?q=ai:farahi.mohammad-hadi"Effati, S."https://zbmath.org/authors/?q=ai:effati.sohrabSummary: We apply a new method to solve fractional partial differential equations (FPDEs) with proportional delays. The method is based on expanding the unknown solution of FPDEs with proportional delays by the basis of Bernstein polynomials with unknown control points and uses operational matrices with the least-squares method to convert the FPDEs with proportional de lays to an algebraic system in terms of Bernstein coefficients (control points) approximating the solution of FPDEs. We use the Caputo derivatives of de gree \(0 < \alpha \leq 1\) as the fractional derivatives in our work. The main advantage of using this technique is that the method can easily be employed to a variety of FPDEs with or without proportional delays, and also the method offers a very simple and flexible framework for direct approximating of the solution of FPDEs with proportional delays. The convergence analysis of the present method is discussed. We show the effectiveness and superiority of the method by comparing the results obtained by our method with the results of some available methods in two numerical examples.On the cohomological equation of a linear contractionhttps://zbmath.org/1496.370032022-11-17T18:59:28.764376Z"Leclercq, Régis"https://zbmath.org/authors/?q=ai:leclercq.regis"Zeggar, Abdellatif"https://zbmath.org/authors/?q=ai:zeggar.abdellatifSummary: In this paper, we study the discrete cohomological equation of a contracting linear automorphism \(A\) of the Euclidean space \(\mathbb{R}^d\). More precisely, if \(\delta\) is the cobord operator defined on the Fréchet space \(E = C^l (\mathbb{R}^d)\) (\(0 \leq l \leq \infty \)) by: \( \delta(h) = h - h \circ A\), we show that:
\begin{itemize}
\item If \(E = C^0(\mathbb{R}^d)\), the range \(\delta(E)\) of \(\delta\) has infinite codimension and its closure is the hyperplane \(E_0\) consisting of the elements of \(E\) vanishing at 0. Consequently, \(H^1 (A, E)\) is infinite dimensional non Hausdorff topological vector space and then the automorphism \(A\) is not cohomologically \(C^0\)-stable.
\item If \(E = C^l(\mathbb{R}^d)\), with \(1 \leq l \leq \infty\), the space \(\delta(E)\) coincides with the closed hyperplane \(E_0\). Consequently, the cohomology space \(H^1 (A, E)\) is of dimension 1 and the automorphism \(A\) is cohomologically \(C^l\)-stable.
\end{itemize}Parametric topological entropy and differential equations with time-dependent impulseshttps://zbmath.org/1496.370122022-11-17T18:59:28.764376Z"Andres, Jan"https://zbmath.org/authors/?q=ai:andres.janThe author studies a parametric topological entropy of nonautonomous dynamical systems and its application to the theory of impulsive differential equations.
One of the main results is a lower estimation of the parametric topological entropy of sequences of self-maps in terms of the Nielsen numbers, the so-called Ivanov-like inequality.
This result is then applied to the following impulsive system on tori:
\begin{align*}
&x' = F(t,x), \\
&x(t_j+) = I_j(x(t_j^-)),\ j \in {\mathbb N},
\end{align*}
where \(F : [t_0,\infty)\times {\mathbb R} \to {\mathbb R}\) is a Carathéodory function, the impulsive functions \(I_j : {\mathbb R}^n \to {\mathbb R}^n\) are equicontinuous, \(t_{j+1} = t_j + \omega_j\), with \(\omega_j > 0\), \(j \in {\mathbb N}\), provided
\[
F(t,\ldots,x_k,\ldots) = F(t,\ldots,x_k +1,\ldots)
\]
and
\[
I_j(\ldots,x_k,\ldots) = I_j(\ldots,x_k + 1,\ldots)\ (\mathrm{mod}\ 1), \ j \in {\mathbb N}
\]
for \(k = 1,\ldots,n\). Sufficient conditions for the positivity of the parametric topological entropy for this impulsive system on the torus \({\mathbb R}^n / {\mathbb Z}^n\) are given in terms of the associated Poincaré translation operators. In the reviewer's opinion this result has no analogy in the related literature.
Reviewer: Jan Tomeček (Olomouc)Homoclinic orbits and chaos in nonlinear dynamical systems: auxiliary systems methodhttps://zbmath.org/1496.370212022-11-17T18:59:28.764376Z"Grechko, D. A."https://zbmath.org/authors/?q=ai:grechko.d-a"Barabash, N. V."https://zbmath.org/authors/?q=ai:barabash.nikita-v"Belykh, V. N."https://zbmath.org/authors/?q=ai:belykh.vladimir-nikitichSummary: The auxiliary systems method in other words the method of two-dimensional comparison systems plays an essential role in the nonlocal bifurcational dynamical systems theory. In this paper we demonstrate this method in a particular case of 4-dimensional nonlinear dynamical system formed by a coupled Van der Pol-Duffing oscillator and a linear oscillator. For this system, using the auxiliary systems method, a rigorous proof of the existence of a homoclinic orbit of a saddle-focus is carried out for which the Shilnikov condition of chaos is satisfied. The paper is dedicated to the memory of Gennady A. Leonov, who made a significant contribution to the development of methods for the analytical study of dynamical systems.Asymptotics of Andronov-Hopf dynamic bifurcationshttps://zbmath.org/1496.370442022-11-17T18:59:28.764376Z"Kalyakin, L. A."https://zbmath.org/authors/?q=ai:kalyakin.leonid-anatolevichSummary: We consider a system of two nonlinear differential equations with a slowly varying parameter \(\mu = \epsilon t \). For a frozen parameter \(\mu = \mathrm{const}\) the system has a focus type equilibrium state the stability of which changes when passing through the value \(\mu = 0\), i.e., we deal with an Andronov-Hopf bifurcation. Using the normal form method combined with the averaging method, we study asymptotics with respect to a small parameter \(\epsilon \rightarrow 0\) for solutions having a narrow transient layer near the bifurcation point in the domain \( |\epsilon t | \ll 1\). We express the leading term of asymptotics in terms of the solution to the Bernoulli equation.Complex periodic bursting structures in the Rayleigh-van der Pol-Duffing oscillatorhttps://zbmath.org/1496.370452022-11-17T18:59:28.764376Z"Ma, Xindong"https://zbmath.org/authors/?q=ai:ma.xindong"Bi, Qinsheng"https://zbmath.org/authors/?q=ai:bi.qinsheng"Wang, Lifeng"https://zbmath.org/authors/?q=ai:wang.lifengSummary: In the present paper, complex bursting patterns caused by the coupling effect of different frequency scales in the Rayleigh-van der Pol-Duffing oscillator (RVDPDO) driven by the external excitation term are presented theoretically. Seven different kinds of bursting, i.e., bursting of compound ``Homoclinic/Homoclinic'' mode via ``Homoclinic/Homoclinic'' hysteresis loop, bursting of compound ``fold/Homoclinic-Homoclinic/Hopf'' mode via ``fold/Homoclinic'' hysteresis loop, bursting of compound ``fold/Homoclinic-Hopf/Hopf'' mode via ``fold/Homoclinic'' hysteresis loop, bursting of ``fold/Homoclinic'' mode via ``fold/Homoclinic'' hysteresis loop, bursting of ``fold/Hopf'' mode via ``fold/fold'' hysteresis loop, bursting of ``Hopf/Hopf'' mode via ``fold/fold'' hysteresis loop and bursting of ``fold/fold'' mode are studied by using the phase diagram, time domain signal analysis, phase portrait superposition analysis and slow-fast analysis. With the help of the Melnikov method, the parameter properties related to the beingness of the Homoclinic and Heteroclinic bifurcations chaos of the periodic excitations are investigated. Then, by acting the external forcing term as a slowly changing state variable, the stability and bifurcation characteristics of the generalized autonomous system are given, which performs a major part in the interpretative principles of different bursting patterns. This paper aims to show the sensitivity of dynamical characteristics of RVDPDO to the variation of parameter \(\mu\) and decide how the choice of the parameters influences the manifold of RVDPDO during the repetitive spiking states. Finally, the validity of the research is tested and verified by the numerical simulations.Chaos explosion and topological horseshoe in three-dimensional impacting hybrid systems with a single impact surfacehttps://zbmath.org/1496.370482022-11-17T18:59:28.764376Z"Wang, Lei"https://zbmath.org/authors/?q=ai:wang.lei.18"Yang, Xiao-Song"https://zbmath.org/authors/?q=ai:yang.xiaosongSummary: For a class of three-dimensional impacting hybrid systems comprising a linear system of ordinary differential equations and a reset map, and having a single impact surface, this paper studies the phenomena of impacting homoclinic bifurcation leading to periodic orbits, horseshoes and chaos explosions. More precisely, it is proved that the homoclinic bifurcation can result in the impacting periodic sinks or impacting periodic saddle orbits when the impact surface is a plane and the reset map satisfies some basic conditions, but it is not easy to find horseshoes in this case. Furthermore, when the single impact surface is not a plane and the reset map has a certain rotational property, it is proved that a topological horseshoe will appear suddenly when bifurcation parameter passes through some threshold, thus, a kind of chaos explosion takes place. These main results are illustrated by several examples, in which it can be seen that the newborn chaotic invariant sets might be chaotic attractors from the perspective of numerical simulation.Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noisehttps://zbmath.org/1496.370552022-11-17T18:59:28.764376Z"Lv, Xiang"https://zbmath.org/authors/?q=ai:lv.xiangSummary: This paper is concerned with the existence of unstable stationary solutions for nonlinear stochastic differential equations (SDEs) with additive white noise. Assume that the nonlinear term \(f\) is monotone (or anti-monotone) and the global Lipschitz constant of \(f\) is smaller than the positive real part of the principal eigenvalue of the competitive matrix \(A\), the random dynamical system (RDS) generated by SDEs has an unstable \(\mathscr{F}_+\)-measurable random equilibrium, which produces a stationary solution for nonlinear SDEs. Here, \(\mathscr{F}_+ = \sigma \{ \omega \mapsto W_t (\omega):t\geq 0\}\) is the future \(\sigma\)-algebra. In addition, we get that the \(\alpha\)-limit set of all pull-back trajectories starting at the initial value \(x(0) = x\in\mathbb{R}^n\) is a single point for all \(\omega\in\Omega\), i.e., the unstable \(\mathscr{F}_+\)-measurable random equilibrium. Applications to stochastic neural network models are given.On essential numerical ranges and essential spectra of Hamiltonian systems with one singular endpointhttps://zbmath.org/1496.370562022-11-17T18:59:28.764376Z"Zhu, Li"https://zbmath.org/authors/?q=ai:zhu.li"Sun, Huaqing"https://zbmath.org/authors/?q=ai:sun.huaqingSummary: This paper is concerned with essential numerical ranges and essential spectra of Hamiltonian systems with one singular endpoint. For semi-bounded systems, the characterization of each element of the essential numerical range in terms of certain singular sequences is given, the concept of form perturbation small at the singular endpoint is introduced, and the stability of the essential numerical range is obtained under this perturbation, which shows the stability of the infimum or supremum of the essential spectrum. Some sufficient conditions for the invariance of the essential numerical range are given in terms of coefficients of Hamiltonian systems.The influence of a parameter that controls the asymmetry of a potential energy surface with an entrance channel and two potential wellshttps://zbmath.org/1496.370572022-11-17T18:59:28.764376Z"Agaoglou, Makrina"https://zbmath.org/authors/?q=ai:agaoglou.makrina"Katsanikas, Matthaios"https://zbmath.org/authors/?q=ai:katsanikas.matthaios"Wiggins, Stephen"https://zbmath.org/authors/?q=ai:wiggins.stephenSummary: In this paper we study an asymmetric valley-ridge inflection point (VRI) potential, whose energy surface (PES) features two sequential index-1 saddles (the upper and the lower), with one saddle having higher energy than the other, and two potential wells separated by the lower index-1 saddle. We show how the depth and the flatness of our potential changes as we modify the parameter that controls the asymmetry as well as how the branching ratio (ratio of the trajectories that enter each well) is changing as we modify the same parameter and its correlation with the area of the lobes as they have been formed by the stable and unstable manifolds that have been extracted from the gradient of the LD scalar fields.Integrability by separation of variableshttps://zbmath.org/1496.370602022-11-17T18:59:28.764376Z"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Ramírez, Rafael"https://zbmath.org/authors/?q=ai:ramirez.rafael-oSummary: We study the integrability in the Jacobi sense (integrability by separation of variables), of the Hamiltonian differential systems using the Levi-Civita Theorem. In particular we solve the Stark problem for \(N > 3\).On locally superquadratic Hamiltonian systems with periodic potentialhttps://zbmath.org/1496.370672022-11-17T18:59:28.764376Z"Ye, Yiwei"https://zbmath.org/authors/?q=ai:ye.yiweiSummary: In this paper, we study the second-order Hamiltonian systems
\[
\ddot{u}-L(t)u+\nabla W(t,u)=0,
\] where \(t\in \mathbb{R}\), \(u\in \mathbb{R}^N\), \(L\) and \(W\) depend periodically on \(t, 0\) lies in a spectral gap of the operator \(-d^2/dt^2+L(t)\) and \(W(t,x)\) is locally superquadratic. Replacing the common superquadratic condition that \(\lim_{|x|\rightarrow \infty}\frac{W(t,x)}{|x|^2}=+\infty\) uniformly in \(t\in \mathbb{R}\) by the local condition that \(\lim_{|x|\rightarrow \infty}\frac{W(t,x)}{|x|^2}=+\infty\) a.e. \(t\in J\) for some open interval \(J\subset \mathbb{R} \), we prove the existence of one nontrivial homoclinic soluiton for the above problem.Ground states for infinite lattices with nearest neighbor interactionhttps://zbmath.org/1496.370782022-11-17T18:59:28.764376Z"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng"Hu, Die"https://zbmath.org/authors/?q=ai:hu.die"Zhang, Yuanyuan"https://zbmath.org/authors/?q=ai:zhang.yuanyuan.1Summary: \textit{J. Sun} and \textit{S. Ma} [J. Differ. Equations 255, No. 8, 2534--2563 (2013; Zbl 1318.37026)]
proved the existence of a nonzero \(T\)-periodic solution for a class of one-dimensional lattice dynamical systems,
\[
\ddot{q_i}=\varPhi_{i-1}^\prime(q_{i-1}-q_i)- \varPhi_i^\prime(q_i-q_{i+1}),\quad i\in \mathbb{Z},
\] where \(q_i\) denotes the co-ordinate of the \(i\)th particle and \(\varPhi_i\) denotes the potential of the interaction between the \(i\)th and the \((i+1)\)th particle. We extend their results to the case of the least energy of nonzero \(T\)-periodic solution under general conditions. Of particular interest is a new and quite general approach. To the best of our knowledge, there is no result for the ground states for one-dimensional lattice dynamical systems.Periodic measures of reaction-diffusion lattice systems driven by superlinear noisehttps://zbmath.org/1496.370832022-11-17T18:59:28.764376Z"Lin, Yusen"https://zbmath.org/authors/?q=ai:lin.yusenSummary: The periodic measures are investigated for a class of reaction-diffusion lattice systems driven by superlinear noise defined on \(\mathbb{Z}^k\). The existence of periodic measures for the lattice systems is established in \(l^2\) by Krylov-Bogolyubov's method. The idea of uniform estimates on the tails of solutions is employed to establish the tightness of a family of distribution laws of the solutions.Dynamical analysis and chaos control of the fractional chaotic ecological modelhttps://zbmath.org/1496.370952022-11-17T18:59:28.764376Z"Mahmoud, Emad E."https://zbmath.org/authors/?q=ai:mahmoud.emad-e"Trikha, Pushali"https://zbmath.org/authors/?q=ai:trikha.pushali"Jahanzaib, Lone Seth"https://zbmath.org/authors/?q=ai:jahanzaib.lone-seth"Almaghrabi, Omar A."https://zbmath.org/authors/?q=ai:almaghrabi.omar-aSummary: In this paper the fractional version of the proposed integer order chaotic ecological system is studied. Here chaos has been observed in the competitive ecological model due to linear and nonlinear interactions among various species considering shortage of food resources. The system being important constituent of the food supply chain is analyzed using tools of dynamics viz. Lyapunov dynamics, bifurcation diagrams, existence and uniqueness of solution, the fixed point analysis and effect of fractional order on the dynamics of the system. In the presence of uncertainties and disturbances the chaos in the F.O. ecological model is controlled using adaptive SMC theory about its two fixed points. Numerical illustrations have been provided using MATLAB.\(q\)-Hamiltonian systemshttps://zbmath.org/1496.390042022-11-17T18:59:28.764376Z"Paşaoğlu, Bilender"https://zbmath.org/authors/?q=ai:pasaoglu.bilender-p"Tuna, Hüseyin"https://zbmath.org/authors/?q=ai:tuna.huseyinSummary: In this paper, we develop the basic theory of linear \(q\)-Hamiltonian systems. In this context, we establish an existence and uniqueness result. Regular spectral problems are studied. Later, we introduce the corresponding maximal and minimal operators for this system. Finally, we give a spectral resolution.Computational and theoretical aspects of Romanovski-Bessel polynomials and their applications in spectral approximationshttps://zbmath.org/1496.420372022-11-17T18:59:28.764376Z"Zaky, Mahmoud A."https://zbmath.org/authors/?q=ai:zaky.mahmoud-a"Abo-Gabal, Howayda"https://zbmath.org/authors/?q=ai:abo-gabal.howayda"Hafez, Ramy M."https://zbmath.org/authors/?q=ai:hafez.ramy-mahmoud"Doha, Eid H."https://zbmath.org/authors/?q=ai:doha.eid-hThe paper under review presents the main properties of a finite class of orthogonal polynomials with respect to the inverse gamma distribution over the positive real line called Romanovski-Bessel polynomials. More precisely, it introduces the related Romanovski-Bessel-Gauss-type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in weighted Sobolev space. It also addresses the relationship between such kinds of finite orthogonal polynomials and other classes of finite and infinite orthogonal polynomials.
Reviewer: M. Abdessadek Saib (Tebessa)New general integral transform via Atangana-Baleanu derivativeshttps://zbmath.org/1496.440042022-11-17T18:59:28.764376Z"Meddahi, M."https://zbmath.org/authors/?q=ai:meddahi.meryem"Jafari, H."https://zbmath.org/authors/?q=ai:jafari.hossein"Ncube, M. N."https://zbmath.org/authors/?q=ai:ncube.mahluli-naisbittSummary: The current paper is about the investigation of a new integral transform introduced recently by \textit{H. Jafari} [``A new general integral transform for solving integral equations'', J. Adv. Res. 32, 133--138 (2021; \url{doi:10.1016/j.jare.2020.08.016})]. Specifically, we explore the applicability of this integral transform on Atangana-Baleanu derivative and the associated fractional integral. It is shown that by applying specific conditions on this integral transform, other integral transforms are deduced. We provide examples to reinforce the applicability of this new integral transform.Qualitative analysis of integro-differential equations with variable retardationhttps://zbmath.org/1496.450072022-11-17T18:59:28.764376Z"Bohner, Martin"https://zbmath.org/authors/?q=ai:bohner.martin-j"Tunç, Osman"https://zbmath.org/authors/?q=ai:tunc.osmanThe paper is concerned with a class of nonlinear time-varying retarded integro-differential equations (RIDEs), which reads as
\[
\frac{\mathrm{d} x}{\mathrm{~d} t}=A(t) x+B F(x(t-\tau(t)))+\int_{t-\tau(t)}^{t} \Omega(t, s) F(x(s)) \mathrm{d} s+G(t, x),
\]
where \(x \in \mathbb{R}^{n},\ t \in \mathbb{R}^{+}:=[0, \infty), \ \tau \in \mathrm{C}^{1}\left(\mathbb{R}^{+}, \ \mathbb{R}^{+}\right), \ A=\left(a_{i j}\right) \in \mathrm{C}\left(\mathbb{R}^{+},\ \mathbb{R}^{n \times n}\right)\), \(\Omega=\left(\Omega_{i j}\right) \in \mathrm{C}\left(\mathbb{R}^{+} \times \mathbb{R}^{+}, \ \mathbb{R}^{n \times n}\right)\), \(B=\left(b_{i j}\right) \in \mathbb{R}^{n \times n}, \ F \in \mathrm{C}\left(\mathbb{R}^{n}, \mathbb{R}^{n}\right), \ F(0)=0\), and \(G\in \mathrm{C}\left(\mathbb{R}^{+} \times \mathbb{R}^{+}, \mathbb{R}^{n}\right)\). The authors focus on the uniform
stability and uniform asymptotic stability of the zero solution and integrability and boundedness of solutions in the case \(G(x,t)=0\). The boundedness of solutions at infinity is discussed for \(G(x,t)\neq 0\) too. Also, the authors provide two illustrative examples. Remarkably the given theorems are not only applicable to time-varying linear RIDEs, but also to nonlinear RIDEs depending on time.
Reviewer: Gaston Vergara-Hermosilla (Dublin)Spectral \(\zeta\)-functions and \(\zeta\)-regularized functional determinants for regular Sturm-Liouville operatorshttps://zbmath.org/1496.470102022-11-17T18:59:28.764376Z"Fucci, Guglielmo"https://zbmath.org/authors/?q=ai:fucci.guglielmo"Gesztesy, Fritz"https://zbmath.org/authors/?q=ai:gesztesy.fritz"Kirsten, Klaus"https://zbmath.org/authors/?q=ai:kirsten.klaus"Stanfill, Jonathan"https://zbmath.org/authors/?q=ai:stanfill.jonathanSummary: The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and \(\zeta\)-functions to efficiently compute values of spectral \(\zeta\)-functions at positive integers associated with regular (three-coefficient) self-adjoint Sturm-Liouville differential expressions \(\tau\). Depending on the underlying boundary conditions, we express the \(\zeta\)-function values in terms of a fundamental system of solutions of \(\tau y=zy\) and their expansions about the spectral point \(z=0\). Furthermore, we give the full analytic continuation of the \(\zeta\)-function through a Liouville transformation and provide an explicit expression for the \(\zeta\)-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schrödinger operators with zero, piecewise constant, and a linear potential on a compact interval.Perspectives on general left-definite theoryhttps://zbmath.org/1496.470402022-11-17T18:59:28.764376Z"Frymark, Dale"https://zbmath.org/authors/?q=ai:frymark.dale"Liaw, Constanze"https://zbmath.org/authors/?q=ai:liaw.constanzeSummary: In 2002, \textit{L. L. Littlejohn} and \textit{R. Wellman} [J. Differ. Equations 181, No. 2, 280--339 (2002; Zbl 1008.47029)] developed a celebrated general left-definite theory for semi-bounded self-adjoint operators with many applications to differential operators. The theory starts with a semi-bounded self-adjoint operator and constructs a continuum of related Hilbert spaces and self-adjoint operators that are intimately related with powers of the initial operator. The development spurred a flurry of activity in the field that is still ongoing today. The main goal of this expository (with the exception of Proposition~1) manuscript is to compare and contrast the complementary theories of general left-definite theory, the Birman-Krein-Vishik (BKV) theory of self-adjoint extensions and singular perturbation theory. In this way, we hope to encourage interest in left-definite theory as well as point out directions of potential growth where the fields are interconnected. We include several related open questions to further these goals.
For the entire collection see [Zbl 1479.47003].On couplings of symmetric operators with possibly unequal and infinite deficiency indiceshttps://zbmath.org/1496.470432022-11-17T18:59:28.764376Z"Mogilevskii, V. I."https://zbmath.org/authors/?q=ai:mogilevskii.vadimThe known results on couplings of symmetric operators \(A_j\), \(j\in\{1,2\}\), defined on the orthogonal sum of the Hilbert spaces, introduced by \textit{A. V. Shtraus} [Sov. Math., Dokl. 3, 779--782 (1962; Zbl 0151.19501); translation from Dokl. Akad. Nauk SSSR 144, 512--515 (1962)] are extended to the case of operators \(A_j\) with arbitrary (possibly unequal and infinite) deficiency indices. In particular, the coupling method based on the theory of boundary triplets is generalized. This makes it possible to obtain the abstract Titchmarsh formula, which gives the representation of the Weyl function of the coupling in terms of Weyl functions of boundary triplets for \(A_1\) and \(A_2\). \par Applications to ordinary differential operators are given.
Reviewer: Anatoly N. Kochubei (Kyïv)Multiplicative operator functions and abstract Cauchy problemshttps://zbmath.org/1496.470692022-11-17T18:59:28.764376Z"Früchtl, Felix"https://zbmath.org/authors/?q=ai:fruchtl.felixSummary: We use the duality between functional and differential equations to solve several classes of abstract Cauchy problems related to special functions. As a general framework, we investigate operator functions which are multiplicative with respect to convolution of a hypergroup. This setting contains all representations of (hyper)groups, and properties of continuity are shown; examples are provided by translation operator functions on homogeneous Banach spaces and weakly stationary processes indexed by hypergroups. Then we show that the concept of a multiplicative operator function can be used to solve a variety of abstract Cauchy problems, containing discrete, compact, and noncompact problems, including \(C_{0}\)-groups and cosine operator functions, and more generally, Sturm-Liouville operator functions.Tripled fixed point theorems and applications to a fractional differential equation boundary value problemhttps://zbmath.org/1496.470802022-11-17T18:59:28.764376Z"Afshari, Hojjat"https://zbmath.org/authors/?q=ai:afshari.hojjat"Kheiryan, Alireza"https://zbmath.org/authors/?q=ai:kheiryan.alirezaApplication measure of noncompactness and operator type contraction for solvability of an infinite system of differential equations in \(\ell_p\)-spacehttps://zbmath.org/1496.471312022-11-17T18:59:28.764376Z"Hazarikaa, Bipan"https://zbmath.org/authors/?q=ai:hazarikaa.bipan"Arab, Reza"https://zbmath.org/authors/?q=ai:arab.reza"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammadSummary: The aim of this paper is to obtain existence results for an infinite system of second order differential equations in the sequence space \(\ell_p\) for \(1 \leq p< \infty\) with the help of a technique associated with measures of noncompactness and contractive condition of operator type. We also provide some illustrative examples in support of our existence theorems.On the applications of a minimax theoremhttps://zbmath.org/1496.490052022-11-17T18:59:28.764376Z"Ricceri, Biagio"https://zbmath.org/authors/?q=ai:ricceri.biagioIn this paper, a minimax theorem is presented and four applications are given: uniquely remotal sets in normed spaces; multiple global minima for the integral functional of the Calculus of Variations; multiple periodic solutions for Lagrangian systems of relativistic oscillators; variational inequalities in balls of Hilbert spaces. A related challenging open problem is also pointed out.
Reviewer: Yang Yang (Wuxi)Polyhedral optimization of second-order discrete and differential inclusions with delayhttps://zbmath.org/1496.490122022-11-17T18:59:28.764376Z"Sağlam, Sevilay Demir"https://zbmath.org/authors/?q=ai:saglam.sevilay-demir"Mahmudov, Elimhan N."https://zbmath.org/authors/?q=ai:mahmudov.elimhan-nadirSummary: The present paper studies the optimal control theory of second-order polyhedral delay discrete and delay differential inclusions with state constraints. We formulate the conditions of optimality for the problems with the second-order polyhedral delay discrete (\(PD_d\)) and the delay differential (\(PC_d\)) in terms of the Euler-Lagrange inclusions and the distinctive ``transversality'' conditions. Moreover, some linear control problem with second-order delay differential inclusions is given to illustrate the effectiveness and usefulness of the main theoretic results.Covariant derivatives for Ehresmann connectionshttps://zbmath.org/1496.530292022-11-17T18:59:28.764376Z"Prince, G. E."https://zbmath.org/authors/?q=ai:prince.geoffrey-eamonn|prince.geoff-e"Saunders, D. J."https://zbmath.org/authors/?q=ai:saunders.david-j|saunders.dylan-jSummary: We deal with the construction of covariant derivatives for some quite general Ehresmann connections on fibre bundles. We show how the introduction of a vertical endomorphism allows construction of covariant derivatives separately on both the vertical and horizontal distributions of the connection which can then be glued together on the total space. We give applications across an important class of tangent bundle cases, frame bundles and the Hopf bundle.Optimal horizontal joinability on the Engel grouphttps://zbmath.org/1496.530422022-11-17T18:59:28.764376Z"Greshnov, Alexander"https://zbmath.org/authors/?q=ai:greshnov.alexandre|greshnov.aleksandr-valerevichAuthor's abstract: On the Engel group we solve the problem of finding the minimal number of segments of integral lines of left-invariant basis horizontal vector fields necessary for joining an arbitrary pair of points. We prove the best version of the Rashevskii-Chow theorem on the Engel group.
Reviewer: Andreea Olteanu (Bucureşti)Resolvent estimates on asymptotically cylindrical manifolds and on the half linehttps://zbmath.org/1496.580102022-11-17T18:59:28.764376Z"Christiansen, Tanya J."https://zbmath.org/authors/?q=ai:christiansen.tanya-j"Datchev, Kiril"https://zbmath.org/authors/?q=ai:datchev.kiril-rIn this article the authors study the spectral and scattering theory for a class of asympotically cylindrical manifolds with sufficiently mild geodesic trapping. One example of such a manifold is the cigar-shaped warped product (\(\mathbb{R}^d\), \(g_0\)), \(d \ge 2\), with metric \(g_0 = dr^2 + F(r) dS\), where \(r\) is the radial variable, \(dS\) is the usual metric on the unit sphere, and \(F(r) = r^2\) near \(r = 0\), while \(F'\) is compactly supported on some interval \([0,R]\) and positive on \((0,R)\). Another example is a convex cocompact hyperbolic surface \((X,g_H)\) for which there is a compact set \(N \subseteq X\) such that
\[
X \setminus N = (0,\infty)_r \times Y_y, \quad g_H\rvert_{X \setminus N} = dr^2 + \cosh^2 r dy^2,
\]
where \(Y\) is a disjoint union of \(k \ge 1\) geodesic circles. Indeed, in the first example, the only trapped geodesics are the circular ones on the cylindrical end (and this is the smallest amount of trapping a manifold with a cylindrical end can have.)
Suppose that trapping is suitably mild, in the sense that, in the presence of complex absorption, the resolvent is bounded polynomially in the spectral parameter \(z\) as \(\text{Re}\, z \to \infty\). Then the authors show that the number of embedded resonances and eigenvalues is finite, and that the cutoff resolvent (without complex absoprtion) is uniformly bounded as \(\text{Re}\, z \to \infty\). This bound is sharp in the setting of the first example described above.
Along the way to their main result, the authors also prove some resolvent estimates for repulsive potentials on the half-line.
Reviewer: Jacob Shapiro (Dayton)Regularization effects of a noise propagating through a chain of differential equations: an almost sharp resulthttps://zbmath.org/1496.600622022-11-17T18:59:28.764376Z"De Raynal, Paul-Éric Chaudru"https://zbmath.org/authors/?q=ai:chaudru-de-raynal.paul-eric"Menozzi, Stéphane"https://zbmath.org/authors/?q=ai:menozzi.stephaneSummary: We investigate the effects of the propagation of a non-degenerate Brownian noise through a chain of deterministic differential equations whose coefficients are rough and satisfy a weak like Hörmander structure (i.e. a non-degeneracy condition w.r.t. the components which transmit the noise). In particular we characterize, through suitable counter-examples, almost sharp regularity exponents that ensure that weak well posedness holds for the associated SDE. As a by-product of our approach, we also derive some density estimates of Krylov type for the weak solutions of the considered SDEs.Periodic averaging theorems for neutral stochastic functional differential equations involving delayed impulseshttps://zbmath.org/1496.600642022-11-17T18:59:28.764376Z"Liu, Jiankang"https://zbmath.org/authors/?q=ai:liu.jiankang"Xu, Wei"https://zbmath.org/authors/?q=ai:xu.wei.1"Guo, Qin"https://zbmath.org/authors/?q=ai:guo.qin"Wang, Jinbin"https://zbmath.org/authors/?q=ai:wang.jinbinSummary: This paper aims at addressing the issue of a periodic averaging method for neutral stochastic functional differential equations with delayed impulses. Two periodic averaging theorems are presented and the approximate equivalence between the solutions to the original systems and those to the reduced averaged systems without impulses is proved. Further, we show a brief framework of extending our main results to Lévy case. At last, an example is given to demonstrate the procedure and validity of the proposed periodic averaging method.Almost periodic and periodic solutions of differential equations driven by the fractional Brownian motion with statistical applicationhttps://zbmath.org/1496.600652022-11-17T18:59:28.764376Z"Marie, Nicolas"https://zbmath.org/authors/?q=ai:marie.nicolas"Raynaud de Fitte, Paul"https://zbmath.org/authors/?q=ai:raynaud-de-fitte.paulSummary: We show that the unique solution to a semilinear stochastic differential equation with almost periodic coefficients driven by a fractional Brownian motion is almost periodic in a sense related to random dynamical systems. This type of almost periodicity allows for the construction of a consistent estimator of the drift parameter in the almost periodic and periodic cases.Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroupshttps://zbmath.org/1496.600742022-11-17T18:59:28.764376Z"Zhang, Xuping"https://zbmath.org/authors/?q=ai:zhang.xuping"Chen, Pengyu"https://zbmath.org/authors/?q=ai:chen.pengyu"Abdelmonem, Ahmed"https://zbmath.org/authors/?q=ai:abdelmonem.ahmed"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: The aim of this paper is to discuss the existence of mild solutions for a class of semilinear stochastic partial differential equation with nonlocal initial conditions and noncompact semigroups in a real separable Hilbert space. Combined with the theory of stochastic analysis and operator semigroups, a generalized Darbo's fixed point theorem and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and a noncompactness measure condition. In addition, the condition of uniformly continuity of the nonlinearity is not required and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example to illustrate the feasibility of the main results is also given.RETRACTED ARTICLE: Existence of weak solutions of stochastic delay differential systems with Schrödinger-Brownian motionshttps://zbmath.org/1496.600752022-11-17T18:59:28.764376Z"Sun, Jianguo"https://zbmath.org/authors/?q=ai:sun.jianguo.2|sun.jianguo|sun.jianguo.1"Kou, Liang"https://zbmath.org/authors/?q=ai:kou.liang"Guo, Gang"https://zbmath.org/authors/?q=ai:guo.gang"Zhao, Guodong"https://zbmath.org/authors/?q=ai:zhao.guodong"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.29Summary: By using new Schrödinger type inequalities appearing in [\textit{Z. Jiang} and \textit{F. M. Usó}, J. Inequal. Appl. 2016, Paper No. 233, 10 p. (2016; Zbl 1350.35070)], we study the existence of weak solutions of stochastic delay differential systems with Schrödinger-Brownian motions.
Editorial remark. This article has been retracted. According to the retraction notice [\textit{J. Sun} et al., J. Inequal. Appl. 2021, Paper No. 109, 1 p. (2021; Zbl 1496.60076)], ``the Editors-in-Chief have retracted this article because it shows evidence of peer review manipulation. Additionally, the article shows significant overlap with an article by different authors that was simultaneously under consideration at another journal [\textit{J. Wang} et al., Bound Value Probl. 2018, Paper No. 74, 13 p. (2018; Zbl 1496.35370)]. Jianguo Sun agrees with the retraction but disagrees with the wording of the retraction notice. The other authors have not responded to the correspondence regarding this retraction.Retraction note: ``Existence of weak solutions of stochastic delay differential systems with Schrödinger-Brownian motions''https://zbmath.org/1496.600762022-11-17T18:59:28.764376Z"Sun, Jianguo"https://zbmath.org/authors/?q=ai:sun.jianguo.2|sun.jianguo.1|sun.jianguo"Kou, Liang"https://zbmath.org/authors/?q=ai:kou.liang"Guo, Gang"https://zbmath.org/authors/?q=ai:guo.gang"Zhao, Guodong"https://zbmath.org/authors/?q=ai:zhao.guodong"Wang, Yong"https://zbmath.org/authors/?q=ai:wang.yong.29Summary: The Editors-in-Chief have retracted this article [\textit{J. Sun}, ibid. 2018, Paper No. 100, 15 p. (2018; Zbl 1496.60075)] because it shows evidence of peer review manipulation. Additionally, the article shows significant overlap with an article by different authors that was simultaneously under consideration at another journal [\textit{J. Wang} et al., Bound Value Probl. 2018, Paper No. 74, 13 p. (2018; Zbl 1496.35370)]. Jianguo Sun agrees with the retraction but disagrees with the wording of the retraction notice. The other authors have not responded to the correspondence regarding this retraction.Error estimate for the approximate solution to multivariate feedback particle filterhttps://zbmath.org/1496.650082022-11-17T18:59:28.764376Z"Dong, Wenhui"https://zbmath.org/authors/?q=ai:dong.wenhui"Gao, Xingbao"https://zbmath.org/authors/?q=ai:gao.xingbaoSummary: In this paper, based on the assumption that the gain function \(K\) has been optimally obtained in the multivariate feedback particle filter (FPF), we focus on the error estimate for the approximate solutions to the particle's density evolution equation, which is actually the forward Kolmogorov equation (FKE) satisfied by the ``particle population''. The approximation is essentially the unnormalized density of the states conditioning on the discrete observations with the given time discretization. Mainly owing to the representation of Brownian bridges for the Brownian motion, and the assumption on the coercivity condition, we prove that the mean square error of the approximate solution is of order equal to the square root of the time interval.Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equationshttps://zbmath.org/1496.650092022-11-17T18:59:28.764376Z"Eisenmann, Monika"https://zbmath.org/authors/?q=ai:eisenmann.monika"Kovács, Mihály"https://zbmath.org/authors/?q=ai:kovacs.mihaly"Kruse, Raphael"https://zbmath.org/authors/?q=ai:kruse.raphael"Larsson, Stig"https://zbmath.org/authors/?q=ai:larsson.stigSummary: In this paper we derive error estimates of the backward Euler-Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable but assumed to be convex. We show that the backward Euler-Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in [\textit{R. H. Nochetto} et al., Commun. Pure Appl. Math. 53, No. 5, 525--589 (2000; Zbl 1021.65047)]. We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic \(p\)-Laplace equation.Strong convergence of a Euler-Maruyama scheme to a variable-order fractional stochastic differential equation driven by a multiplicative white noisehttps://zbmath.org/1496.650122022-11-17T18:59:28.764376Z"Yang, Zhiwei"https://zbmath.org/authors/?q=ai:yang.zhiwei"Zheng, Xiangcheng"https://zbmath.org/authors/?q=ai:zheng.xiangcheng"Zhang, Zhongqiang"https://zbmath.org/authors/?q=ai:zhang.zhongqiang"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.1Summary: We prove the existence and uniqueness of the solution to a variable-order fractional stochastic differential equation driven by a multiplicative white noise, which describes the random phenomena with nonlocal effect. We further develop a Euler-Maruyama scheme and prove the strong convergence of the scheme. Numerical experiments are presented to substantiate the mathematical analysis.A continuous kernel functions method for mixed-type functional differential equationshttps://zbmath.org/1496.650822022-11-17T18:59:28.764376Z"Gao, Y."https://zbmath.org/authors/?q=ai:gao.yuxuan|gao.yunjiao|gao.yitian|gao.yinmin|gao.yongsheng|gao.yongchun|gao.yunbao|gao.yuhong|gao.yune|gao.yisheng|gao.yongliang|gao.yingcai|gao.yaru|gao.yan|gao.yubing|gao.yuanqi|gao.ya|gao.yanpu|gao.yanxin|gao.yunlong|gao.yiming|gao.yunyuan|gao.yanxiang|gao.yansha|gao.yong|gao.yongyi|gao.yuanda|gao.yuanyuan|gao.yanbo|gao.yangjun|gao.yuling|gao.yanqiu|gao.yanfeng|gao.yongxiang|gao.yiping|gao.yingfang|gao.yuqiang|gao.yueyue|gao.yunlan|gao.yuxiang|gao.yicong|gao.yun|gao.yingjie|gao.yunliang|gao.yuzhuo|gao.yinghui|gao.yuping|gao.yazhou|gao.yanan|gao.yihan|gao.yufeng|gao.yongchan|gao.yueli|gao.yieping|gao.yannan|gao.yunjun|gao.yixian|gao.yueyuan|gao.yanfang|gao.yunxiao|gao.yanliang|gao.yunpeng|gao.yukui|gao.yongxin|gao.yuetian|gao.yufen|gao.yanbin|gao.yingxin|gao.yulin|gao.yingchun|gao.yunfei|gao.yujing|gao.yao|gao.yongdong|gao.yongqiang|gao.yuliang|gao.yunxin|gao.yuxiao|gao.yuan|gao.yongshuai|gao.yin-zhu|gao.yubin|gao.youxing|gao.yanyu|gao.yumin|gao.yanli|gao.yibo|gao.yanqing|gao.yitan|gao.yichen|gao.yongxi|gao.yaodong|gao.yuanboyi|gao.yuqiong|gao.yunkai|gao.yongbin|gao.yaqian|gao.yinling|gao.youmei|gao.yidi|gao.yuechao|gao.yakun|gao.yunzhu|gao.yanjun|gao.yahe|gao.yingying|gao.yuchen|gao.yongfeng|gao.yanping|gao.yuxin|gao.yuli|gao.yihang|gao.yanwei|gao.yonghua|gao.yachun|gao.yongfei|gao.yuhuan|gao.youtao|gao.yanfei|gao.yueqin|gao.yufei|gao.yajing|gao.yushan|gao.yali|gao.yanshan|gao.yongling|gao.yangen|gao.yunshi|gao.yuying|gao.yuerang|gao.yiqing|gao.yujie|gao.yunhui|gao.yanchao|gao.yongcun|gao.yanling|gao.yangqing|gao.yanyan|gao.yaxin|gao.yingchao|gao.yakui|gao.yuzhao|gao.yunfu|gao.yitiann|gao.yifan|gao.yihong|gao.yabin|gao.yaowen|gao.yizhe|gao.yuqing|gao.yue|gao.yi|gao.yaozong|gao.yunchuan|gao.yandong|gao.yingbin|gao.yuefeng|gao.yonghong|gao.yin|gao.yinzhi|gao.yunfeng|gao.yaping|gao.yinping|gao.yaling|gao.youwu|gao.yetian|gao.yongjiu|gao.yichao|gao.yajun|gao.yanhui|gao.yansong|gao.yonghui|gao.yanmin|gao.yulong|gao.yabing|gao.yipeng|gao.yuanwen|gao.yangyang|gao.yukun|gao.yingmin|gao.yulan|gao.you|gao.yudong|gao.yuyang|gao.yuefang|gao.ye|gao.yu|gao.yougang|gao.yaguang|gao.yuxia|gao.yubo|gao.yujia|gao.yanqun|gao.yuanning|gao.yachao|gao.yijin|gao.ying|gao.yanhong|gao.yuyan|gao.yuanjun|gao.yingtao|gao.yiu|gao.yufang|gao.yanni|gao.yili|gao.yuelin|gao.yang|gao.yunshu"Li, X. Y."https://zbmath.org/authors/?q=ai:li.xiaoyin|li.xingya|li.xiaoyun|li.xin-you|li.xuanya|li.xiaoyao|li.xiangyi|li.xueyun|li.xueyan|li.xuyong|li.xingyu|li.xingyong|li.xueyuan|li.xinyang|li.xinyan|li.xianyu|li.xian-ying|li.xiayun|li.xiaoyan|li.xiaoya|li.xiayang|li.xiangyong|li.xiangyang|li.xiaoyang|li.xinya|li.xiaoyong|li.xinyong|li.xiuyu|li.xinyun|li.xinying|li.xiayan|li.xiyao|li.xinyin|li.xiyan|li.xiangyou|li.xiuyun|li.xiuyong|li.xiangyu|li.xinyu|li.xiongya|li.xingyuan|li.xueyou|li.xiaoyi|li.xiyang|li.xianyin|li.xuyang|li.xiyue|li.xingyang|li.xiye|li.xinye|li.xuanying|li.xueyong|li.xueyao|li.xiaoyue|li.xueyang|li.xinyue|li.xiuying|li.xinyuan|li.xiangye|li.xiangyin|li.xiaoying|li.xiaoyuan|li.xiyu|li.xiangyun|li.xianyue|li.xiangyue|li.xingye|li.xianyong|li.xianyi|li.xiaoyu|li.xiaoye|li.xiongying|li.xianyue.1|li.xingyi|li.xiuyan|li.xinyi"Wu, B. Y."https://zbmath.org/authors/?q=ai:wu.bang-ye|wu.bo-yang|wu.baoyi|wu.boying|wu.baiyu|wu.bingye|wu.bangying|wu.bin-yi|wu.bingyang|wu.bingyao|wu.baoyuan|wu.baiyi|wu.bi-yi|wu.bangyuSummary: The goal of the paper is to propose a continuous technique for dealing with first-order linear mixed-type functional differential equations. The approach is established via the employment of the reproducing kernel functions and their nice property. The error results of the tests demonstrate that this approach can give good continuous approximations to the considerable problems.A novel numerical technique for fractional ordinary differential equations with proportional delayhttps://zbmath.org/1496.650832022-11-17T18:59:28.764376Z"Liaqat, Muhammad Imran"https://zbmath.org/authors/?q=ai:liaqat.muhammad-imran"Khan, Adnan"https://zbmath.org/authors/?q=ai:khan.adnan-qadir|khan.adnan-a"Akgül, Ali"https://zbmath.org/authors/?q=ai:akgul.ali"Ali, Md. Shajib"https://zbmath.org/authors/?q=ai:ali.md-shajib(no abstract)Semi-analytic shooting methods for Burgers' equationhttps://zbmath.org/1496.650852022-11-17T18:59:28.764376Z"Gie, Gung-Min"https://zbmath.org/authors/?q=ai:gie.gung-min"Jung, Chang-Yeol"https://zbmath.org/authors/?q=ai:jung.changyeol"Lee, Hoyeon"https://zbmath.org/authors/?q=ai:lee.hoyeonSummary: We implement new semi-analytic shooting methods for the stationary viscous Burgers' equation by modifying the classical time differencing methods. When the viscosity is small, a very stiff boundary layer appears and this boundary layer causes significant difficulties to approximate the solution for Burgers' equation. To overcome this issue and improve the numerical quality of the shooting methods with the classical Integrating Factor (IF) methods and Exponential Time Differencing (ETD) methods, we first employ the singular perturbation analysis for Burgers' equation, and derived the so-called correctors that approximate the stiff part of the solution. Then, we build our new semi-analytic shooting methods for the stationary viscous Burgers' equation by embedding these correctors into the IF and ETD methods. By performing numerical simulations, we verify that our new schemes, enriched with the correctors, give much better approximations, compared with the classical schemes.An effective finite element method with shifted fractional powers bases for fractional boundary value problemshttps://zbmath.org/1496.650922022-11-17T18:59:28.764376Z"Fu, Taibai"https://zbmath.org/authors/?q=ai:fu.taibai"Du, Changfa"https://zbmath.org/authors/?q=ai:du.changfa"Xu, Yufeng"https://zbmath.org/authors/?q=ai:xu.yufengSummary: In this paper, an effective finite element method with shifted fractional powers bases is developed for fractional convection diffusion equations involving a Riemann-Liouville derivative of order \(\alpha \in (3/2,2)\). A Petrov-Galerkin variational formulation is constructed on the domain \(\tilde{H}^{\alpha -1}(\Omega)\times \tilde{H}^1(\Omega)\), based on which the finite element approximation scheme is developed by employing shifted fractional power functions and continuous piecewise polynomials of degree up to \(m \, (m\in \mathbb{N}^+)\) for trial and test finite element spaces, respectively. The approximation property of trial finite element space and \(\inf -\sup\) condition for discrete variational form are derived, which enables us to derive the error estimates in \(L^2(\Omega)\) and \(H^{\alpha -1}(\Omega)\) norms. Numerical examples are included to verify the theoretical findings and demonstrate an actual convergence rate of order \(\alpha -1+m\), where \(m\) equals to 1 or 2.Novel chaotic systems with fractional differential operators: numerical approacheshttps://zbmath.org/1496.650942022-11-17T18:59:28.764376Z"Sweilam, N. H."https://zbmath.org/authors/?q=ai:sweilam.nasser-hassan"AL-Mekhlafi, S. M."https://zbmath.org/authors/?q=ai:al-mekhlafi.seham-mahyoub"Mohamed, D. G."https://zbmath.org/authors/?q=ai:mohamed.d-gSummary: The purpose of this paper is to study numerically the behavior of two novel different classes of fractional order chaotic systems. These systems are; the fractal-fractional hyperchaotic finance system and the fractal-fractional Bloch system with time delay. The fractal-fractional derivatives are defined in the Caputo and Riemann-Liouville senses. Two Grünwald-Letnikov nonstandard finite difference schemes are presented to study the proposed chaotic systems. Moreover the stability analysis of the used methods are proved. In order to show the simplicity and effectively of the proposed methods, numerical simulations and comparative studies are given.A numerical solution for fractional linear quadratic optimal control problems via shifted Legendre polynomialshttps://zbmath.org/1496.650972022-11-17T18:59:28.764376Z"Nezhadhosein, Saeed"https://zbmath.org/authors/?q=ai:nezhadhosein.saeed"Ghanbari, Reza"https://zbmath.org/authors/?q=ai:ghanbari.reza"Ghorbani-Moghadam, Khatere"https://zbmath.org/authors/?q=ai:ghorbani-moghadam.khatere(no abstract)Effective numerical technique for solving variable order integro-differential equationshttps://zbmath.org/1496.651772022-11-17T18:59:28.764376Z"El-Gindy, Taha M."https://zbmath.org/authors/?q=ai:el-gindy.taha-m"Ahmed, Hoda F."https://zbmath.org/authors/?q=ai:ahmed.hoda-f"Melad, Marina B."https://zbmath.org/authors/?q=ai:melad.marina-bSummary: In this article, an effective numerical technique for solving the variable order Fredholm-Volterra integro-differential equations (VO-FV-IDEs), systems of VO-FV-IDEs and variable order Volterra partial integro-differential equations (VO-V-PIDEs) is given. The suggested technique is built on the combination of the spectral collocation method with some types of operational matrices of the variable order fractional differentiation and integration of the shifted fractional Gegenbauer polynomials (SFGPs). The proposed technique reduces the considered problems to systems of algebraic equations that are straightforward to solve. The error bound estimation of using SFGPs is discussed. Finally, the suggested technique's authenticity and efficacy are tested via presenting several numerical applications. Comparisons between the outcomes achieved by implementing the proposed method with other numerical methods in the existing literature are held, the obtained numerical results of these applications reveal the high precision and performance of the proposed method.An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problemhttps://zbmath.org/1496.652052022-11-17T18:59:28.764376Z"Zhang, Hui"https://zbmath.org/authors/?q=ai:zhang.hui.22"Liu, Zixin"https://zbmath.org/authors/?q=ai:liu.zixin"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.11Summary: In this article, an effective finite element method based on dimension reduction scheme is proposed for a fourth-order Steklov eigenvalue problem in a circular domain. By using the Fourier basis function expansion and variable separation technique, the original problem is transformed into a series of radial one-dimensional eigenvalue problems with boundary eigenvalue. Then we introduce essential polar conditions and establish the discrete variational form for each radial one-dimensional eigenvalue problem. Based on the minimax principle and the approximation property of the interpolation operator, we prove the error estimates of approximation eigenvalues. Finally, some numerical experiments are provided, and the numerical results show the efficiency of the proposed algorithm.Discontinuous dynamical behaviors in a 2-DOF friction collision system with asymmetric dampinghttps://zbmath.org/1496.700102022-11-17T18:59:28.764376Z"Cao, Jing"https://zbmath.org/authors/?q=ai:cao.jing"Fan, Jinjun"https://zbmath.org/authors/?q=ai:fan.jinjunSummary: By using the flow switchability theory in discontinuous dynamical systems, this paper deals with the discontinuous dynamical behaviors of a two degrees of freedom system with asymmetric damping, where considering that friction and impact coexist and the static and dynamic friction coefficients are different. Because of the particularity of friction force, the flow barriers on the velocity boundary that affect the leaving flow are considered in this paper. Based on discontinuity that is caused by the sudden change of friction force or the collision between two objects, the phase space of motion for the object is divided into several different domains and boundaries; and with the help of the analysis of vector fields and G-functions on the corresponding discontinuous boundaries or in domains, the analytical conditions for all possible motions are obtained, which is used to determine the switching of motion state in this system. Finally, numerical simulations are presented to better understand the analytical conditions of the stick, grazing, impact, stuck and periodic motions.On the gradient flow formulation of the Lohe matrix model with high-order polynomial couplingshttps://zbmath.org/1496.700112022-11-17T18:59:28.764376Z"Ha, Seung-Yeal"https://zbmath.org/authors/?q=ai:ha.seung-yeal"Park, Hansol"https://zbmath.org/authors/?q=ai:park.hansolSummary: We present a first-order aggregation model for a homogeneous Lohe matrix ensemble with higher order couplings via a gradient flow approach. For homogeneous free flow with the same Hamiltonian, it is well known that the Lohe matrix model with cubic couplings can be recast as a gradient system with a potential which is a squared Frobenius norm of of averaged state. In this paper, we further derive a generalized Lohe matrix model with higher-order couplings via gradient flow approach for a polynomial potential. For the proposed model, we also provide a sufficient framework in terms of coupling strengths and initial data leading to the emergent dynamics of a homogeneous ensemble.Partial stability criterion for a heterogeneous power grid with hub structureshttps://zbmath.org/1496.780092022-11-17T18:59:28.764376Z"Khramenkov, Vladislav"https://zbmath.org/authors/?q=ai:khramenkov.vladislav"Dmitrichev, Aleksei"https://zbmath.org/authors/?q=ai:dmitrichev.aleksei"Nekorkin, Vladimir"https://zbmath.org/authors/?q=ai:nekorkin.vladimir-iSummary: One of the priority tasks in studying power grids is to find the conditions of their stable operation. The fundamental requirement is synchronizing all the elements (generators and consumers) of a power grid. However, various disturbances can destroy the synchronization. Moreover, desynchronization occurring in a small part of the grid can lead to severe large-scale power outages (or blackouts) due to numerous cascading failures. Here we consider the model of a heterogeneous power grid with hub structures (subgrids), taking into account arbitrary lengths and impedances of grid's transmission lines and also their arbitrary amount. Using the auxiliary comparison systems approach, we analyze the dynamics of a hub subgrid. Based on the findings, we develop a novel criterion of partial stability of power grids featuring hub structures. The criterion makes it possible to identify the regions of safe operation of individual hub subgrid elements. First, the criterion allows obtaining parameters' values that guarantee existence and local stability of either synchronous or quasi-synchronous modes in individual hub subgrid elements, i.e. steady-state stability. Second, the criterion allows obtaining the safe values of abrupt frequency and phase disturbances that eventually vanish, i.e. imply transient stability with respect to state disturbance. Third, the criterion permits determining the safe ranges of parameters' disturbances that do not lead to catastrophic desynchronizing effects, i.e. imply transient stability with respect to parameters' disturbance. Also, we discover typical dependences of the safe regions on the parameters of transmission lines. We demonstrate the applicability of the criterion on two test power grids with arbitrary distributions of powers as well as effective lengths and impedances of transmission lines. The results may help optimize stability and contribute to developing new real-time control schemes for smart grids that can automatically recover from failures.Bifurcation of solitary and periodic waves of an extended cubic-quintic Schrödinger equation with nonlinear dispersion effects governing modulated waves in a bandpass inductor-capacitor networkhttps://zbmath.org/1496.780132022-11-17T18:59:28.764376Z"Deffo, Guy Roger"https://zbmath.org/authors/?q=ai:deffo.guy-roger"Yamgoué, Serge Bruno"https://zbmath.org/authors/?q=ai:yamgoue.serge-bruno"Pelap, François Beceau"https://zbmath.org/authors/?q=ai:pelap.francois-beceauSummary: The present work describes the behavior of solitary and periodic waves in a nonlinear electrical transmission line with linear dispersion. Based on the semidiscrete approximation, we show that the dynamics of modulated wave in the system can be described by an extended cubic-quintic nonlinear Schrödinger equation. Using a simple transformation, we reduce the given equation to a cubic-quintic Duffing oscillator equation. By means of the method of dynamical systems, we obtain bifurcations of the phase portraits of the traveling wave under different parameter conditions. Corresponding to the various phase portrait trajectories, we derive possible exact explicit parametric representations of solutions. The results of our study demonstrate that the additional imprint phase in the signal voltage leads to a number of interesting solitary-wave solutions, e.g., gray soliton and anti-gray soliton, which have not been observed for the same model without this parameter. These new obtained solutions are useful in better understanding of the dynamic of the considered network as well as of other systems that can be governed by a cubic-quintic nonlinear Schrödinger equation model.Laplace transform method in one dimensional quantum mechanics on the semi infinite axishttps://zbmath.org/1496.810512022-11-17T18:59:28.764376Z"Chung, Wonsang"https://zbmath.org/authors/?q=ai:chung.won-sang"Kim, Yeounju"https://zbmath.org/authors/?q=ai:kim.yeounju"Kwon, Jeongmin"https://zbmath.org/authors/?q=ai:kwon.jeongminSummary: In this paper we discuss the Laplace transform method for solving one dimensional Schrödinger equation in a semi infinite axis. As examples we discuss the delta potential, quantum bouncer, Coulomb-like potential and half harmonic potential.Exact solutions of an asymmetric double well potentialhttps://zbmath.org/1496.810552022-11-17T18:59:28.764376Z"Sun, Guo-Hua"https://zbmath.org/authors/?q=ai:sun.guohua"Dong, Qian"https://zbmath.org/authors/?q=ai:dong.qian"Bezerra, V. B."https://zbmath.org/authors/?q=ai:bezerra.valdir-b"Dong, Shi-Hai"https://zbmath.org/authors/?q=ai:dong.shihaiSummary: The analytical solutions of an asymmetric double well potential \(V(x)=a\, x^2-b\, x^3+c\, x^4\) are found to be a triconfluent Heun function \(H_T(\alpha , \beta , \gamma ; z)\). It should be emphasized that these potential parameters are taken arbitrarily without any restriction on them. The wave functions which depend on the potential parameters are shrunk toward to the origin for given \(b\) and \(c\) when the parameter \(a\) increases, while they are moved far from the origin and toward to the left when the parameter \(b\) increases for given \(a\) and \(c\). Also, when the parameter \(c\) increases for given \(a\) and \(b\) they have the similar property to the case when the parameter \(a\) increases.Explosive synchronization induced by environmental couplinghttps://zbmath.org/1496.810742022-11-17T18:59:28.764376Z"Ramesan, Gayathri"https://zbmath.org/authors/?q=ai:ramesan.gayathri"Shajan, Emilda"https://zbmath.org/authors/?q=ai:shajan.emilda"Shrimali, Manish Dev"https://zbmath.org/authors/?q=ai:shrimali.manish-devSummary: The occurrence of explosive synchronization transition in a system of limit-cycle oscillators in the presence of two types of coupling; direct mean field diffusive and indirect environmental couplings, both operating simultaneously, is reported. The dynamics of coupled nonlinear Van der Pol and Rayleigh oscillators are studied in detail as a function of the distribution of intrinsic parameters of the oscillators. This explosive synchronization transition depends on the strength of indirect coupling and is irreversible giving rise to a characteristic hysteresis region. The different routes to synchronization observed in these coupled oscillators are studied in detail with the help of effective frequency and time series analysis. We have investigated the efficiency of the proposed scheme in various other topologies such as random, scale-free, and two-community networks as well.Muon \(g - 2\) anomaly and non-localityhttps://zbmath.org/1496.810992022-11-17T18:59:28.764376Z"Capolupo, A."https://zbmath.org/authors/?q=ai:capolupo.antonio"Lambiase, G."https://zbmath.org/authors/?q=ai:lambiase.gaetano"Quaranta, A."https://zbmath.org/authors/?q=ai:quaranta.antonella|quaranta.anna-graziaSummary: We show that the discrepancy between the observed value of the muon anomalous moment and the standard model prediction can be explained in the framework of nonlocal theories. We compute the leading order and next to leading order nonlocal correction to the anomalous magnetic moment \(\alpha_{NL}\) and we find that it depends on the nonlocality scale \(M_f\) and the fermion mass \(m_f\) as \(\alpha_{NL} \propto \frac{m_f^2}{M_f^2}\). Such a dependence of the anomalous magnetic moment allows to explain, in a flavor-blind nonlocality scale, why the observed anomalous magnetic moment of the electron is much closer to the standard model prediction, and permits to predict a large anomaly that should exist for the \(\tau\) particle. We also determine the lower bounds on the nonlocality scale, for both flavor-blind and flavor-dependent scenarios.Running primal-dual gradient method for time-varying nonconvex problemshttps://zbmath.org/1496.900682022-11-17T18:59:28.764376Z"Tang, Yujie"https://zbmath.org/authors/?q=ai:tang.yujie"Dall'Anese, Emiliano"https://zbmath.org/authors/?q=ai:dallanese.emiliano"Bernstein, Andrey"https://zbmath.org/authors/?q=ai:bernstein.andrey"Low, Steven"https://zbmath.org/authors/?q=ai:low.steven-hEvolutionary dynamics in the rock-paper-scissors system by changing community paradigm with population flowhttps://zbmath.org/1496.910192022-11-17T18:59:28.764376Z"Park, Junpyo"https://zbmath.org/authors/?q=ai:park.junpyoSummary: Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.Stochastic model of innovation diffusion that takes into account the changes in the total market volumehttps://zbmath.org/1496.910522022-11-17T18:59:28.764376Z"Parphenova, Alena Yu."https://zbmath.org/authors/?q=ai:parphenova.alena-yu"Saraev, Leonid A."https://zbmath.org/authors/?q=ai:saraev.leonid-aleksandrovichSummary: The article proposes a stochastic mathematical model of the diffusion of consumer innovations, which takes into account changes over time in the total number of potential buyers of an innovative product. A stochastic differential equation is constructed for a random value of the number of consumers of an innovative product. The interaction of random changes in the number of consumers with changes in the total market volume of the product under consideration is investigated. Following the Euler-Maruyama method, an algorithm for the numerical solution of the stochastic differential equation for the diffusion of innovations is constructed. For each implementation of this algorithm, the corresponding stochastic trajectories are constructed for a random function of the number of consumers of an innovative product. A variant of the method for calculating the mathematical expectation of a random function of the number of consumers of an innovative product is developed and the corresponding differential equation is obtained. It is shown that the numerical solution of this equation and the average value of the function of the number of consumers calculated for all the implemented implementations of stochastic trajectories give practically the same results. Numerical analysis of the developed model showed that taking into account an external random disturbing factor in the stochastic model leads to significant deviations from the classical deterministic model of smooth market development with innovative goods.Dynamics and control of delayed rumor propagation through social networkshttps://zbmath.org/1496.910692022-11-17T18:59:28.764376Z"Ghosh, Moumita"https://zbmath.org/authors/?q=ai:ghosh.moumita"Das, Samhita"https://zbmath.org/authors/?q=ai:das.samhita"Das, Pritha"https://zbmath.org/authors/?q=ai:das.prithaSummary: Investigation of rumor spread dynamics and its control in social networking sites (SNS) has become important as it may cause some serious negative effects on our society. Here we aim to study the rumor spread mechanism and the influential factors using epidemic like model. We have divided the total population into three groups, namely, ignorant, spreader and aware. We have used delay differential equations to describe the dynamics of rumor spread process and studied the stability of the steady-state solutions using the threshold value of influence which is analogous to the basic reproduction number in disease model. Global stability of rumor prevailing state has been proved by using Lyapunov function. An optimal control system is set up using media awareness campaign to minimize the spreader population and the corresponding cost. Hopf bifurcation analyses with respect to time delay and the transmission rate of rumors are discussed here both analytically and numerically. Moreover, we have derived the stability region of the system corresponding to change of transmission rate and delay values.Pricing of financial derivatives based on the Tsallis statistical theoryhttps://zbmath.org/1496.910932022-11-17T18:59:28.764376Z"Zhao, Pan"https://zbmath.org/authors/?q=ai:zhao.pan"Pan, Jian"https://zbmath.org/authors/?q=ai:pan.jian"Yue, Qin"https://zbmath.org/authors/?q=ai:yue.qin"Zhang, Jinbo"https://zbmath.org/authors/?q=ai:zhang.jinboSummary: Asset return distributions usually have peaks, fat tails and skewed tails, because of the impact of extreme events outside financial markets. The Tsallis distribution has the peak and fat-tail characteristic, and the asymmetric jump process can fit the skewed tail of returns. Therefore, to accurately describe asset returns, we propose a price model by the use of the Tsallis distribution and a Poisson jump process, which can characterize the long-term memory and the skewness of asset returns. Moreover, using the stochastic differential theory and the martingale method, we obtain an explicit solution for pricing European options.Reliability index and Asian barrier option pricing formulas of the uncertain fractional first-hitting time model with Caputo typehttps://zbmath.org/1496.911012022-11-17T18:59:28.764376Z"Jin, Ting"https://zbmath.org/authors/?q=ai:jin.ting"Ding, Hui"https://zbmath.org/authors/?q=ai:ding.hui"Xia, Hongxuan"https://zbmath.org/authors/?q=ai:xia.hongxuan"Bao, Jinfeng"https://zbmath.org/authors/?q=ai:bao.jinfengSummary: This paper mainly studies the pricing problem of arithmetic average Asian barrier options in the continuous case and analyzes the corresponding reliability index. Owing to the fact that the return function of the option is closely related to the average price of the underlying asset in a certain period, which can effectively alleviate the market speculation, the Asian barrier option is widely active in the financial market as a derivative product. Firstly, considering that there exists difficult to predict and measure reliability based on historical data, traditional stock models relied on stochastic theory are abandoned, and further assumes that the underlying assets follow an uncertain process. Then, we introduce the uncertain fractional differential equations with Caputo-type to describe the dynamic changes of risky asset prices. Secondly, in order to analyze the impact on the first hitting time of the barrier being triggered and the option execution result theoretically, a novel first-hitting time model is established to measure the relationship between the option reliability index and the value of risky assets. Meanwhile, the pricing formulas of four Asian barrier options and the corresponding reliability index under the first-hitting time model are derived, respectively. Finally, numerical algorithms and corresponding methods are designed to verify the results of our model.Adaptation mechanisms in phosphorylation cycles by allosteric binding and gene autoregulationhttps://zbmath.org/1496.920182022-11-17T18:59:28.764376Z"Fang, Zhou"https://zbmath.org/authors/?q=ai:fang.zhou"Jayawardhana, Bayu"https://zbmath.org/authors/?q=ai:jayawardhana.bayu"van der Schaft, Arjan"https://zbmath.org/authors/?q=ai:van-der-schaft.arjan-j"Gao, Chuanhou"https://zbmath.org/authors/?q=ai:gao.chuanhouEditorial remark: No review copy delivered.The impact of distributed time delay in a tumor-immune interaction systemhttps://zbmath.org/1496.920212022-11-17T18:59:28.764376Z"Sardar, Mrinmoy"https://zbmath.org/authors/?q=ai:sardar.mrinmoy"Biswas, Santosh"https://zbmath.org/authors/?q=ai:biswas.santosh"Khajanchi, Subhas"https://zbmath.org/authors/?q=ai:khajanchi.subhasSummary: The impact of continuously distributed delay has been investigated in this paper to describe the interaction among tumor cells, tumor-specific CD8+T cells, helper T cells and immuno-stimulatory cytokine interleukin-2 (IL-2) through a system of coupled nonlinear ordinary differential equations. We analyze the qualitative properties of the model such as positivity of the solutions and the existence of biologically feasible equilibrium points. Next, we discuss the local asymptotic stability for the delayed and non-delayed system. Our model system experiences Hopf bifurcation with respect to the activation rate \(\lambda_1\) of tumor-specific CD8+T cells. The effect of continuously distributed delay involved in immune-activation on the system dynamics of the tumor is demonstrated. Our study reveals that the activation rate of CD8+T cells can prevent the oscillation of tumor-presence equilibria as well as tumor-free equilibria of the system. Then we performed some numerical results and interpret their biological implications to validate our analytical findings.The kinetic space of multistationarity in dual phosphorylationhttps://zbmath.org/1496.920282022-11-17T18:59:28.764376Z"Feliu, Elisenda"https://zbmath.org/authors/?q=ai:feliu.elisenda"Kaihnsa, Nidhi"https://zbmath.org/authors/?q=ai:kaihnsa.nidhi"de Wolff, Timo"https://zbmath.org/authors/?q=ai:de-wolff.timo"Yürük, Oğuzhan"https://zbmath.org/authors/?q=ai:yuruk.oguzhanIn biology and biochemistry, phosphorylation (as well as its inverse, dephosphorylation) is a fundamental modification process consisting in the attachment (or removal, resp.) of a phosphate group. It is important in cell signaling and is a special case of post-translational modification (PTM) in that it, e.g., changes the behavior of a compound w.r.t. a membrane. It is managed by an enzyme reaction open to quantitative description by the ODE system of a mass action reaction network. The standard machinery is applied to the enzyme reaction of a dual phosphorylation, involving a substrate with two phosphorylation sites and two enzymes, resulting in a polynomial ODE system with nine species concentrations and 12 rate constants (parameters). The objective is to study the parameter space w.r.t. points of multistationarity. In the situation investigated here, real algebraic geometry allows rather precise statements on the parameter areas of monostationarity and multistationarity, their boundaries and their connectedness. Suitable polynomials, their signs, Newton polytopes and cylindrical algebraic decompositions play a decisive role, as well as application of symbolic algorithms. The approach explored here is relevant not only for the system itself, but lends itself also to test the examination of similar models.
Reviewer: Dieter Erle (Dortmund)Quasi-steady-state and singular perturbation reduction for reaction networks with noninteracting specieshttps://zbmath.org/1496.920292022-11-17T18:59:28.764376Z"Feliu, Elisenda"https://zbmath.org/authors/?q=ai:feliu.elisenda"Lax, Christian"https://zbmath.org/authors/?q=ai:lax.christian"Walcher, Sebastian"https://zbmath.org/authors/?q=ai:walcher.sebastian"Wiuf, Carsten"https://zbmath.org/authors/?q=ai:wiuf.carstenSimplification of systems of ODE, in the presence of fast and slow variables, is frequently achieved by dimension reduction. Quasi-steady state approximation, originally justified heuristically, meanwhile has been put in the framework of singular perturbation. This can be looked up e.g. in reference [\textit{A. Goeke} et al., Physica D 345, 11--26 (2017; Zbl 1378.34033)]. Building on that, the authors study systems of ODE where the variables to be removed occur linearly. They develop conditions under which quasi-steady state reduction and singular perturbation reduction coincide. In more detail, in a suitable situation, two rather technical so-called blanket conditions allow for singular perturbation reduction and quasi-steady state reduction, guarantee coincidence and provide a formula for the reduced system. The results are applied to reaction networks based on mass action kinetics with noninteracting species. In a typical case, a set of species is noninteracting if ``no species in the set appear together on the same side of a reaction, and all appear with coefficient at most one in any complex''. Examples display the dependence of the blanket conditions on the reaction constants. From the part of the reaction network involving only the noninteracting species, a directed multidigraph is constructed that permits checking the blanket conditions. In this context, applications deal with the classical Michaelis-Menten system and a generalization, namely post-translational modification systems, furthermore a predator-prey system and a two substrate mechanism.
Reviewer: Dieter Erle (Dortmund)Impact of enzyme turnover on the dynamics of the Michaelis-Menten modelhttps://zbmath.org/1496.920302022-11-17T18:59:28.764376Z"Peletier, Lambertus A."https://zbmath.org/authors/?q=ai:peletier.lambertus-adrianus"Gabrielsson, Johan"https://zbmath.org/authors/?q=ai:gabrielsson.johanThe authors continue their investigations on the open Michaelis-Menten enzyme system begun in reference [\textit{J. Gabrielsson} and \textit{L. Peletier}, ``Michaelis-Menten from an in vivo perspective: open versus closed systems'', AAPS Journal 20, Paper No. 102, 13 p. (2018; \url{doi:10.1208/s12248-018-0256-z})]. In the closed system, the total pool of enzyme, free and bound, is constant. Inserting synthesis and degeneration of enzyme (enzymatic turnover) into the model leads to the open version. This model is closer to what happens in vivo, where proteins are steadily synthesized and eliminated. It also allows to keep track of the development over time after constant and after bolus supply of substrate. Various details of the open system are uncovered. An example is the fact that the maximal substrate value turns out to depend quadratically on the infusion rate. Also, the effect of the rates of enzyme turnover and of catalysis on the dynamics of the open system is discussed, and the limit behavior of the open system to the closed one if the turnover tends to zero is studied. Graphical simulation diagrams illustrate numerous results.
Reviewer: Dieter Erle (Dortmund)Effects of delays in mathematical models of cancer chemotherapyhttps://zbmath.org/1496.920312022-11-17T18:59:28.764376Z"Abdulrashid, Ismail"https://zbmath.org/authors/?q=ai:abdulrashid.ismail"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Han, Xiaoying"https://zbmath.org/authors/?q=ai:han.xiaoyingSummary: Two mathematical models of chemotherapy cancer treatment are studied and compared, one modeling the chemotherapy agent as the predator and the other modeling the chemotherapy agent as the prey. In both models constant delay parameters are introduced to incorporate the time lapsed from the instant the chemotherapy agent is injected to the moment it starts to be effective. For each model, the existence and uniqueness of non-negative bounded solutions are first established. Then both local and Lyapunov stability for all steady states are investigated. In particular, sufficient conditions dependent of the delay parameters under which each steady state is asymptotically stable are constructed. Numerical simulations are presented to illustrate the theoretical results.Dynamical characteristics and signal flow graph of nonlinear fractional smoking mathematical modelhttps://zbmath.org/1496.920352022-11-17T18:59:28.764376Z"Mahdy, A. M. S."https://zbmath.org/authors/?q=ai:mahdy.amr-m-s"Mohamed, M. S."https://zbmath.org/authors/?q=ai:mohamed.mohamed-saad"Gepreel, K. A."https://zbmath.org/authors/?q=ai:gepreel.khaled-a"AL-Amiri, A."https://zbmath.org/authors/?q=ai:alamiri.abdalla|al-amiri.abdalla-m"Higazy, M."https://zbmath.org/authors/?q=ai:higazy.m-shSummary: In this article, we study approximate analytic solutions for one of the famous models in biomathematics, namely the nonlinear fractional mathematical smoking model. When alpha=1, we deduce the analytical solution using the Reduced differential transforms method (RDTM) for the nonlinear ordinary mathematical smoking model. The disease-free equilibrium point, the stability of the equilibrium point, and the reproduction number are all discussed for the fractional mathematical smoking model. We use mathematical software packages such as Mathematica to find more iteration when calculating the approximate solution. Results are presented graphically to illustrate the behavior of the obtained approximate solutions. The system is presented by a proposed novel signal flow graph and simulated via SIMULINK/MATLAB. The graph of signal flow is used to calculate some of the model invariants such as the adjacency matrix, model energy, and Estrada index. Also, the novelty and significance of the results are clear using a 3D plot.Is the Allee effect relevant to stochastic cancer model?https://zbmath.org/1496.920372022-11-17T18:59:28.764376Z"Sardar, Mrinmoy"https://zbmath.org/authors/?q=ai:sardar.mrinmoy"Khajanchi, Subhas"https://zbmath.org/authors/?q=ai:khajanchi.subhas(no abstract)Bistability in deterministic and stochastic SLIAR-type models with imperfect and waning vaccine protectionhttps://zbmath.org/1496.920412022-11-17T18:59:28.764376Z"Arino, Julien"https://zbmath.org/authors/?q=ai:arino.julien"Milliken, Evan"https://zbmath.org/authors/?q=ai:milliken.evanSummary: Various vaccines have been approved for use to combat COVID-19 that offer imperfect immunity and could furthermore wane over time. We analyze the effect of vaccination in an SLIARS model with demography by adding a compartment for vaccinated individuals and considering disease-induced death, imperfect and waning vaccination protection as well as waning infections-acquired immunity. When analyzed as systems of ordinary differential equations, the model is proven to admit a backward bifurcation. A continuous time Markov chain (CTMC) version of the model is simulated numerically and compared to the results of branching process approximations. While the CTMC model detects the presence of the backward bifurcation, the branching process approximation does not. The special case of an SVIRS model is shown to have the same properties.Global Hopf branches in a delayed model for immune response to HTLV-1 infections: coexistence of multiple limit cycleshttps://zbmath.org/1496.920432022-11-17T18:59:28.764376Z"Li, Michael Y."https://zbmath.org/authors/?q=ai:li.michael-yi"Lin, Xihui"https://zbmath.org/authors/?q=ai:lin.xihui"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.4(no abstract)Competition in the chemostat: some remarkshttps://zbmath.org/1496.920462022-11-17T18:59:28.764376Z"de Leenheer, Patrick"https://zbmath.org/authors/?q=ai:de-leenheer.patrick"Li, Bingtuan"https://zbmath.org/authors/?q=ai:li.bingtuan"Smith, Hal L."https://zbmath.org/authors/?q=ai:smith.hal-leslie(no abstract)Bifurcations in the symmetric viability model with weak selectionhttps://zbmath.org/1496.920522022-11-17T18:59:28.764376Z"Li, Yingtao"https://zbmath.org/authors/?q=ai:li.yingtao"Zhang, Weinian"https://zbmath.org/authors/?q=ai:zhang.weinianThis paper is concerned with the symmetric viability model in population genetics, which considers the genetic structure and evolution of populations when having four alleles at two loci. The authors focus on ``the non-hyperbolic case of equilibria in the symmetric viability model with weak selection and their bifurcations''. They show ``that the system has at most 5 interior equilibria, 4 edge equilibria and 4 corner equilibria and give a complete qualitative analysis for all equilibria in both hyperbolic cases and non-hyperbolic cases, including the case of fully null degeneracy''. They display ``all bifurcations of equilibria such as pitchfork bifurcation, transcritical bifurcation and bifurcations at corner equilibria of fully null degeneracy, which is of codimension 2 but none of known results of codimension 2 can be applied to''. They further discuss ``the unfolding system of fully null degeneracy, displaying all its bifurcations and giving bifurcation curves analytically''.
Reviewer: Thomas Jiaxian Li (Charlottesville)Stability of three species symbiosis model with delay and stochastic perturbationshttps://zbmath.org/1496.920792022-11-17T18:59:28.764376Z"Abbas, Syed"https://zbmath.org/authors/?q=ai:abbas.syed"Shaikhet, Leonid"https://zbmath.org/authors/?q=ai:shaikhet.leonid-eSummary: In this paper a three-species symbiosis population model with delay and stochastic perturbations is considered. The model is modified by considering more general rate which ensure the existence of at least one nontrivial equilibrium. Conditions for the existence of positive equilibrium of the considered model are obtained. New sufficient conditions of stability in probability for the obtained positive equilibrium are formulated in the terms of linear matrix inequalities (LMIs), which can be investigated by virtue of MATLAB. Besides some necessary stability conditions are formulated in the form of simple analytical inequalities. The results obtained are illustrated via numerical simulations of a solution of the considered model.An efficient numerical technique for a biological population model of fractional orderhttps://zbmath.org/1496.920812022-11-17T18:59:28.764376Z"Attia, Nourhane"https://zbmath.org/authors/?q=ai:attia.nourhane"Akgül, Ali"https://zbmath.org/authors/?q=ai:akgul.ali"Seba, Djamila"https://zbmath.org/authors/?q=ai:seba.djamila"Nour, Abdelkader"https://zbmath.org/authors/?q=ai:nour.abdelkaderSummary: In the present paper, a biological population model of fractional order (FBPM) with one carrying capacity has been examined with the help of reproducing kernel Hilbert space method (RKHSM). This important fractional model arises in many applications in computational biology. It is worth noting that, the considered FBPM is used to provide the changes that is made on the densities of the predator and prey populations by the fractional derivative. The technique employed to construct new numerical solutions for the FBPM which is considered of a system of two nonlinear fractional ordinary differential equations (FODEs). In the proposed investigation, the utilised fractional derivative is the Caputo derivative. The most valuable advantages of the RKHSM is that it is easily and fast implemented method. The solution methodology is based on the use of two important Hilbert spaces, as well as on the construction of a normal basis through the use of Gram-Schmidt orthogonalization process. We illustrate the high competency and capacity of the suggested approach through the convergence analysis. The computational results, which are compared with the homotopy perturbation Sumudu transform method (HPSTM), clearly show: On the one hand, the effect of the fractional derivative in the obtained outcomes, and on the other hand, the great agreement between the mentioned methods, also the superior performance of the RKHSM. The numerical computational are presented in illustrated graphically to show the variations of the predator and prey populations for various fractional order derivatives and with respect to time.Impact of predator incited fear and prey refuge in a fractional order prey predator modelhttps://zbmath.org/1496.920822022-11-17T18:59:28.764376Z"Barman, Dipesh"https://zbmath.org/authors/?q=ai:barman.dipesh"Roy, Jyotirmoy"https://zbmath.org/authors/?q=ai:roy.jyotirmoy-sinha"Alrabaiah, Hussam"https://zbmath.org/authors/?q=ai:alrabaiah.hussam"Panja, Prabir"https://zbmath.org/authors/?q=ai:panja.prabir"Mondal, Sankar Prasad"https://zbmath.org/authors/?q=ai:mondal.sankar-prasad"Alam, Shariful"https://zbmath.org/authors/?q=ai:alam.sharifulSummary: In this article, a predator-prey model has been evolved in the form of a system of fractional order differential equations incorporating two important factors, namely, fear factor and prey refuge factor. Here, the fractional calculus has been taken into consideration to investigate the dynamical behaviour of the solutions of the proposed model system as the changes in life cycle of prey species are of memory bound. Biological validation and well-posedness such as positivity and boundedness of solutions of the model system have been proved analytically. Stability analysis of all the feasible equilibrium points of the model system has been performed in a systematic way. Some important dynamical features of the model system (such as transition of stability of the system) have been demonstrated through rigorous numerical simulation. It is observed that our proposed model system experiences Hopf-bifurcation around the interior equilibrium point with respect to both the parameters \(f\) and \(m_1\), which are linked with amount of predator induced fear and rate of prey refuge, respectively. The system dynamics is more likely to be stable in the framework of fractional order derivative in comparison to integer-order derivative. The high amount of predator induced fear \(f\) and prey refuge rate \(m_1\) are independently capable to make the system dynamics to be stable in integer order model system. On the other hand, the dynamics of the model system shifts towards the stability from its unstable behaviour when we continuously reduce the order of the model system; especially under the scenario of low level of predator induced fear and prey refuge rate. Thus, our comprehensive mathematical findings reveal the fact that fading memory can play a contributory role towards stable coexistence of the predator-prey system whereas strong memory of the species deteriorates the stable coexistence of the model system.Note on the persistence and stability property of a commensalism model with Michaelis-Menten harvesting and Holling type II commensalistic benefithttps://zbmath.org/1496.920852022-11-17T18:59:28.764376Z"Chen, Fengde"https://zbmath.org/authors/?q=ai:chen.fengde"Chen, Yuming"https://zbmath.org/authors/?q=ai:chen.yuming"Li, Zhong"https://zbmath.org/authors/?q=ai:li.zhong.1"Chen, Lijuan"https://zbmath.org/authors/?q=ai:chen.lijuanSummary: In this paper, we revisit the commensalism model proposed and analyzed recently by \textit{S. Jawad} [``Study the dynamics of commensalism interaction with Michaels-Menten type prey harvesting'', Al-Nahrain J. Sci. 25, No. 1, 45--50 (2022; \url{doi:10.22401/ANJS.25.1.08})]. By applying the standard comparison theorem and the theory of asymptotically autonomous systems, we completely describe the partial survival, permanence, and global stability of the positive equilibrium of the system. These results not only complement but also essentially improve the corresponding ones of Jawad.Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvestinghttps://zbmath.org/1496.920862022-11-17T18:59:28.764376Z"Chen, Jialin"https://zbmath.org/authors/?q=ai:chen.jialin"Zhu, Zhenliang"https://zbmath.org/authors/?q=ai:zhu.zhenliang"He, Xiaqing"https://zbmath.org/authors/?q=ai:he.xiaqing"Chen, Fengde"https://zbmath.org/authors/?q=ai:chen.fengdeSummary: In this paper, a discrete Leslie-Gower predator-prey system with Michaelis-Menten type harvesting is studied. Conditions on the existence and stability of fixed points are obtained. It is shown that the system can undergo fold bifurcation, flip bifurcation, and Neimark-Sacker bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations are presented to illustrate the main theoretical results. Compared to the continuous analog, the discrete system here possesses much richer dynamical behaviors including orbits of period-16, 21, 35, 49, 54, invariant cycles, cascades of period-doubling bifurcation in orbits of period-2, 4, 8, and chaotic sets.Bifurcation in a ratio-dependent predator-prey model with prey harvestinghttps://zbmath.org/1496.920872022-11-17T18:59:28.764376Z"Chen, Lili"https://zbmath.org/authors/?q=ai:chen.lili"Li, Yilong"https://zbmath.org/authors/?q=ai:li.yilong"Xiao, Dongmei"https://zbmath.org/authors/?q=ai:xiao.dongmeiSummary: The dynamics and bifurcations of a class of ratio-dependent predator-prey models with prey harvesting are investigated. The existence of all ecological feasible equilibria for this model is determined and the topological classifications of these equilibria are derived. It is proved that the model can undergo Hopf bifurcation and Bogdanov-Takens bifurcation near the corresponding positive equilibrium as some parameters of the model vary, and unstable oscillation can be observed. Some numerical simulations are provided to support our theoretical results. These theoretical conclusions reveal the effect of constant harvesting rate on the coexistence of the two species, which not only provides predication whether the two species will suffer from extinction one after the other but also gets insight into the optimal management of exploitation resources.Delay induced nonlinear dynamics of oxygen-plankton interactionshttps://zbmath.org/1496.920902022-11-17T18:59:28.764376Z"Gökçe, Aytül"https://zbmath.org/authors/?q=ai:gokce.aytul"Yazar, Samire"https://zbmath.org/authors/?q=ai:yazar.samire"Sekerci, Yadigar"https://zbmath.org/authors/?q=ai:sekerci.yadigarSummary: The present investigation deals with a generic oxygen-plankton model with constant time delays using the combinations of analytical and numerical methods. First, a two-component delayed model: the interaction between the concentration of dissolved oxygen and the density of the phytoplankton is examined in terms of the local stability and Hopf bifurcation analysis around the positive steady state. Then, a three-component model (oxygen-phytoplankton-zooplankton system) is investigated. The prime objective of this trio model is to explore how a constant time delay in growth response of phytoplankton and in the gestation time of zooplankton affects the dynamics of interaction between the concentration of oxygen and the density of plankton. The analytical and numerical investigations reveal that the positive steady states for both models are stable in the absence of time delays for a given hypothetical parameter space. Analysing eigenvalues of the characteristic equation which depends on the delay parameters, the conditions for linear stability and the existence of delay-induced Hopf bifurcation threshold are studied for all possible cases. As the delay rate increases, stability of coexistence state switches from stable to unstable. To support the analytical results, detailed numerical simulations are performed. Our findings show that time delay has a significant impact on the dynamics and may provide useful insights into underlying ecological oxygen-plankton interactions.Weight functions and the uniqueness of limit cycles in predator-prey systemhttps://zbmath.org/1496.920912022-11-17T18:59:28.764376Z"Hasík, Karel"https://zbmath.org/authors/?q=ai:hasik.karel(no abstract)Analysis of the dynamics of phytoplankton nutrient and whooping cough models with nonsingular kernel arising in the biological systemhttps://zbmath.org/1496.920922022-11-17T18:59:28.764376Z"Jena, Rajarama Mohan"https://zbmath.org/authors/?q=ai:jena.rajarama-mohan"Chakraverty, Snehashish"https://zbmath.org/authors/?q=ai:chakraverty.snehashish"Jena, Subrat Kumar"https://zbmath.org/authors/?q=ai:jena.subrat-kumarSummary: In this study, the dynamics of the phytoplankton nutrient and whooping cough models have been examined. Mechanisms of transmission of whooping cough and phytoplankton nutrient models are defined in the Atangana-Baleanu-Caputo (ABC) fractional derivative sense. The first biological system is concerned with the dynamics of phytoplankton-nutrient interaction in the recycling of nutrients, and the second is the modeling of whooping cough in the human population. The essential characteristics of the titled models have been presented, and further, the transmissions of the models defined in the ABC sense are addressed. The concept of fixed point theory is used to derive the existence and uniqueness results of the titled models. Solutions are obtained using the homotopy perturbation Elzaki transform method (HPETM), and numerical results are computed. Graphical analysis of the effect of arbitrary order derivatives has been investigated in detail.Dynamics of fractional-order delay differential model of prey-predator system with Holling-type III and infection among predatorshttps://zbmath.org/1496.920952022-11-17T18:59:28.764376Z"Rihan, F. A."https://zbmath.org/authors/?q=ai:rihan.fathalla-a"Rajivganthi, C."https://zbmath.org/authors/?q=ai:rajivganthi.chinnathambiSummary: In this work, we study the dynamics of a fractional-order delay differential model of prey-predator system with Holling-type III and predator population is infected by an infectious disease. We use Laplace transform, Lyapunov functional, and stability criterion to establish new sufficient conditions that ensure the asymptotic stability of the steady states of the system. Existence of Hopf bifurcation is investigated. The model undergoes Hopf bifurcation, when the feedback time-delays passes through the critical values \(\tau_1^*\) and \(\tau_2^*\). Fractional-order improves the dynamics of the model; while time-delays play a considerable influence on the creation of Hopf bifurcation and stability of the system. Some numerical simulations are provided to validate the theoretical results.A time-delay model for prey-predator growth with stage structurehttps://zbmath.org/1496.920972022-11-17T18:59:28.764376Z"Saito, Yasuhisa"https://zbmath.org/authors/?q=ai:saito.yasuhisa"Takeuchi, Yasuhiro"https://zbmath.org/authors/?q=ai:takeuchi.yasuhiroSummary: This paper studies a stage-structured prey-predator model based on the model proposed by \textit{W. G. Aiello} and \textit{H. I. Freedman} [Math. Biosci. 101, No. 2, 139--153 (1990; Zbl 0719.92017)], where the stage structure was introduced by considering a time to maturity for the predator as a time delay. We establish conditions for the local asymptotic stability and global attractivity of an interior equilibrium of the model.Stability in distribution of a stochastic predator-prey system with S-type distributed time delayshttps://zbmath.org/1496.920982022-11-17T18:59:28.764376Z"Wang, Sheng"https://zbmath.org/authors/?q=ai:wang.sheng"Hu, Guixin"https://zbmath.org/authors/?q=ai:hu.guixin"Wei, Tengda"https://zbmath.org/authors/?q=ai:wei.tengda"Wang, Linshan"https://zbmath.org/authors/?q=ai:wang.linshanThe paper studies stability of the following Lotka-Volterra stochastic system
\[
d x_1(t)=x_1(t)\left(r_1-a_{11}x_1(t)-\int_{-\tau_{11}}^0 x_1(t+\theta) d \mu_{11}(\theta)-a_{12}x_2(t)\right.
\]
\[
\left.-\int_{-\tau_{12}}^0 x_2(t+\theta) d \mu_{12}(\theta)\right)dt+\sigma_1x_1(t)d B_1(t)
\]
\[
d x_2(t)=x_2(t)\left(r_2-a_{22}x_2(t)-\int_{-\tau_{22}}^0 x_2(t+\theta) d \mu_{22}(\theta)-a_{21}x_1(t)\right.
\]
\[
\left.-\int_{-\tau_{21}}^0 x_1(t+\theta) d \mu_{21}(\theta)\right)dt+\sigma_2x_2(t)d B_2(t),
\]
where \(B_i(t)\) are mutually independent standard Wiener processes, \(\tau_{ij}>0\).
Reviewer: Leonid Berezanski (Be'er Sheva)Stationary distribution of a stochastic ratio-dependent predator-prey system with regime-switchinghttps://zbmath.org/1496.920992022-11-17T18:59:28.764376Z"Wang, Zhaojuan"https://zbmath.org/authors/?q=ai:wang.zhaojuan"Deng, Meiling"https://zbmath.org/authors/?q=ai:deng.meiling"Liu, Meng"https://zbmath.org/authors/?q=ai:liu.mengSummary: This article explores a stochastic ratio-dependent predator-prey model with regime-switching. We testify that the model admits a unique stationary distribution, and demonstrate that the transition probability of the solution of the model converges to the stationary distribution in exponent rate. We also discuss the biological implications of the results by aid of some numerical simulations.Mathematical perspective of Covid-19 pandemic: disease extinction criteria in deterministic and stochastic modelshttps://zbmath.org/1496.921002022-11-17T18:59:28.764376Z"Adak, Debadatta"https://zbmath.org/authors/?q=ai:adak.debadatta"Majumder, Abhijit"https://zbmath.org/authors/?q=ai:majumder.abhijit"Bairagi, Nandadulal"https://zbmath.org/authors/?q=ai:bairagi.nandadulalSummary: The world has been facing the biggest virological invasion in the form of Covid-19 pandemic since the beginning of the year 2020. In this paper, we consider a deterministic epidemic model of four compartments based on the health status of the populations of a given country to capture the disease progression. A stochastic extension of the deterministic model is further considered to capture the uncertainty or variation observed in the disease transmissibility. In the case of a deterministic system, the disease-free equilibrium will be globally asymptotically stable if the basic reproduction number is less than unity, otherwise, the disease persists. Using Lyapunov functional methods, we prove that the infected population of the stochastic system tends to zero exponentially almost surely if the basic reproduction number is less than unity. The stochastic system has no interior equilibrium, however, its asymptotic solution is shown to fluctuate around the endemic equilibrium of the deterministic system under some parametric restrictions, implying that the infection persists. A case study with the Covid-19 epidemic data of Spain is presented and various analytical results have been demonstrated. The epidemic curve in Spain clearly shows two waves of infection. The first wave was observed during March-April and the second wave started in the middle of July and not completed yet. A real-time reproduction number has been given to illustrate the epidemiological status of Spain throughout the study period. Estimated cumulative numbers of confirmed and death cases are 1,613,626 and 42,899, respectively, with case fatality rate 2.66\% till the deadly virus is eliminated from Spain.SEAIR epidemic spreading model of COVID-19https://zbmath.org/1496.921032022-11-17T18:59:28.764376Z"Basnarkov, Lasko"https://zbmath.org/authors/?q=ai:basnarkov.laskoSummary: We study Susceptible-Exposed-Asymptomatic-Infectious-Recovered (SEAIR) epidemic spreading model of COVID-19. It captures two important characteristics of the infectiousness of COVID-19: delayed start and its appearance before onset of symptoms, or even with total absence of them. The model is theoretically analyzed in continuous-time compartmental version and discrete-time version on random regular graphs and complex networks. We show analytically that there are relationships between the epidemic thresholds and the equations for the susceptible populations at the endemic equilibrium in all three versions, which hold when the epidemic is weak. We provide theoretical arguments that eigenvector centrality of a node approximately determines its risk to become infected.Dynamics of a stochastic coronavirus (COVID-19) epidemic model with Markovian switchinghttps://zbmath.org/1496.921042022-11-17T18:59:28.764376Z"Boukanjime, Brahim"https://zbmath.org/authors/?q=ai:boukanjime.brahim"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"El Fatini, Mohamed"https://zbmath.org/authors/?q=ai:el-fatini.mohamed"El Khalifi, Mohamed"https://zbmath.org/authors/?q=ai:el-khalifi.mohamedSummary: In this paper, we analyze a stochastic coronavirus (COVID-19) epidemic model which is perturbed by both white noise and telegraph noise incorporating general incidence rate. Firstly, we investigate the existence and uniqueness of a global positive solution. Then, we establish the stochastic threshold for the extinction and the persistence of the disease. The data from Indian states, are used to confirm the results established along this paper.Dynamics of epidemics: impact of easing restrictions and control of infection spreadhttps://zbmath.org/1496.921082022-11-17T18:59:28.764376Z"de Souza, Silvio L. T."https://zbmath.org/authors/?q=ai:de-souza.silvio-luiz-thomaz"Batista, Antonio M."https://zbmath.org/authors/?q=ai:batista.antonio-marcos"Caldas, Iberê L."https://zbmath.org/authors/?q=ai:caldas.ibere-l"Iarosz, Kelly C."https://zbmath.org/authors/?q=ai:iarosz.kelly-c"Szezech, José D. jun."https://zbmath.org/authors/?q=ai:szezech.jose-danilo-junSummary: During an infectious disease outbreak, mathematical models and computational simulations are essential tools to characterize the epidemic dynamics and aid in design public health policies. Using these tools, we provide an overview of the possible scenarios for the COVID-19 pandemic in the phase of easing restrictions used to reopen the economy and society. To investigate the dynamics of this outbreak, we consider a deterministic compartmental model (SEIR model) with an additional parameter to simulate the restrictions. In general, as a consequence of easing restrictions, we obtain scenarios characterized by high spikes of infections indicating significant acceleration of the spreading disease. Finally, we show how such undesirable scenarios could be avoided by a control strategy of successive partial easing restrictions, namely, we tailor a successive sequence of the additional parameter to prevent spikes in phases of low rate of transmissibility.Global stability of a 9-dimensional HSV-2 epidemic modelhttps://zbmath.org/1496.921102022-11-17T18:59:28.764376Z"Feng, Zhilan"https://zbmath.org/authors/?q=ai:feng.zhilan"Qiu, Zhipeng"https://zbmath.org/authors/?q=ai:qiu.zhipeng"Sang, Zi"https://zbmath.org/authors/?q=ai:sang.ziSummary: This paper focuses on the global stability of a 9-dimensional epidemiological model for the transmission dynamics of HSV-2. The model incorporates heterosexual interactions in which a single male population and two groups of female populations with different activity levels are considered. The method of global Lyapunov functions as well as the LaSalle Invariance Principle are used to show that the basic reproduction number provides a sharp threshold which completely determines the global dynamics of the model. That is, in the case when the production number is less than or equal to one, the disease-free equilibrium is globally asymptotically stable; whereas in the case when the reproduction number is greater than one, a unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region and the disease will persist at the endemic equilibrium if it is initially present.Stability analysis of a fractional-order novel hepatitis B virus model with immune delay based on Caputo-Fabrizio derivativehttps://zbmath.org/1496.921112022-11-17T18:59:28.764376Z"Gao, Fei"https://zbmath.org/authors/?q=ai:gao.fei"Li, Xiling"https://zbmath.org/authors/?q=ai:li.xiling"Li, Wenqin"https://zbmath.org/authors/?q=ai:li.wenqin"Zhou, Xianjin"https://zbmath.org/authors/?q=ai:zhou.xianjinSummary: In mathematical epidemiology, mathematical models play a vital role in understanding the dynamics of infectious diseases. Therefore, in this paper, a novel mathematical model for the hepatitis B virus (HBV) based on the Caputo-Fabrizio fractional derivative with immune delay is introduced, while taking care of the dimensional consistency of the proposed model. Initially, the existence and uniqueness of the model solutions are proved by Laplace transform and the fixed point theorem. The positivity and boundedness of the solutions are also discussed. Sumudu transform and Picard iteration were used to analyze the stability and iterative solution of the fractional order model of HBV. Further, using the stability theory of fractional order system, the stability and bifurcation of equilibrium point are discussed. Finally, results are presented for different values of the fractional parameter.Modeling and dynamic analysis of novel coronavirus pneumonia (COVID-19) in Chinahttps://zbmath.org/1496.921122022-11-17T18:59:28.764376Z"Guo, Youming"https://zbmath.org/authors/?q=ai:guo.youming"Li, Tingting"https://zbmath.org/authors/?q=ai:li.tingting(no abstract)Parameter estimation in epidemic models: simplified formulashttps://zbmath.org/1496.921132022-11-17T18:59:28.764376Z"Hadeler, K. P."https://zbmath.org/authors/?q=ai:hadeler.karl-peterSummary: We consider the problem of identifying the time-dependent transmission rate from incidence data and from prevalence data in epidemic SIR, SIRS, and SEIRS models. We show closed representation formulas avoiding the computation of higher derivatives of the data or solving differential equations. We exhibit the connections between the formulas given in several recent papers. In particular we explain the difficulties to estimate the initial number of susceptible or, equivalently, the initial transmission rate.A stochastic SIR epidemic evolution model with non-concave force of infection: mathematical modeling and analysishttps://zbmath.org/1496.921172022-11-17T18:59:28.764376Z"Lahrouz, A."https://zbmath.org/authors/?q=ai:lahrouz.aadil"Settati, A."https://zbmath.org/authors/?q=ai:settati.adel"Jarroudi, M."https://zbmath.org/authors/?q=ai:jarroudi.m"Mahjour, H."https://zbmath.org/authors/?q=ai:mahjour.h"Fatini, M."https://zbmath.org/authors/?q=ai:fatini.m-el"Merzguioui, M."https://zbmath.org/authors/?q=ai:merzguioui.m"Tridane, A."https://zbmath.org/authors/?q=ai:tridane.abdessamadForecasting of COVID-19 pandemic: from integer derivatives to fractional derivativeshttps://zbmath.org/1496.921222022-11-17T18:59:28.764376Z"Nabi, Khondoker Nazmoon"https://zbmath.org/authors/?q=ai:nabi.khondoker-nazmoon"Abboubakar, Hamadjam"https://zbmath.org/authors/?q=ai:abboubakar.hamadjam"Kumar, Pushpendra"https://zbmath.org/authors/?q=ai:kumar.pushpendraSummary: In this work, a new compartmental mathematical model of COVID-19 pandemic has been proposed incorporating imperfect quarantine and disrespectful behavior of citizens towards lockdown policies, which are evident in most of the developing countries. An integer derivative model has been proposed initially and then the formula for calculating basic reproductive number, \(\mathcal{R}_0\) of the model has been presented. Cameroon has been considered as a representative for the developing countries and the epidemic threshold, \(\mathcal{R}_0\) has been estimated to be \(\sim 3.41(95\% \text{CI}:2.2-4.4)\) as of July 9, 2020. Using real data compiled by the Cameroonian government, model calibration has been performed through an optimization algorithm based on renowned trust-region-reflective (TRR) algorithm. Based on our projection results, the probable peak date is estimated to be on August 1, 2020 with approximately \(1073(95\%\text{CI}:714-1654)\) daily confirmed cases. The tally of cumulative infected cases could reach \(\sim 20,100(95\%\text{CI}:17,343-24,584)\) cases by the end of August 2020. Later, global sensitivity analysis has been applied to quantify the most dominating model mechanisms that significantly affect the progression dynamics of COVID-19. Importantly, Caputo derivative concept has been performed to formulate a fractional model to gain a deeper insight into the probable peak dates and sizes in Cameroon. By showing the existence and uniqueness of solutions, a numerical scheme has been constructed using the Adams-Bashforth-Moulton method. Numerical simulations have enlightened the fact that if the fractional order \(\alpha\) is close to unity, then the solutions will converge to the integer model solutions, and the decrease of the fractional-order parameter \((0<\alpha<1)\) leads to the delaying of the epidemic peaks.Global dynamics of a two-strain disease model with latency and saturating incidence ratehttps://zbmath.org/1496.921242022-11-17T18:59:28.764376Z"Rahman, S. M. Ashrafur"https://zbmath.org/authors/?q=ai:ashrafur-rahman.s-m"Zou, Xingfu"https://zbmath.org/authors/?q=ai:zou.xingfu(no abstract)Global dynamics of a time-delayed dengue transmission modelhttps://zbmath.org/1496.921252022-11-17T18:59:28.764376Z"Wang, Zhen"https://zbmath.org/authors/?q=ai:wang.zhen|wang.zhen.2|wang.zhen.6|wang.zhen.1|wang.zhen.7|wang.zhen.3|wang.zhen.5"Zhao, Xiao-Qiang"https://zbmath.org/authors/?q=ai:zhao.xiao-qiang|zhao.xiaoqiangSummary: We present a time-delayed dengue transmission model. We first introduce the basic reproduction number for this model and then show that the disease persists when \(\mathcal R_0>1\). It is also shown that the disease will die out if \(\mathcal R_0<1\), provided that the invasion intensity is not strong. We further establish a set of sufficient conditions for the global attractivity of the endemic equilibrium by the method of fluctuations. Numerical simulations are performed to illustrate our analytic results.Stochastic analysis of a SIRI epidemic model with double saturated rates and relapsehttps://zbmath.org/1496.921262022-11-17T18:59:28.764376Z"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.5"Gao, Shujing"https://zbmath.org/authors/?q=ai:gao.shujing"Chen, Shihua"https://zbmath.org/authors/?q=ai:chen.shihuaSummary: Infectious diseases have for centuries been the leading causes of death and disability worldwide and the environmental fluctuation is a crucial part of an ecosystem in the natural world. In this paper, we proposed and discussed a stochastic SIRI epidemic model incorporating double saturated incidence rates and relapse. The dynamical properties of the model were analyzed. The existence and uniqueness of a global positive solution were proven. Sufficient conditions were derived to guarantee the extinction and persistence in mean of the epidemic model. Additionally, ergodic stationary distribution of the stochastic SIRI model was discussed. Our results indicated that the intensity of relapse and stochastic perturbations greatly affected the dynamics of epidemic systems and if the random fluctuations were large enough, the disease could be accelerated to extinction while the stronger relapse rate were detrimental to the control of the disease.An investigation of delay induced stability transition in nutrient-plankton systemshttps://zbmath.org/1496.921312022-11-17T18:59:28.764376Z"Thakur, Nilesh Kumar"https://zbmath.org/authors/?q=ai:thakur.nilesh-kumar"Ojha, Archana"https://zbmath.org/authors/?q=ai:ojha.archana"Tiwari, Pankaj Kumar"https://zbmath.org/authors/?q=ai:tiwari.pankaj-kumar"Upadhyay, Ranjit Kumar"https://zbmath.org/authors/?q=ai:kumar-upadhyay.ranjitSummary: In this paper, a nutrient-plankton interaction model is proposed to explore the characteristic of plankton system in the presence of toxic phytoplankton and discrete time delay. Anti-predator efforts of phytoplankton by toxin liberation act as a prominent role on plankton dynamics. Toxicity controls the system dynamics and reduces the grazing rate of zooplankton. The toxic substance released by phytoplankton is not an instantaneous process, it requires some time for maturity. So, a discrete time delay is incorporated in the toxin liberation by the phytoplankton. The choice of functional response is important to understand the toxin liberation and it depends on the nonlinearity of the system, which follows the Monod-Haldane type functional response. Theoretically, we have studied the boundedness condition along with all the feasible equilibria analysis and stability criteria of delay free system. We have explored the local stability conditions of delayed system. The existence criterion for stability and direction of Hopf-bifurcation are also derived by using the theory of normal form and center manifold arguments. The essential features of time delay are studied by time series, phase portrait and bifurcation diagram. We perform a global sensitivity analysis to identify the important parameters of the model having a significant impact on zooplankton. Our numerical investigation reveals that the toxin liberation delay switches the stability of the system from stable to limit cycle and after a certain interval chaotic dynamics is observed. High rate of toxic substances production shows extinction of zooplankton. Further, the negative and positive impacts of other control parameters are studied. Moreover, to support the occurrence of chaos, the Poincaré map is drawn and the maximum Lyapunov exponents are also computed.Modeling effects of impulsive control strategies on the spread of mosquito borne disease: role of latent periodhttps://zbmath.org/1496.921322022-11-17T18:59:28.764376Z"Sisodiya, Omprakash Singh"https://zbmath.org/authors/?q=ai:sisodiya.omprakash-singh"Misra, O. P."https://zbmath.org/authors/?q=ai:misra.om-prakash"Dhar, Joydip"https://zbmath.org/authors/?q=ai:dhar.joydipSummary: Controlling the mosquito population is a big challenge for humans. In this paper, we have studied the effects of impulsive control strategies on the spread of mosquito-borne diseases considering the latent period. Therefore, we proposed and analyzed a mosquito-borne disease model governed by a system of impulsive delay differential equations. The proposed mosquito-borne disease model also accounts for three different impulsive control strategies, namely vaccination, pesticides, and adulticides. Two thresholds \(R_1\) and \(R_2\) established for the global attractivity of the disease-free state and the persistence of the endemic state. The non-trivial disease-free solution of the proposed model is globally asymptotically stable if \(R_1\) and \(R_2\) less than one. It is shown that a unique positive endemic periodic solution exists only when \(R_1\) and \(R_2\) greater than unity, which makes for the persistence of the disease. Numerical simulation supports the analytical finding and shows the effectiveness of the impulse control strategies.Controlling swarms toward flocks and millshttps://zbmath.org/1496.930192022-11-17T18:59:28.764376Z"Carrillo, José A."https://zbmath.org/authors/?q=ai:carrillo.jose-antonio"Kalise, Dante"https://zbmath.org/authors/?q=ai:kalise.dante"Rossi, Francesco"https://zbmath.org/authors/?q=ai:rossi.francesco"Trélat, Emmanuel"https://zbmath.org/authors/?q=ai:trelat.emmanuelReach control problem for a class of convex differential inclusions on simpliceshttps://zbmath.org/1496.930222022-11-17T18:59:28.764376Z"Lv, Dejing"https://zbmath.org/authors/?q=ai:lv.dejing"Ge, Bin"https://zbmath.org/authors/?q=ai:ge.bin"Wu, Mingze"https://zbmath.org/authors/?q=ai:wu.mingzeSummary: This paper focuses on the reach control problem for systems expressed as a class of convex differential inclusions. The purpose is to find affine feedback for the trajectories of the systems to reach and leave an unrestricted facet of a given simplex in a finite time. It is proved that the condition for strong reachability is sufficient and necessary. The sufficient condition for weak reachability is obtained based on the solvability of the reach control problem for linear affine systems. At last, algorithms are designed and numerical examples are given to verify the validity of the results.Quasi feedback forms for differential-algebraic systemshttps://zbmath.org/1496.930452022-11-17T18:59:28.764376Z"Berger, Thomas"https://zbmath.org/authors/?q=ai:berger.thomas-k|berger.thomas-r"Ilchmann, Achim"https://zbmath.org/authors/?q=ai:ilchmann.achim"Trenn, Stephan"https://zbmath.org/authors/?q=ai:trenn.stephanSummary: We investigate feedback forms for linear time-invariant systems described by differential-algebraic equations. Feedback forms are representatives of certain equivalence classes. For example, state space transformations, invertible transformations from the left and proportional state feedback constitute an equivalence relation. The representative of such an equivalence class, which we call proportional feedback form for the above example, allows to read off relevant system theoretic properties. Our main contribution is to derive a \textit{quasi} proportional feedback form. This form is advantageous since it provides some geometric insight and is simple to compute, but still allows to read off the relevant structural properties of the control system. We also derive a \textit{quasi} proportional and derivative feedback form. Similar advantages hold.Simplified synchronizability scheme for a class of nonlinear systems connected in chain configuration using contractionhttps://zbmath.org/1496.930582022-11-17T18:59:28.764376Z"Anand, Pallov"https://zbmath.org/authors/?q=ai:anand.pallov"Sharma, Bharat Bhushan"https://zbmath.org/authors/?q=ai:sharma.bharat-bhushanSummary: This paper derives results for the stabilizing and synchronizing controller for a generalized class of nonlinear systems connected in chain configuration. The proposed procedure utilizes contraction based backstepping approach blended with Gershgorin theorem instead of Lyapunov stability based backstepping technique for designing controllers for such systems. A systematic step by step strategy is adopted to obtain a single controller to achieve stabilization of states of systems. Further, results are extended to synchronize the systems belonging to the given generalized class of nonlinear systems. The proposed procedure leads to quite a simple controller for targeted synchronization task in comparison to existing controllers in literature for such class of systems. The systems among which the synchronization has to be done are assumed to be connected in chain formation through one-way coupling. To verify the efficacy of the proposed approach, chaotic systems such as Lorenz-Stenflo, Chen, Lü and Lorenz systems have been considered and detailed numerical validations are presented appropriately.Synchronization of second-order chaotic systems with uncertainties and disturbances using fixed-time adaptive sliding mode controlhttps://zbmath.org/1496.930952022-11-17T18:59:28.764376Z"Yao, Qijia"https://zbmath.org/authors/?q=ai:yao.qijiaSummary: In this paper, a novel fixed-time adaptive sliding mode control scheme is proposed for the synchronization of second-order chaotic systems with system uncertainties and external disturbances. First, a new type of fixed-time nonsingular sliding mode surface is developed based on the bi-limit homogeneous theory. Then, on the basis of the fixed-time sliding mode surface, the fixed-time adaptive sliding mode controller is designed by integrating the fixed-time terminal sliding mode control with the parametric adaptation technique. Finally, rigorous theoretical analysis for the semi-global fixed-time stability of the resulting closed-loop system is provided. A distinct feature of the proposed controller is that it can guarantee the synchronization tracking errors converge to the arbitrarily small neighbourhood of zero in fixed time even in the presence of lumped disturbances. To the best of the authors' knowledge, there are relatively few existing controllers can achieve such excellent performance in the same conditions. Two simulation examples are performed to illustrate the effectiveness and benefits of the proposed control scheme.Exponential stability of nonlinear systems involving partial unmeasurable states via impulsive controlhttps://zbmath.org/1496.930982022-11-17T18:59:28.764376Z"Li, Mingyue"https://zbmath.org/authors/?q=ai:li.mingyue"Chen, Huanzhen"https://zbmath.org/authors/?q=ai:chen.huanzhen"Li, Xiaodi"https://zbmath.org/authors/?q=ai:li.xiaodiSummary: This paper investigates the stability problem of partial unmeasurable nonlinear systems under impulsive control. Some sufficient conditions are given to guarantee exponential stability of systems using transition matrix method coupled with dimension expansion technique, where the possibility of the effects of partial unmeasurable states is fully considered. In our proposed method, we not only allow systems to have incomplete states, but also relax restrictions on measurable states, which has a wider range of applications in practice. Finally, two illustrative examples are presented, with their numerical simulations, to demonstrate the effectiveness of main results.Finite-time \(H_\infty\) control of linear singular fractional differential equations with time-varying delayhttps://zbmath.org/1496.931052022-11-17T18:59:28.764376Z"Niamsup, Piyapong"https://zbmath.org/authors/?q=ai:niamsup.piyapong"Thanh, Nguyen T."https://zbmath.org/authors/?q=ai:thanh.nguyen-trung|thanh.nguyen-thi"Phat, Vu N."https://zbmath.org/authors/?q=ai:vu-ngoc-phat.Summary: In this paper, we propose an efficient analytical approach based on fractional calculus and singularity value theory to designing the finite-time \(H_\infty\) controller for linear singular fractional differential equations with time-varying delay. By introducing new fractional-order \(H_\infty\) norm, the state feedback controller is designed to guarantee that the closed-loop system is singular, impulse-free and finite-time stable with prescribed \(H_\infty\) performance. New sufficient conditions for designing the \(H_\infty\) finite-time controller are presented. The results of this paper improve the corresponding ones of integer-order singular systems with time-varying delay. Finally, a numerical example demonstrates the validity and effectiveness of the proposed theoretical results.