Recent zbMATH articles in MSC 34Chttps://zbmath.org/atom/cc/34C2024-05-13T19:39:47.825584ZWerkzeugAn introduction to dynamical systems and chaoshttps://zbmath.org/1532.340022024-05-13T19:39:47.825584Z"Layek, G. C."https://zbmath.org/authors/?q=ai:layek.g-cPublisher's description: This book discusses continuous and discrete nonlinear systems in systematic and sequential approaches. The unique feature of the book is its mathematical theories on flow bifurcations, nonlinear oscillations, Lie symmetry analysis of nonlinear systems, chaos theory, routes to chaos and multistable coexisting attractors. The logically structured content and sequential orientation provide readers with a global overview of the topic. A systematic mathematical approach has been adopted, featuring a multitude of detailed worked-out examples alongside comprehensive exercises. The book is useful for courses in dynamical systems and chaos and nonlinear dynamics for advanced undergraduate, graduate and research students in mathematics, physics and engineering.
The second edition of the book is thoroughly revised and includes several new topics: center manifold reduction, quasi-periodic oscillations, Bogdanov-Takens, period-bubbling and Neimark-Sacker bifurcations, and dynamics on circle. The organized structures in bi-parameter plane for transitional and chaotic regimes are new active research interest and explored thoroughly. The connections of complex chaotic attractors with fractals cascades are explored in many physical systems. Chaotic attractors may attain multiple scaling factors and show scale invariance property. Finally, the ideas of multifractals and global spectrum for quantifying inhomogeneous chaotic attractors are discussed.
See the review of the first edition in [Zbl 1354.34001].The method of adapted energies for second order evolution equations with dissipationhttps://zbmath.org/1532.340032024-05-13T19:39:47.825584Z"Haraux, Alain"https://zbmath.org/authors/?q=ai:haraux.alainSummary: This survey paper emphasizes the method of adapted energy functions to obtain sharp information on various stability questions: exponential stability, strong stability with explicit decay rates, ultimate bounds of solutions and so on. The efficiency of the method, basically related to Lyapunov's and Barbashin-Krasovskii-LaSalle's approaches, very often allows one to obtain close-to-optimal estimates, as demonstrated on a variety of examples connected to mechanics via differential equations of second order with respect to the time variable.
For the entire collection see [Zbl 1504.74002].On the modeling and numerical discretizations of a chaotic system via fractional operators with and without singular kernelshttps://zbmath.org/1532.340172024-05-13T19:39:47.825584Z"Sene, Ndolane"https://zbmath.org/authors/?q=ai:sene.ndolaneSummary: A new chaotic system with Caputo, Atangana-Baleanu, and Caputo-Fabrizio derivatives has been presented. The conditions for the existence and uniqueness of the new fractional chaotic system solutions have been provided for the Caputo, Atangana-Baleanu, and Caputo-Fabrizio operators. The bifurcation maps to detect chaotic regions according to the variations in the model's parameters have been proposed. The Lyapunov exponents for the fractional-order chaotic systems have been calculated to characterize the behaviors of the dynamics of the considered fractional-order system. The stability analysis of the equilibrium points of the considered model has been investigated with two methods. The phase portraits of the fractional chaotic model studied in this paper have been obtained via the fractional linear multistep method and Adams-Basford method. The fractional operators in our modeling are the Caputo derivative, the fractional derivative with the Mittag-Leffler kernel, and the Caputo-Fabrizio fractional derivative. The circuit schematics for the fractional version of our presented model regarding resistors and capacitors have been proposed to confirm the theoretical results.Upper bounds for the blow-up time of a system of fractional differential equations with Caputo derivativeshttps://zbmath.org/1532.340202024-05-13T19:39:47.825584Z"Villa-Morales, José"https://zbmath.org/authors/?q=ai:villa-morales.joseSummary: The article provides upper bounds for the blow-up time of a system of fractional differential equations in the Caputo sense.Liouvillian solutions for second order linear differential equations with Laurent polynomial coefficienthttps://zbmath.org/1532.340232024-05-13T19:39:47.825584Z"Acosta-Humánez, Primitivo B."https://zbmath.org/authors/?q=ai:acosta-humanez.primitivo-belen"Blázquez-Sanz, David"https://zbmath.org/authors/?q=ai:blazquez-sanz.david"Venegas-Gómez, Henock"https://zbmath.org/authors/?q=ai:venegas-gomez.henockSummary: This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at 0 and \(\infty\) of the Laurent polynomial, is seen as an affine algebraic variety. We prove that the set of Picard-Vessiot integrable differential equations in the family is an enumerable union of algebraic subvarieties. We compute explicitly the algebraic equations of its components. We give some applications to well known subfamilies, such as the doubly confluent and biconfluent Heun equations, and to the theory of algebraically solvable potentials of Shrödinger equations. Also, as an auxiliary tool, we improve a previously known criterium for a second order linear differential equations to admit a polynomial solution.Slow-fast normal forms arising from piecewise smooth vector fieldshttps://zbmath.org/1532.340272024-05-13T19:39:47.825584Z"Perez, Otavio Henrique"https://zbmath.org/authors/?q=ai:perez.otavio-henrique"Rondón, Gabriel"https://zbmath.org/authors/?q=ai:rondon.gabriel"da Silva, Paulo Ricardo"https://zbmath.org/authors/?q=ai:da-silva.paulo-ricardoIn this paper, the authors study the relation between slow-fast and piecewise smooth systems of vector fields (with the latter also known as Fillipov systems). More precisely, the authors study how typical SF-singularities (singularities of slow-fast systems) arises from different types of regularization of PS-singularities (singularities of piecewise smooth systems). They consider linear and non-linear regularizations, with monotonic and non-monotonic transitions functions. The paper has tree main results.
In the first main result they study the linear regularizations, proving that critical points of the transition function results in non normally hyperbolic points of the SF-system. Moreover, under certain conditions the regularized sliding region is greater then the classical Fillipov sliding region. In the second main result the authors prove that normally-hyperbolic points, SF-fold and SF-transcritical singularities are realizable by linear regularizations of PS-systems, while SF-pitchfork singularities are not. In the third main result the authors prove SF-pitchfork singularities are realizable by non-linear regularizations.
Reviewer: Paulo Santana (São José do Rio Preto)A class of quintic Kolmogorov systems with explicit non-algebraic limit cyclehttps://zbmath.org/1532.340462024-05-13T19:39:47.825584Z"Bendjeddou, Ahmed"https://zbmath.org/authors/?q=ai:bendjeddou.ahmed"Grazem, Mohamed"https://zbmath.org/authors/?q=ai:grazem.mohamedSummary: Various physical, ecological, economic, etc phenomena are governed by planar differential systems. Subsequently, several research studies are interested in the study of limit cycles because of their interest in the understanding of these systems. The aim of this paper is to investigate a class of quintic Kolmogorov systems, namely systems of the form
\[\begin{aligned}
\dot{x}=xP_4\left( x,y\right),\\ \dot{y}=yQ_4\left( x,y\right), \end{aligned}\]
where \(P_4\) and \(Q_4\) are quartic polynomials. Within this class, our attention is restricted to study the limit cycle in the realistic quadrant
\(\left \{ (x,y)\in\mathbb{R}^2;\, x>0,\, y>0\right \} \). According to the hypothesises, the existence of algebraic or non-algebraic limit cycle is proved. Furthermore, this limit cycle is explicitly given in polar coordinates. Some examples are presented in order to illustrate the applicability of our result.Limit cycles for a class of polynomial differential systems via averaging theoryhttps://zbmath.org/1532.340472024-05-13T19:39:47.825584Z"Bendjeddou, Ahmed"https://zbmath.org/authors/?q=ai:bendjeddou.ahmed"Berbache, Aziza"https://zbmath.org/authors/?q=ai:berbache.aziza"Kina, Abdelkrim"https://zbmath.org/authors/?q=ai:kina.abdelkrimSummary: In this paper, we consider the limit cycles of a class of polynomial differential systems of the form
\[\begin{cases}
\dot{x}=y-\varepsilon (g_{11}\left( x\right) y^{2\alpha +1}+f_{11}\left( x\right) y^{2\alpha })-\varepsilon^2(g_{12}\left( x\right) y^{2\alpha +1}+f_{12}\left( x\right) y^{2\alpha }) ,\\
\dot{y}=-x-\varepsilon (g_{21}\left( x\right) y^{2\alpha +1}+f_{21}\left( x\right) y^{2\alpha })-\varepsilon^2(g_{22}\left( x\right) y^{2\alpha +1}+f_{22}\left( x\right) y^{2\alpha }), \end{cases}\]
where \(m\), \(n\), \(k\), \(l\) and \(\alpha\) are positive integers, \(g_{1\kappa }\), \(g_{2\kappa }\), \(f_{1\kappa }\) and \(f_{2\kappa }\) have degree \(n\), \(m\), \(l\) and \(k\), respectively for each \(\kappa =1\), 2, and \(\varepsilon\) is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center \(\dot{x}=y\), \(\dot{y}=-x\) using the averaging theory of first and second order.On the well-posedness of a singular problem for linear differential equationshttps://zbmath.org/1532.340482024-05-13T19:39:47.825584Z"Uteshova, R. E."https://zbmath.org/authors/?q=ai:uteshova.rosa-e|uteshova.roza-e"Kokotova, Ye. V."https://zbmath.org/authors/?q=ai:kokotova.ye-vLinear differential system
\[
x'=A(t)x+f(t),\quad t\in (-\infty,\infty)
\]
is considered provided that \(A(t)\) and \(f(t)\) are continuous,
\[
\|A(t)\|=\max_j\sum_{k=1}^n|a_{jk}(t)|=\alpha(t)>0,
\]
and
\[
\lim_{t\to\pm\infty}\alpha(t)=0,\quad \int_{-\infty}^0\alpha(t)dt= \int^{\infty}_0\alpha(t)dt=\infty.
\]
If \(f\) is bounded than, under above conditions, the system has not always a bounded on \((-\infty,\infty)\) solution. Assuming additionally that \(f\) is bounded with the weight \(1/\alpha(t)\), the problem of existence of a bounded solution on \((-\infty,\infty)\) is studied. This problem is solved by the parametrization method with nonuniform partition and is transformed into an equivalent problem with a parameter. Then an algorithm, based on iterative processes, is suggested. This procedure permits to prove the existence of a unique solution of the problem. It is also proved that the problem is well-posed.
Reviewer: Josef Diblík (Brno)Equivariant Pyragas control of discrete waveshttps://zbmath.org/1532.340492024-05-13T19:39:47.825584Z"de Wolff, Babette A. J."https://zbmath.org/authors/?q=ai:de-wolff.b-a-jSummary: Equivariant Pyragas control is a delayed feedback method that aims to stabilize spatio-temporal patterns in systems with symmetries. In this article, we apply equivariant Pyragas control to discrete waves, which are periodic solutions that have a finite number of spatio-temporal symmetries. We prove sufficient conditions under which a discrete wave can be stabilized via equivariant Pyragas control. The result is applicable to a broad class of discrete waves, including discrete waves that are far away from a bifurcation point. Key ingredients of the proof are an adaptation of Floquet theory to systems with symmetries, and the use of characteristic matrix functions to reduce the infinite dimensional eigenvalue problem to a one dimensional zero finding problem.Limit cycles of discontinuous piecewise differential systems separated by a straight line and formed by cubic reversible isochronous centers having rational first integralshttps://zbmath.org/1532.340502024-05-13T19:39:47.825584Z"Benabdallah, Imane"https://zbmath.org/authors/?q=ai:benabdallah.imane"Benterki, Rebiha"https://zbmath.org/authors/?q=ai:benterki.rebiha"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaumeSummary: A lot of attention has been paid in recent years to the study of piecewise differential systems, and more especially in studying the maximum number of limit cycles that these systems can exhibit. In this paper we consider all classes of discontinuous piecewise differential systems with cubic reversible isochronous centers having rational first integrals separated by the straight line \(x= 0\).
First, we solve the extension of the second part of the sixteenth Hilbert problem for each of the three classes of discontinuous piecewise differential systems formed by an arbitrary linear center and one of the three cubic reversible isochronous centers. We establish that, depending on the class presented, the maximum number of limit cycles of these classes varies between one and two. Second, by combining the three types of cubic reversible isochronous centers, we obtain six classes of discontinuous piecewise differential systems formed by two cubic reversible isochronous centers. So we solve the extended sixteenth Hilbert problem for all these classes and find the maximum number of limit cycles that such classes can exhibit. Moreover we have reinforced our results by giving examples for each class.A mechanism for detecting normally hyperbolic invariant tori in differential equationshttps://zbmath.org/1532.340512024-05-13T19:39:47.825584Z"Pereira, Pedro C. C. R."https://zbmath.org/authors/?q=ai:pereira.pedro-c-c-r"Novaes, Douglas D."https://zbmath.org/authors/?q=ai:novaes.douglas-duarte"Cândido, Murilo R."https://zbmath.org/authors/?q=ai:candido.murilo-rConsider differential systems in the so-called standard form for averaging \(x'=\varepsilon F(t,x,\varepsilon)\), where \(F\) is \(T\)-periodic in the first variable (\(T>0\) is fixed). Using first order averaging theory, it is known that the existence of \(T\)-periodic solutions is related to the existence of equilibrium points of the averaged equation \(x'=f_1(x)\) where \(f_1(x)=\int_0^TF(t,x,0)dt\). D. Novaes worked in collaboration with other authors to generalize this result in the framework of higher order averaging theory.
Results that go back to Bogoliubov-Mitropolskii and Hale assure that the existence of invariant tori is related to the existence of limit cycles of the averaged equation. In this article, the authors generalise these results in the framework of higher order averaging theory. They work in dimension \(2\). The proof of their main result is an adaptation of the method of continuation employed in [\textit{C. Chicone} and \textit{W. Liu}, SIAM J. Math. Anal. 31, No. 2, 386--415 (2000; Zbl 0942.34035)].
Reviewer: Adriana Buică (Cluj-Napoca)Invariant hyperplane sections of vector fields on the product of sphereshttps://zbmath.org/1532.340522024-05-13T19:39:47.825584Z"Benny, Joji"https://zbmath.org/authors/?q=ai:benny.joji"Sarkar, Soumen"https://zbmath.org/authors/?q=ai:sarkar.soumenSummary: Let \(S_{p,q}\) be the hypersurface in \(\mathbb{R}^n\), where \(n=p+q+1\), defined by the following:
\[
S_{p,q}:= \left\{ (x_1,\ldots, x_n) \in \mathbb{R}^n \quad \big| \quad\left( \sum_{i=1}^{p+1} x_i^2 - a^2 \right)^2 + \sum_{j=p+2}^n x_j^2 = 1 \right\}
\]
where \(a > 1\). We show that \(S_{p,q}\) is homeomorphic to the product \(S^p \times S^q\). We classify all degree one and two polynomial vector fields on \(S_{p,q}\). We consider the polynomial vector field \(\mathcal{X} = (R_1, \ldots, R_{p+1}, R_{p+2}, \ldots, R_n)\) in \(\mathbb{R}^{p+q+1}\) which keeps \(S_{p,q}\) invariant. Then, we study the number of certain invariant algebraic subsets of \(S_{p,q}\) for the vector field \(\mathcal{X}\) if either \(p>1\) or \(q>1\).Stability analysis of tempered fractional nonlinear Mathieu type equation model of an ion motion with octopole-only imperfectionshttps://zbmath.org/1532.340532024-05-13T19:39:47.825584Z"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-o"Selvam, A. George Maria"https://zbmath.org/authors/?q=ai:selvam.a-george-maria"Vignesh, Dhakshinamoorthy"https://zbmath.org/authors/?q=ai:vignesh.dhakshinamoorthy"Etemad, Sina"https://zbmath.org/authors/?q=ai:etemad.sina"Rezapour, Shahram"https://zbmath.org/authors/?q=ai:rezapour.shahramSummary: The development of laser-based cooling and spectroscopic methods has produced unprecedented growth in the ion trapping industry. Mathieu equation, a differential equation with periodic coefficients, is employed to develop models of ion motions under the influence of fields. Ion traps with octopole field is described with nonlinear Mathieu equation with cubic term. This article aims at considering motion of ions under the electric potential with negative octopole field with damping caused by the collision of the ions with Helium buffer gas modeled with tempered fractional derivative. Schaefer's fixed point theorem and Banach's contraction principle are employed to establish the existence of unique solution for the considered tempered fractional nonlinear Mathieu equation model of an ion motion. Further, the analysis of stability is performed in the sense of Hyers and Ulam. The feasibility of the obtained theoretical results are numerically confirmed for suitable parametric values, and simulations are performed supporting them.
{{\copyright} 2023 John Wiley \& Sons, Ltd.}Evolutionary stable strategies and cubic vector fieldshttps://zbmath.org/1532.340542024-05-13T19:39:47.825584Z"Bastos, Jefferson"https://zbmath.org/authors/?q=ai:bastos.jefferson-l-r"Buzzi, Claudio"https://zbmath.org/authors/?q=ai:buzzi.claudio-aguinaldo"Santana, Paulo"https://zbmath.org/authors/?q=ai:santana.pauloSummary: The introduction of concepts of Game Theory and Ordinary Differential Equations into Biology gave birth to the field of Evolutionary Stable Strategies, with applications in Biology, Genetics, Politics, Economics and others. In special, the model composed by two players having two pure strategies each results in a planar cubic vector field with an invariant octothorpe. Therefore, in this paper we study such class of vector fields, suggesting the notion of genericity and providing the global phase portraits of the generic systems with a singularity at the central region of the octothorpe.Saddle-node bifurcation and Bogdanov-Takens bifurcation of a SIRS epidemic model with nonlinear incidence ratehttps://zbmath.org/1532.340552024-05-13T19:39:47.825584Z"Cui, Wenzhe"https://zbmath.org/authors/?q=ai:cui.wenzhe"Zhao, Yulin"https://zbmath.org/authors/?q=ai:zhao.yulinSummary: The Bogdanov-Takens bifurcation of the SIRS epidemic model with nonlinear incidence rate was studied by \textit{S. Ruan} and \textit{W. Wang} [J. Differ. Equations 188, No. 1, 135--163 (2003; Zbl 1028.34046)], \textit{Y. Tang} et al. [SIAM J. Appl. Math. 69, No. 2, 621--639 (2008; Zbl 1171.34033)] and \textit{M. Lu} et al. [J. Differ. Equations 267, No. 3, 1859--1898 (2019; Zbl 1421.92034)] in recent years. The results in the mentioned papers showed that the SIRS epidemic model with nonlinear incidence rate \(kI^2/(1 + \omega I^2)\) can undergo a Bogdanov-Takens bifurcation of codimension two. In this paper we study the SIRS epidemic model with nonlinear incidence rate \(k I^p /(1 + \omega I^q)\) for general \(p\) and \(q\). The bifurcation analysis indicates that there is a saddle-node or a cusp of codimension two for various parameter values and the model can undergo a saddle-node bifurcation or a Bogdanov-Takens bifurcation of codimension two if suitable bifurcation parameters are selected. It means that there are some SIRS epidemic models which have a limit cycle or a homoclinic loop. Moreover, it is also shown that the codimension of Bogdanov-Takens bifurcation is at most two.Periodic orbits in a seasonal \textit{SIRS} model with both incidence and treatment generalized rateshttps://zbmath.org/1532.340562024-05-13T19:39:47.825584Z"Guerrero-Flores, Shaday"https://zbmath.org/authors/?q=ai:guerrero-flores.shaday"Osuna, Osvaldo"https://zbmath.org/authors/?q=ai:osuna.osvaldo"Villavicencio Pulido, José Geiser"https://zbmath.org/authors/?q=ai:villavicencio-pulido.jose-geiserSummary: In this work, we prove that a seasonal-dependent \textit{SIRS} model with general incidence and treatment rates has periodic solutions. This generalized model is analyzed using Leray-Schauder degree theory to prove the existence of a periodic solution. Finally, numerical simulations are shown to illustrate the theoretical results.On a fractal-fractional-based modeling for influenza and its analytical resultshttps://zbmath.org/1532.340572024-05-13T19:39:47.825584Z"Khan, Hasib"https://zbmath.org/authors/?q=ai:khan.hasib"Rajpar, Altaf Hussain"https://zbmath.org/authors/?q=ai:rajpar.altaf-hussain"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-o"Aslam, Muhammad"https://zbmath.org/authors/?q=ai:aslam.muhammad.3"Etemad, Sina"https://zbmath.org/authors/?q=ai:etemad.sina"Rezapour, Shahram"https://zbmath.org/authors/?q=ai:rezapour.shahramSummary: There have been reports of influenza virus resistance in the past, and because this virus has the potential of resistance to cause several pandemics and also is lethal, we investigate the conditions under which the strains coexist as a result. The non-resistant strain undergoes mutation, giving rise to the resistant strain. The incidence rates of the non-resistant and saturated-resistant strains are bi-linear and saturated, respectively. In this study, two flu strain models (resistant and non-resistant) are investigated in a fractal-fractional sense, and the presence of solutions, stability, and numerical simulations are examined for various orders and derivative dimensions. Using numerical values from freely accessible open resources, a numerical technique that is based on Lagrange's interpolation polynomial is constructed and validated for a particular example.Super-explosion and inverse canard explosion in a piecewise-smooth slow-fast Leslie-Gower modelhttps://zbmath.org/1532.340582024-05-13T19:39:47.825584Z"Zhang, Huiping"https://zbmath.org/authors/?q=ai:zhang.huiping"Cai, Yuhua"https://zbmath.org/authors/?q=ai:cai.yuhua"Shen, Jianhe"https://zbmath.org/authors/?q=ai:shen.jianheSummary: In this paper, we study a slow-fast Leslie-type predator-prey model with piecewise-linear functional response. Our approach is based on the geometric singular perturbation theory and the canard theory. When the fold point of the critical curve is lower than the transcritical bifurcation point, theoretical and numerical analyses show that a supercritical super-explosion occurs near the non-smooth corner followed by an inverse canard explosion close to the smooth fold. The critical values of parameters corresponding to these dynamical behaviors are obtained. Moreover, a stable relaxation oscillation is generated by the supercritical super-explosion, which will vanish as the occurrence of the inverse canard explosion.Probability of existence of limit cycles for a family of planar systemshttps://zbmath.org/1532.340592024-05-13T19:39:47.825584Z"Coll, B."https://zbmath.org/authors/?q=ai:coll.bartomeu"Gasull, A."https://zbmath.org/authors/?q=ai:gasull.armengol"Prohens, R."https://zbmath.org/authors/?q=ai:prohens.rafelTheorems concerning existence and number of limit cycles of deterministic \(2\times 2\) systems of nonlinear differential equations are proved. The probability of existence of limit cycles of such systems is studied by using the associated system of random differential equations \[\dot{x}=Af(x)+ By,\qquad \dot{y}= Cf(x)+ Dy,\] where \(f\) is a smooth function with \(f(0)=0\), and \(A\), \(B\), \(C\), \(D\) are independent standard normal random variables.
When \(f\) is a judiciously chosen polynomial, theorems and corollaries are proved that specify or bound probabilities such as that of having no limit cycles, an odd number of limit cycles, at most a finite number of limit cycles, and exactly (or no or at most) one limit cycle.
Reviewer: Melvin D. Lax (Long Beach)The existence and averaging principle for stochastic fractional differential equations with impulseshttps://zbmath.org/1532.340602024-05-13T19:39:47.825584Z"Zou, Jing"https://zbmath.org/authors/?q=ai:zou.jing"Luo, Danfeng"https://zbmath.org/authors/?q=ai:luo.danfeng"Li, Mengmeng"https://zbmath.org/authors/?q=ai:li.mengmeng(no abstract)On fully nonlinear equations with fractional time derivative: local existence and uniqueness, stable manifoldhttps://zbmath.org/1532.340622024-05-13T19:39:47.825584Z"Guidetti, Davide"https://zbmath.org/authors/?q=ai:guidetti.davideSummary: We prove a maximal regularity result for abstract linear evolution autonomous equations with a fractional time derivative in the sense of Caputo. We employ it to show theorem of existence and uniqueness of local solutions for fully nonlinear equations and a theorem of existence of a stable manifold which is analogous to well known results in the case of a derivative of order one. We conclude with some examples and applications to mixed Cauchy-Dirichlet and Cauchy-Neumann problems.On the convergence behaviour of solutions of certain system of second order nonlinear delay differential equationshttps://zbmath.org/1532.340802024-05-13T19:39:47.825584Z"Olutimo, A. L."https://zbmath.org/authors/?q=ai:olutimo.a-lSummary: Convergence criteria for the solutions of certainsystem of two nonlinear delay differential equations with con-tinuous deviating argument \(\varrho (t)\) using a suitable Lyapunov-Krasovskii's functional are established in this study. The newresult attained extends and updates some results mentioned in the literature. A numerical illustration is given to show the va-lidity of the result as well geometric analysis to describe thebehavior of solutions of the system.Positive periodic solutions of nonautonomous Lotka-Volterra dynamic systems with a general attack rate on time scaleshttps://zbmath.org/1532.340942024-05-13T19:39:47.825584Z"Bordj, Belkis"https://zbmath.org/authors/?q=ai:bordj.belkis"Ardjouni, Abdelouaheb"https://zbmath.org/authors/?q=ai:ardjouni.abdelouaheb\textit{V. Volterra} [Mem. Accad. naz. Lincei, Cl. Sci. fis. mat. nat. (6) 2, 31--113 (1927; JFM 52.0450.06)] constructed the known Lotka-Volterra model based on the assumption that fish and sharks are in a predator-prey relationship. Moreover, \textit{A. J. Lotka} [The elements of physical biology. Baltimore, Williams \& Wilkins Co.; London, Baillière, Tindall \& Cox (1925; JFM 51.0416.06)] constructed a similar model in a different context around the same time, so it is known as Lotka-Volterra. The Lotka-Volterra model is applied to describe evolutionary processes in various areas of human activity. In the real world, some processes vary continuously (modeled by differential equations) while others vary discretely (modeled by difference equations), or both continuously and discretely (times scales). In this sense, times scales \(\mathbb{T}\) are considered to be periodic (additively periodic) such that \(0\in{\mathbb T}\). The main purpose of this paper is to extend the \textit{C. Lois-Prados} and \textit{R. Precup} results [Nonlinear Anal., Real World Appl. 52, Article ID 103024, 17 p. (2020; Zbl 1433.34066)] to dynamic systems of the form
\[
\begin{cases} x^{\Delta}(t)=\ a(t)x(t)g(x(t))-f(t,x(t),y(t))x(t)y(t),\\
y^{\Delta}(t)=\ -b(t)y^{\sigma}(t)+c(t)f(t,x(t),y(t))x(t)y(t) \end{cases}\tag{1}
\]
on time scales \({\mathbb T}\). Particularly, the authors obtain a discrete analogue to the continuous result and they simultaneously prove both continuous and discrete results, and use the Krasnoselskii-type homotopy fixed-point theorem to prove the existence of positive periodic solutions of nonautonomous Lotka-Volterra dynamic systems with a general attack rate on time scales. In Section 2, some preliminary results on the calculus on time scales are presented. In Section 3, Krasnoselskii-type homotopy fixed-point theorem [\textit{D. O'Regan} and \textit{R. Precup}, Theorems of Leray-Schauder-type and applications. London: Gordon and Breach Science Publishers (2001; Zbl 1045.47002)] is stated which is the main tool of the paper. Then System (1) is transformed into an integral system and explores the existence of positive periodic solutions. In the final part of the paper, some particular cases of system (1) is discussed to emphasize the obtained results.
Reviewer: Abdullah Özbekler (Ankara)Weighted Stepanov-like pseudo almost periodicity on time scales and applicationshttps://zbmath.org/1532.340952024-05-13T19:39:47.825584Z"Es-saiydy, Mohssine"https://zbmath.org/authors/?q=ai:es-saiydy.mohssine"Zitane, Mohamed"https://zbmath.org/authors/?q=ai:zitane.mohamedIn this paper, the authors introduce some new classes of functions called weighted Stepanov-like pseudo almost periodic functions on time scales \(\mathbb{T}\) (or weighted pseudo almost periodic functions in the sense of Stepanov on time scales) and explore its properties. By using the measure theory on time scales, the notion of the so-called ergodic component is hereby enlarged. Among other things, it is shown that these new functions generalize in a natural fashion the classical notion of almost periodicity and its various extensions. The introduced notion is equivalent to the usual considered notion of weighted Stepanov-like pseudo-almost periodicity introduced by Diagana when \(\mathbb{T=R}\). If \(\mu\equiv 1\), then the space of Stepanov-like pseudo almost periodic functions on time scales considered by Tang and Li is a particular case of this newly introduced space.
Furthermore, some applications are given to show that if the input function is a weighted Stepanov-like pseudo almost periodic on a time scale \(\mathbb{T}\), then some nonlinear dynamic equations have a weighted pseudo almost periodic solutions on \(\mathbb{T}\).
Reviewer: Eze Raymond Nwaeze (Montgomery)Correction to: ``Serrin's overdetermined problem on \(\mathbb{S}^N \times\mathbb{R}\)''https://zbmath.org/1532.353152024-05-13T19:39:47.825584Z"Morabito, Filippo"https://zbmath.org/authors/?q=ai:morabito.filippoCorrection to the author's paper [ibid. 33, No. 10, Paper No. 327, 17 p. (2023; Zbl 1528.35080)].Qualitative behaviors of a four-dimensional Lorenz systemhttps://zbmath.org/1532.370392024-05-13T19:39:47.825584Z"Zhang, Fuchen"https://zbmath.org/authors/?q=ai:zhang.fuchen"Xu, Fei"https://zbmath.org/authors/?q=ai:xu.fei.4|xu.fei|xu.fei.3|xu.fei.2|xu.fei.1"Zhang, Xu"https://zbmath.org/authors/?q=ai:zhang.xuSummary: In this paper, the qualitative behaviors of an important four-dimensional Lorenz system with wild pseudohyperbolic attractor that proposed in [\textit{S. Gonchenko} et al., Nonlinearity 34, No. 4, 2018--2047 (2021; Zbl 1472.34106)] are considered. Here, we prove that the four-dimensional Lorenz system with varying parameters is global bounded according to Lyapunov's direct method. Furthermore, we provide a collection of global absorbing sets, where in addition we obtain the rate of the trajectories going from the exterior to the global absorbing set. In particular, we solve the critical case \(k\to 0^+\) that cannot be resolved by using the previous methods. The fundamental qualitative behaviors are analyzed theoretically and numerically. We present bifurcation diagrams to further explore the complicated dynamical behaviors of this system. The period-doubling bifurcation phenomenon is found. To illustrate the efficiency of our method, we present numerical simulations to show the validity of our research results. Finally, we present some applications of our research results in this paper.
{{\copyright} 2024 IOP Publishing Ltd}Normalization flowhttps://zbmath.org/1532.370552024-05-13T19:39:47.825584Z"Treschev, Dmitry V."https://zbmath.org/authors/?q=ai:treshchev.dmitrij-vThe author proposes a continuous normalization for Hamiltonian systems near a nonresonant elliptic singular point. The system under consideration has the following form:
\[\dot{z} = i \partial_{\bar{z}} \hat{H} , \qquad \dot{\bar{z}} = -i \partial_{\bar{z}} \hat{H},\]
with
\( \hat{H} = \hat{H} (z, \bar{z})\), where \( z = (z_1,\dots, z_n)\) and \(\bar{z} = (\bar{z_1}, \dots, \bar{z_n})\) are independent coordinates on \(\mathbb{C}^{2n}\). The Hamiltonian function has the form \(\hat{H} = H_2 + \hat{H}_0\) with \(\hat{H}_0 = O_3(z, \bar{z})\) and \[H_2(z,\bar{z}) = \sum_{j=1}^{n} \omega_j z_j \bar{z}_j,\] where \((\omega_1, \dots, \omega_n)\) are real numbers and \(O_3\) means \(O(|z|^3 + |\bar{z}^3|)\).
Assuming that the frequency \(\omega\) is nonresonant, there exists a formal canonical near-identity change of variables \((z, \bar{z}) \rightarrow (Z, \bar{Z})\) with \(d\bar{z} \wedge dz = d\bar{Z} \wedge dZ\) so that the new Hamiltonian takes the form \(H(z, \bar{z}) = N(Z_1\bar{Z}_1, \dots, Z_n\bar{Z}_n)\). The change of variables is formal because \(Z = Z(z,\bar{z})\), \(\bar{Z} = \bar{Z}(z, \bar{z})\), and \(N\) are power series in general. The problem of the convergence or divergence of the normalizing transformation assuming that \(\hat{H}\) is analytic is here a central question.
With \(\mathcal{F}\) defined as the space of all power series in the variables \(z\) and \(\bar{z}\), the author studies a flow on \(\mathcal{F}\), called normalization flow. In the shift \[H_2 + \hat{H}_0 \rightarrow H_2 + \phi^\delta(\hat{H}_0)\] for \(\hat{H}_0 \in \mathcal{F}\), the normalization flow is \(\phi^\delta\) with \( \delta > 0\); it leads to a canonical change of variables. The flow \(\phi^\delta\) is determined by an ordinary differential equation in \(\mathcal {F}\) given by \(\partial_\delta {H_0} = - \{ {\xi H_0}, H_2 + H_0 \} \) with \(H_0 = \hat{H}\) when \(\delta = 0\) where \(\{ , \}\) is the Poisson bracket and \(\xi\) is a linear operator on \(\mathcal{F}\). The author argues that solutions of this equation map Hamiltonian functions to their normal forms.
Reviewer: William J. Satzer Jr. (St. Paul)Numerical computation of transverse homoclinic orbits for periodic solutions of delay differential equationshttps://zbmath.org/1532.370692024-05-13T19:39:47.825584Z"Hénot, Olivier"https://zbmath.org/authors/?q=ai:henot.olivier"Lessard, Jean-Philippe"https://zbmath.org/authors/?q=ai:lessard.jean-philippe"James, Jason D. Mireles"https://zbmath.org/authors/?q=ai:mireles-james.jason-dSummary: We present a computational method for studying transverse homoclinic orbits for periodic solutions of delay differential equations, a phenomenon that we refer to as the \textit{Poincaré scenario.} The strategy is geometric in nature and consists of viewing the connection as the zero of a nonlinear map, such that the invertibility of its Fréchet derivative implies the transversality of the intersection. The map is defined by a projected boundary value problem (BVP), with boundary conditions in the (finite dimensional) unstable and (infinite dimensional) stable manifolds of the periodic orbit. The parameterization method is used to compute the unstable manifold, and the BVP is solved using a discrete time dynamical system approach (defined via the \textit{method of steps}) and Chebyshev series expansions. We illustrate this technique by computing transverse homoclinic orbits in the cubic Ikeda and Mackey-Glass systems.Fractional stochastic differential equations driven by \(G\)-Brownian motion with delayshttps://zbmath.org/1532.601112024-05-13T19:39:47.825584Z"Saci, Akram"https://zbmath.org/authors/?q=ai:saci.akram"Redjil, Amel"https://zbmath.org/authors/?q=ai:redjil.amel"Boutabia, Hacene"https://zbmath.org/authors/?q=ai:boutabia.hacene"Kebiri, Omar"https://zbmath.org/authors/?q=ai:kebiri.omarSummary: This paper consists of two parts. In part I, existence and uniqueness of solution for fractional stochastic differential equations driven by \(G\)-Brownian motion with delays (\(G\)-FSDEs for short) is established. In part II, the averaging principle for this type of equations is given. We prove under some assumptions that the solution of \(G\)-FSDE can be approximated by solution of its averaged stochastic system in the sense of mean square.Quantitative coarse-graining of Markov chainshttps://zbmath.org/1532.601572024-05-13T19:39:47.825584Z"Hilder, Bastian"https://zbmath.org/authors/?q=ai:hilder.bastian"Sharma, Upanshu"https://zbmath.org/authors/?q=ai:sharma.upanshuSummary: Coarse-graining techniques play a central role in reducing the complexity of stochastic models and are typically characterized by a mapping which projects the full state of the system onto a smaller set of variables which captures the essential features of the system. Starting with a continuous-time Markov chain, in this work we propose and analyze an \textit{effective dynamics,} which approximates the dynamical information in the coarse-grained chain. Without assuming explicit scale-separation, we provide sufficient conditions under which this effective dynamics stays close to the original system and provide quantitative bounds on the approximation error. We also compare the effective dynamics and corresponding error bounds to the averaging literature on Markov chains which involve explicit scale-separation. We demonstrate our findings on an illustrative test example.Contribution of a \(\mathrm{Ca}^{2+}\)-activated \(\mathrm{K}^+\) channel to neuronal bursting activities in the Chay modelhttps://zbmath.org/1532.920162024-05-13T19:39:47.825584Z"Feng, Danqi"https://zbmath.org/authors/?q=ai:feng.danqi"Chen, Yu"https://zbmath.org/authors/?q=ai:chen.yu.6|chen.yu.1|chen.yu.2|chen.yu.4|chen.yu.18|chen.yu.3|chen.yu.8|chen.yu.15|chen.yu.12|chen.yu.10|chen.yuqun"Ji, Quanbao"https://zbmath.org/authors/?q=ai:ji.quanbao(no abstract)Cusp bifurcation in a metastatic regulatory networkhttps://zbmath.org/1532.920242024-05-13T19:39:47.825584Z"Delamonica, Brenda"https://zbmath.org/authors/?q=ai:delamonica.brenda"Balázsi, Gábor"https://zbmath.org/authors/?q=ai:balazsi.gabor"Shub, Michael"https://zbmath.org/authors/?q=ai:shub.michaelSummary: Understanding the potential for cancers to metastasize is still relatively unknown. While many predictive methods may use deep learning or stochastic processes, we highlight a long standing mathematical concept that may be useful for modeling metastatic breast cancer systems. Ordinary differential equations (ODEs) can model cell state transitions by considering the pertinent environmental variables as well as the paths systems take over time. Bifurcation theory is a branch of dynamical systems which studies changes in the behavior of an ODE system while one or more parameters are varied. Many studies have applied concepts in one-parameter bifurcation theory to model biological network dynamics, and cell division. However, studies of two-parameter bifurcations are much more rare. Two-parameter bifurcations have not been studied in metastatic systems. Here we show how a specific two-parameter bifurcation phenomenon called a cusp bifurcation separates two qualitatively different metastatic cell state transitions modalities and propose a new perspective on defining such transitions based on mathematical theory. We hope the observations and verification methods detailed here may help in the understanding of metastatic potential from a basic biological perspective and in clinical settings.Modelling of pathogens impact on the human disease transmission with optimal control strategieshttps://zbmath.org/1532.920692024-05-13T19:39:47.825584Z"Melese, Abdisa Shiferaw"https://zbmath.org/authors/?q=ai:melese.abdisa-shiferawSummary: This study concentrates on a nonlinear deterministic mathematical model for the impact of pathogens on human disease transmission with optimal control strategies. Both pathogen-free and coexistence equilibria are computed. The basic reproduction number \(R_0\), which plays a vital role in mathematical epidemiology, was derived. The qualitative analysis of the model revealed the scenario for both pathogen-free and coexistence equilibria together with \(R_0\). The local stability of the equilibria is established via the Jacobian matrix and Routh-Hurwitz criteria, while the global stability of the equilibria is proven by using an appropriate Lyapunov function. Also, the normalized sensitivity analysis has been performed to observe the impact of different parameters on \(R_0\). The proposed model is extended into optimal control problem by incorporating three control variables, namely, preventive measure variable based on separation of susceptible from contacting the pathogens, integrated vector management based on chemical, biological control, etc. to kill pathogens and their carriers, and supporting infective medication variable based on the care of the infected individual in quarantine center. Optimal disease control analysis is examined using Pontryagin minimum principle. Numerical simulations are performed depending on analytical results and discussed quantitatively.Stability and boundedness analysis of a prey-predator system with predator cannibalismhttps://zbmath.org/1532.920752024-05-13T19:39:47.825584Z"Olutimo, A. L."https://zbmath.org/authors/?q=ai:olutimo.a-l"Adams, D. O."https://zbmath.org/authors/?q=ai:adams.daniel-oluwasegun|adams.daniel-o"Abdurasid, A. A."https://zbmath.org/authors/?q=ai:abdurasid.a-aSummary: The prey-predator system with predator cannibalism is considered in this papper. We employ the Lyapunov's direct method for the prey-predator systemand demonstrate its efficacy. This method is built upon theoretical Lyapunov's function that is constructed such that the scalar function and its derivative ispositive and negative definite respectively to determine the dynamic behaviour of the system considered including stability and boundedness. The results show that the density functions describing the prey-predator system is better rapidly converging under certain sufficient conditions obtained by the Lyapunov functional. We give numeric example to support our findings.Effect of nonlinear prey refuge on predator-prey dynamicshttps://zbmath.org/1532.920762024-05-13T19:39:47.825584Z"Samaddar, Shilpa"https://zbmath.org/authors/?q=ai:samaddar.shilpa"Dhar, Mausumi"https://zbmath.org/authors/?q=ai:dhar.mausumi"Bhattacharya, Paritosh"https://zbmath.org/authors/?q=ai:bhattacharya.paritoshSummary: A mathematical model on predator-prey dynamics is analyzed in this study. In traditional models, prey refuge is usually taken constant which is nearly impossible in real-life scenario. We have considered nonlinear prey refuge which depends on both prey and predator. We have performed various dynamical studies incorporating Holling type-II functional response. The system can perceive at most three equilibria. The boundedness of all the solutions, stability-instability conditions, and bifurcation analysis are demonstrated in this work. All the analytical findings are verified with numerical simulations. Additionally, a model comparison is performed which helps to understand the dynamical changes due to nonlinear refuge.
For the entire collection see [Zbl 1531.00060].Bifurcation analysis in a discrete predator-prey model with herd behaviour and group defensehttps://zbmath.org/1532.920802024-05-13T19:39:47.825584Z"Xia, Jie"https://zbmath.org/authors/?q=ai:xia.jie"Li, Xianyi"https://zbmath.org/authors/?q=ai:li.xianyi(no abstract)A multi-strain model for COVID-19https://zbmath.org/1532.920942024-05-13T19:39:47.825584Z"Ghosh, Samiran"https://zbmath.org/authors/?q=ai:ghosh.samiran.1|ghosh.samiran"Banerjee, Malay"https://zbmath.org/authors/?q=ai:banerjee.malaySummary: The main objective of this work is to propose and analyze a multi-compartment ordinary differential equation model for multi-strain epidemic disease. The proposed model mainly focuses on the epidemic disease spread due to SARS-CoV-2, and the recurrent outbreaks are due to the emergence of a new strain. The possibility of reinfection of the recovered individuals is considered in the model. The multi-strain model is validated with the help of strain-specific daily infection data from France and Italy.
For the entire collection see [Zbl 1531.00060].Dynamics on hepatitis B virus infection in vivo with interval delayhttps://zbmath.org/1532.921172024-05-13T19:39:47.825584Z"Zhong, Haonan"https://zbmath.org/authors/?q=ai:zhong.haonan"Wang, Kaifa"https://zbmath.org/authors/?q=ai:wang.kaifaThis paper studies dynamics on hepatitis B virus infection in vivo with interval delay. It proposes a delay differential equation model of hepatitis B virus infection within-host, where a kernel function is utilized to depict the delay occurring in the past subinterval, rather than a fixed point (discrete delay) or an entire interval (distributed delay). It presents the stability analysis of equilibria, including virus-free equilibrium and virus-present equilibrium, which is also known as the chronic infection equilibrium. Some conditions of the existence of a pure imaginary root for the characteristic equation at virus-present equilibrium are derived. The Hopf bifurcations of the delay center and delay radius are established. Numerical simulations are presented to illustrate the theoretical prediction using the detected virus load series of two untreated chronic hepatitis B patients.
Reviewer: Yilun Shang (Newcastle upon Tyne)Stoichiometric microplastics models in natural and laboratory environmentshttps://zbmath.org/1532.921212024-05-13T19:39:47.825584Z"Wang, Tianxu"https://zbmath.org/authors/?q=ai:wang.tianxu"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.25|wang.hao.4|wang.hao.14|wang.hao.12|wang.hao.29|wang.hao.37|wang.hao.27|wang.hao.22|wang.hao.7|wang.hao.1|wang.hao.24|wang.hao.26Summary: Microplastics pose a severe threat to marine ecosystems; however, relevant mathematical modeling and analysis are lacking. This paper formulates two stoichiometric producer-grazer models to investigate the interactive effects of microplastics, nutrients, and light on population dynamics under different settings. One model incorporates optimal microplastic uptake and foraging behavior based on nutrient availability for natural settings, while the other model does not include foraging in laboratory settings. We establish the well-posedness of the models and examine their long-term behaviors. Our results reveal that in natural environments, producers and grazers exhibit higher sensitivity to microplastics, and the system may demonstrate bistability or tristability. Moreover, the influences of microplastics, nutrients, and light intensity are highly intertwined. The presence of microplastics amplifies the constraints on grazer growth related to food quality and quantity imposed by extreme light intensities, while elevated phosphorus input enhances the system's resistance to intense light conditions. Furthermore, higher environmental microplastic levels do not always imply elevated microplastic body burdens in organisms, as organisms are also influenced by nutrients and light. We also find that grazers are more vulnerable to microplastics, compared to producers. If producers can utilize microplastics for growth, the system displays significantly greater resilience to microplastics.Modeling and analysis of release strategies of sterile mosquitoes incorporating stage and sex structure of wild oneshttps://zbmath.org/1532.921222024-05-13T19:39:47.825584Z"Huang, Mingzhan"https://zbmath.org/authors/?q=ai:huang.mingzhan"Yu, Xiaohuan"https://zbmath.org/authors/?q=ai:yu.xiaohuan"Liu, Shouzong"https://zbmath.org/authors/?q=ai:liu.shouzong(no abstract)Effects of additional food availability and pulse control on the dynamics of a Holling-\(( p +1)\) type pest-natural enemy modelhttps://zbmath.org/1532.921242024-05-13T19:39:47.825584Z"Yan, Xinrui"https://zbmath.org/authors/?q=ai:yan.xinrui"Tian, Yuan"https://zbmath.org/authors/?q=ai:tian.yuan"Sun, Kaibiao"https://zbmath.org/authors/?q=ai:sun.kaibiao(no abstract)Dynamics of a three-molecule autocatalytic Schnakenberg model with cross-diffusion: Turing patterns of spatially homogeneous Hopf bifurcating periodic solutionshttps://zbmath.org/1532.921282024-05-13T19:39:47.825584Z"Li, Weiyu"https://zbmath.org/authors/?q=ai:li.weiyu"Wang, Hongyan"https://zbmath.org/authors/?q=ai:wang.hongyan(no abstract)