On the equivalence of two differential equations by means of reflecting functions coinciding

In this article, we discuss the equivalence of two differential systems by using the method of reflecting functions. We obtain some necessary and sufficient conditions under which certain differential equations are equivalent. Given these results, new types of differential systems equivalent to the given systems can be found. We also discussed the qualitative behavior of the periodic solutions of such differential systems. These results are new, in the sense that they generalize previous discussions on the equivalence of differential systems.     

On the equivalence of differential systems

In this article. First, we construct some nonlinear differential systems which are equivalent to some known systems. Second, we discuss, in a different method, the equivalence between some linear differential systems. And then we apply the obtained results to the study of the qualitative properties of these systems simultaneously.     

On the reflecting function and the qualitative behavior of solutions of some non-autonomous differential equations

In this article, we use the Mironenko''s method to discuss the qualitative behavior of some non-autonomous differential equations. We study the structure of the reflecting functions of the simplest differential equations, and obtain some sufficient conditions under which these equations have the rational reflecting functions. We apply the obtained results to discuss the numbers of periodic solutions of the non-autonomous differential systems and derive some sufficient conditions for a critical point of theirs to be a center.     

On the qualitative behavior of periodic solutions of differential systems

This article deals with the reflective function of differential systems. The obtained results are applied to studying the existence and stability of the periodic solutions of some linear and nonlinear periodic differential systems.     

Conditional stability for periodic solutions of periodic systems containing a small parameter

We treat periodic solutions to periodic systems of ordinary differential equations, having a small parameter. Conditional stability and the dimensionality of the asymptotically stable manifolds are determined by the degree of stability of rest points of related iterated average systems. Finally, rates of decay for solutions on these manifolds are found, and for linear systems, the Floquet multipliers are determined.     

ON THE EQUIVALENCE OF AUTONOMOUS AND NONAUTONOMOUS DIFFERENTIAL SYSTEMS

Using reflecting function of Mironenko we construct some nonautonomous differential systems which are equivalent to the given autonomous differential systems. The results are applied to discuss the existence and stability of periodic solution of these nonautonomous nonlinear differential systems.     

两种群非自治Lotka-Volterra竞争扩散系统的概周期解

本文讨论两种群概周期竞争扩散系统，利用微分不等式，证明了系统概周期解的存在、唯一性及其在壳扰动下的稳定性．     

Search for Periodic Solutions of Highly Nonlinear Dynamical Systems

Numerical-analytical methods for finding periodic solutions of highly nonlinear autonomous and nonautonomous systems of ordinary differential equations are considered. Algorithms for finding initial conditions corresponding to a periodic solution are proposed. The stability of the found periodic solutions is analyzed using corresponding variational systems. The possibility of studying the evolution of periodic solutions in a strange attractor zone and on its boundaries is discussed, and interactive software implementations of the proposed algorithms are described. Numerical examples are given.     

一类无穷时滞微分系统的周期解和全局渐近稳定性

利用重合度理论中的延拓定理和微分不等式讨论一类无穷时滞微分系统的周期解的存在性和全局渐近稳定性,获得了简便的判别条件.     

一类泛函微分方程的周期解的存在性、唯一性及全局吸引性

该文研究了一类具有分布滞量的微分系统的周期解的存在性、唯一性及全局吸引性等问题.利用不动点方法和Lyapunov泛函方法,建立了保证该类系统周期解的存在性、唯一性、一致稳定性及全局吸引性的充分条件.     

微分系统的反射函数与周期解

This article deals with the reflective function of the mth-order nonlinear differential systems. The results are applied to discussing the stability property of periodic solutions of these systems.     

The differential systems with the same reflecting function

Using reflecting function, we give the conditions under which some complex differential systems are equivalent to the relatively simple differential system. The obtained results are applied to study the behavior of solutions of these complex differential systems.     

Dynamics in three cells with multiple time delays

We consider systems of delay differential equations representing the models containing three cells with any time-delayed connections. Global stability, delay-independent and delay-dependent local stability are studied, the existence of local and global periodic solutions is investigated. We give the stability conditions, respectively, and show that the local periodic solutions can be extended globally after certain critical values of delay.     

Equivalence of Differential System

Using reflecting function of Mironenko we construct some differential systems which are equivalentto the given differential system.This gives us an opportunity to find out the monodromic matrix of these periodicsystems which are not integrable in finite terms.     

Periodic traveling-wave-type solutions in circular chains of unidirectionally coupled equations

We consider special systems of ordinary differential equations, so-called circular unidirectional chains. For this class of systems, we develop a new method for studying the existence and stability problems for periodic solutions. A feature of the proposed approach is that some auxiliary systems with delay are used in both seeking cycles and analyzing their stability properties. We illustrate the relevance of this approach with a concrete example of a circular Hopfield neural network     

Morse Decompositions for Periodic General Dynamical Systems and Differential Inclusions

We first establish the Morse decomposition theory of periodic invariant sets for non-autonomous periodic general dynamical systems (set-valued dynamical systems). Then we discuss the stability of Morse decompositions of periodic uniform forward attractors. We also apply the abstract results to non-autonomous periodic differential inclusions with only upper semi-continuous right-hand side. We show that Morse decompositions are robust with respect to both internal and external perturbations (upper semi-continuity of Morse sets). Finally as an application we study the effect of small time delays to asymptotic behavior of control systems from the point of view of Morse decompositions.     

Dynamics in numerics: on two different finite difference schemes for ODEs

We compare two finite difference schemes for Kolmogorov type of ordinary differential equations: Euler's scheme (a derivative approximation scheme) and an integral approximation (IA) scheme, from the view point of dynamical systems. Among the topics we investigate are equilibria and their stability, periodic orbits and their stability, and topological chaos of these two resulting nonlinear discrete dynamical systems.     

Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems

The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given.     

A Branching Method for Studying Stability of a Solution to a Delay Differential Equation

We study stability of antisymmetric periodic solutions to delay differential equations. We introduce a one-parameter family of periodic solutions to a special system of ordinary differential equations with a variable period. Conditions for stability of an antisymmetric periodic solution to a delay differential equation are stated in terms of this period function.     

Critical stability of solutions to linear ordinary differential equations with large high-frequency terms

The stability problem is considered for certain classes of systems of linear ordinary differential equations with almost periodic coefficients. These systems are characterized by the presence of rapidly oscillating terms with large amplitudes. For each class of equations, a procedure for analyzing the critical stability of solutions is constructed on the basis of the Shtokalo-Kolesov method. A verification scheme is described. The theory proposed is illustrated by using a linearized stability problem for the upper equilibrium of a pendulum with a vibrating suspension point.