Stability criteria for a nonlinear nonautonomous system with delays

This paper further develops a method, originally introduced by Mori et al., for proving local stability of steady states in linear systems of delay differential equations. A nonlinear nonautonomous system of delay differential equations with several delays is considered. Explicit delay-independent sufficient conditions for global attractivity of the solutions with an extremely simple form are provided. The above-mentioned conditions make the stability test quite practical. We illustrate application of this test to the Hopfield neural network models. The results obtained were also applied to a new marine protected areas model with delay that describes the ecological linkage between the reserve and fishing ground.     

具有可变时滞的非线性非自治差分方程的线性化全局渐近稳定性及其应用

证明了在一定条件下,具有可变时滞的非线性非自治差分方程的全局渐近稳定性可由某种线性差分方程的渐近稳定性确定,给出了这类差分方程全局渐近稳定的充分条件.作为实例,获得了具有可变时滞的离散型非自治广义Log istic方程的全局吸收性判别准则.     

Some sufficient conditions for the observability of nonlinear systems

Some sufficient conditions are presented for the observability of systems described by nonlinear, ordinary differential equations with linear observations. The point which the authors especially emphasize is the fact that the results obtained are global in some sense. In the beginning, the observability problem is reduced to the uniqueness problem of solutions of some nonlinear integral equations for general nonlinear systems with linear observations. Then, for some restricted systems, a simple sufficient condition is derived. The relation between global and local observability for nonlinear systems is also considered.The authors wish to thank their colleagues in Nagoya University.     

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations, including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations.     

Stability of Steady-State Solutions of Systems of Nonlinear Nonautonomous Delay Differential Equations

Sufficient conditions for the stability of steady-state solutions of systems of nonautonomous linear and nonlinear differential equations with time-dependent delay are obtained in terms of coefficients. These sufficient conditions are written as inequalities relating quantities that can be calculated directly from the right-hand side of the system of equations.     

On the topological structure of singular attractors of nonlinear systems of differential equations

We study the topological structure of singular (in the sense of the Feigenbaum-Sharkovskii-Magnitskii theory) attractors of nonlinear dissipative systems of differential equations. We show that any such attractor is a stable nonperiodic trajectory lying on a two-dimensional infinitely folded heteroclinic separatrix manifold generated by the unstable two-dimensional invariant manifold of the original singular cycle as the bifurcation parameter of the system varies. The results obtained for two-dimensional nonautonomous and three-dimensional autonomous dissipative systems are generalized to autonomous multi- and infinite-dimensional dissipative systems as well as to conservative (in particular, Hamiltonian) systems.     

Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point

We establish conditions for the existence of solutions vanishing at a singular point for various classes of systems of quasilinear differential equations appearing in the investigation of the asymptotic behavior of solutions of essentially nonlinear nonautonomous differential equations of higher orders.     

Existence Theorems and Estimates of Solutions for Equations of Principal Resonance

In this paper, we study systems of nonlinear, nonautonomous, ordinary differential equations that appear in the theory of averaging of nonlinear oscillations. We prove existence theorems for them and obtain conditions under which variables of the type of amplitude (or energy) are uniformly bounded with respect to time.     

Global stability for separable nonlinear delay differential systems

In this paper, we consider separable nonlinear delay differential systems and we establish conditions for global asymptotic stability of the zero solution. Applying these, we offer improved 3/2-type criteria for global asymptotic stability of nonautonomous Lotka-Volterra systems with delays.     

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.     

Positivity conditions and bounds for Green’s functions for higher order two-point BVP

We consider a linear nonautonomous higher order ordinary differential equation and establish the positivity conditions and two-sided bounds for Green’s function for the two-point boundary value problem. Applications of the obtained results to nonlinear equations are also discussed.     

Invariant set and attractivity of nonlinear differential equations with delays

The aim of this paper is to study the attracting set and attraction basin of the nonlinear and nonautonomous delay differential equations. Some new criteria for the attracting set and the attraction basin are obtained. Several examples are also worked out to demonstrate the advantages of our results.     

Method of limiting equations for the stability analysis of equations with infinite delay in the Carathéodory conditions: II

We consider systems of nonautonomous nonlinear differential equations with the infinite delay. We study the stability properties and the limiting equations whose right-hand sides are defined as the limit points of some sequence in the introduced function space. By using the method of limiting equations, we obtain new sufficient conditions for the asymptotic stability of the zero solution of the considered class of equations.     

Uniformly attracting solutions of nonautonomous differential equations

Understanding the structure of attractors is fundamental in nonautonomous stability and bifurcation theory. By means of clarifying theorems and carefully designed examples we highlight the potential complexity of attractors for nonautonomous differential equations that are as close to autonomous equations as possible. We introduce and study bounded uniform attractors and repellors for nonautonomous scalar differential equations, in particular for asymptotically autonomous, polynomial, and periodic equations. Our results suggest that uniformly attracting or repelling solutions are the true analogues of attracting or repelling fixed points of autonomous systems. We provide sharp conditions for the autonomous structure to break up and give way to a bewildering diversity of nonautonomous bifurcations.     

非线性中立抛物型偏微分方程系统的振动性定理

研究一类非线性中立抛物型时滞偏微分方程系统解的振动性质,利用积分不等式和泛函微分方程的某些结果,获得了该类系统在第一类边值条件下所有解振动的若干充分条件.结论充分表明振动是由时滞量引起的.     

Periodic solutions of second-order differential inclusions systems with p-Laplacian

Some existence results are obtained for periodic solutions of nonautonomous second-order differential inclusions systems with p-Laplacian.     

约束力学系统的联络及其运动方程的测地性质

用现代整体微分几何方法研究非定常约束力学系统运动方程的测地性质，得到非定常力学系统的动力学流关于１＿射丛上的联络具有测地性质的充分必要条件·非定常情形下的动力学流关于无挠率的联络总具有测地性质，因此任何非定常约束力学系统在外力作用下的运动总可以表示为关于１＿射丛上无挠率的动力学联络的测地运动，这与定常力学的情形有所区别·     

具有可变时滞的非线性非自治差分方程的振动性及其应用

本文获得了一类具有可变时滞的非线性非自治差分方程的振动准则,建立了这类差分方程的几个线性化振动性定理,并得到了具有可变时滞的离散型非自治广义Logistic方程关于其正平衡点振动的一系列充分条件.     

Banach空间中非线性脉冲Volterra型积分方程组的可解性

在较弱的条件下，利用锥理论和单调迭代方法首先建立了Banach空间中一类非线性算子方程组最小最大解的存在性定理；然后作为应用，利用一个新的比较结果，得到了Banach空间中非线性脉冲Volterra型积分方程组的整体解，改进了最近的许多结果．     

Estimates for Solutions to a Class of Nonautonomous Systems of Neutral Type with Unbounded Delay

Siberian Mathematical Journal - Under consideration is the class of nonlinear systems of nonautonomous differential equations of neutral type with a variable delay that can be unbounded. Using a...