STABILITY OF LYAPUNOV EXPONENTS, WEAK INTEGRAL SEPARATION AND NONUNIFORM DICHOTOMY SPECTRUM

In this paper, a necessary and sufficient condition for the stability of Lyapunov exponents of linear differential system is proved in the sense that the equations satisfy the weaker form of integral separation instead of its classical one. Furthermore, the existence of full nonuniform exponential dichotomy spectrum under the condition of weak integral separateness is also presented.

Uniform Dichotomy Concepts for Discrete-Time Skew Evolution Cocycles in Banach Spaces

In the present paper, we consider the problem of dichotomic behaviors of dynamical systems described by discrete-time skew evolution cocycles in Banach spaces. We study two concepts of uniform dichotomy: uniform exponential dichotomy and uniform polynomial dichotomy. Some characterizations of these notions and connections between these concepts are given.

Permanence of nonuniform nonautonomous hyperbolicity for infinite-dimensional differential equations

In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturbation. Moreover, we provide conditions to obtain uniqueness and continuous dependence of projections associated with nonuniform exponential dichotomies. We also present an example of evolution process in a Banach space that admits nonuniform exponential dichotomy and study the permanence of the nonuniform hyperbolicity under perturbation. Finally, we prove persistence of nonuniform hyperbolic solutions for nonlinear evolution processes under perturbations.

Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

Remotely Almost Periodic Solutions of Ordinary Differential Equations

In this paper, we analyze the existence and uniqueness of remotely almost periodic solutions for systems of ordinary differential equations. The existence and uniqueness of remotely almost periodic solutions are achieved by using the results about the exponential dichotomy and the Bi-almost remotely almost periodicity of the homogeneous part of the corresponding systems of ordinary differential equations under our consideration.

Almost periodic solutions of a discrete Lotka-Volterra model via exponential dichotomy theory

<abstract><p>In this paper, we consider a discrete non-autonomous Lotka-Volterra model. Under some assumptions, we prove the existence of positive almost periodic solutions. Our analysis relies on the exponential dichotomy for the difference equations and the Banach fixed point theorem. Furthermore, by constructing a Lyapunov function, the exponential convergence is proved. Finally, a numerical example illustrates the effectiveness of the results.</p></abstract>