On The Dynamical Behavior Of Some Nonautonomous Systems

This paper is concerned with the long time dynamical behavior of some nonau-tonomous systems. Our aims are two.One is to make a discussion on the dynamics of an autonomous system with nonau-tonomous perturbation by using the attractor of the autonomous system. We first extend the stability of the Morse decompositions of attractors in infinite dynamical sys-tem via constructing Lipschitz Morse-Lyapunov functions. Then based on this result, we consider the following perturbed autonomous system on a Banach space X: and prove that any solution of the perturbed system will ultimately enter and stay in the e-neighborhood of some Morse set of the autonomous system provided g(t) sufficiently small.The other is to study a general nonautonomous problem. By using the theory of pullback attractors, we investigate the following nonautonomous abstract evolution equation with multiple time delays: where A:D(A)(?) Hâ†'H is a positive definite selfadjoint operator with compact resolvent, F:D(Aα)nâ†'H (0≤α≤1/2) is a locally Lipschitz mapping satisfying a linear growth condition, g(t)∈C(R; H) is a bounded function, and r1,..., rn≥0are constant time delays. First, we prove the dissipativity and establish the existence of pullback attractors under appropriate conditions. Then, we introduce the concept of locally almost periodic functions and show that if g is locally almost periodic/periodic, then the retarded equation has at least a locally almost periodic/periodic solution. Finally, we are interested in the synchronizing phenomena of the retarded equation.