Almost Periodic Dynamics of Perturbed Infinite-Dimensional Dynamical Systems

TL;DR

This paper studies how small almost periodic perturbations affect the dynamics of infinite-dimensional gradient systems, showing the persistence of almost periodic solutions and describing the structure of the pullback attractor.

Contribution

It proves that under perturbations, the number of almost periodic solutions remains unchanged and characterizes the pullback attractor as a union of unstable manifolds of these solutions.

Findings

The perturbed system retains the same number of almost periodic solutions.

The pullback attractor is the union of unstable manifolds of these solutions.

Application to the Chafee-Infante equation demonstrates the theory.

Abstract

This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions. An application of the result to the Chafee-Infante equation is discussed.