STRUCTURE AND BIFURCATION OF PULLBACK ATTRACTORS IN A NON-AUTONOMOUS CHAFEE-INFANTE EQUATION

The Chafee-Infante equation is one of the canonical infinite-dimensional dynamical systems for which a complete description of the global attractor is available. In this paper we study the structure of the pullback attractor for a non-autonomous version of this equation, u(t) = u(xx) + lambda(xx) - lambda u beta(t)u(3), and investigate the bifurcations that this attractor undergoes as A is varied. We are able to describe these in some detail, despite the fact that our model is truly non-autonomous; i.e., we do not restrict to 'small perturbations' of the autonomous case.

Pullback exponential attractors for evolution processes in Banach spaces: properties and applications

This article is a continuation of our previous work [5], where we formulated general existence theorems for pullback exponential attractors for asymptotically compact evolution processes in Banach spaces and discussed its implications in the autonomous case. We now study properties of the attractors and use our theoretical results to prove the existence of pullback exponential attractors in two examples, where previous results do not apply.

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

An estimate on the fractal dimension of attractors of gradient-like dynamical systems

This paper is dedicated to estimate the fractal dimension of exponential global attractors of some generalized gradient-like semigroups in a general Banach space in terms of the maximum of the dimension of the local unstable manifolds of the isolated invariant sets, Lipschitz properties of the semigroup and the rate of exponential attraction. We also generalize this result for some special evolution processes, introducing a concept of Morse decomposition with pullback attractivity. Under suitable assumptions, if (A, A*) is an attractor-repeller pair for the attractor A of a semigroup {T(t) : t >= 0}, then the fractal dimension of A can be estimated in terms of the fractal dimension of the local unstable manifold of A*, the fractal dimension of A, the Lipschitz properties of the semigroup and the rate of the exponential attraction. The ingredients of the proof are the notion of generalized gradient-like semigroups and their regular attractors, Morse decomposition and a fine analysis of the structure of the attractors. As we said previously, we generalize this result for some evolution processes using the same basic ideas. (C) 2012 Elsevier Ltd. All rights reserved.