Global an cocycle atractor for non-utonomous reaction-difussion equations . The case of null upper Lyapunov exponente

TL;DR

This paper characterizes the structure of global and cocycle attractors for non-autonomous reaction-diffusion equations with null Lyapunov exponents, revealing simple or complex attractor types depending on the boundedness of associated cocycles.

Contribution

It provides a detailed description of attractors in scalar linear-dissipative parabolic problems under minimal ergodic conditions, including bifurcation scenarios and chaos in measure.

Findings

Attractors are simple in periodic cases and complex in aperiodic cases.

Chaotic attractors can occur in measure for certain equations.

A non-autonomous bifurcation scenario is identified for concave equations.

Abstract

In this paper we obtain a detailed description of the global and cocycle attractors for the skew-product semiflows induced by the mild solutions of a family of scalar linear-dissipative parabolic problems over a minimal and uniquely ergodic flow. We consider the case of null upper Lyapunov exponent for the linear part of the problem. Then, two different types of attractors can appear, depending on whether the linear equations have a bounded or an unbounded associated real cocycle. In the first case (e.g.~in periodic equations), the structure of the attractor is simple, whereas in the second case (which occurs in aperiodic equations), the attractor is a pinched set with a complicated structure. We describe situations when the attractor is chaotic in measure in the sense of Li-Yorke. Besides, we obtain a non-autonomous discontinuous pitchfork bifurcation scenario for concave equations, applicable for instance to a linear-dissipative version of the Chafee-Infante equation.