Pullback, forward and chaotic dynamics in 1D non-autonomous linear-dissipative equations

The global attractor of a skew product semiflow for a non-autonomous differential equation describes the asymptotic behaviour of the model. This attractor is usually characterized as the union, for all the parameters in the base space, of the associated cocycle attractors in the product space. The continuity of the cocycle attractor in the parameter is usually a difficult question. In this paper we develop in detail a 1D non-autonomous linear differential equation and show the richness of non-autonomous dynamics by focusing on the continuity, characterization and chaotic dynamics of the cocycle attractors. In particular, we analyse the sets of continuity and discontinuity for the parameter of the attractors, and relate them with the eventually forward behaviour of the processes. We will also find chaotic behaviour on the attractors in the Li-Yorke and Auslander-Yorke senses. Note that they hold for linear 1D equations, which shows a crucial difference with respect to the presence of chaotic dynamics in autonomous systems.