Structure of non-autonomous attractors for a class of diffusively coupled ode

In this work we will study the structure of the skew-product attractor for a planar diffusively coupled ordinary differential equation, given by (x) over dot = k(y - x) x - beta(t)x(3) and (y) over dot = k(x - y) y - beta(t)y(3), t >= 0. We identify the non-autonomous structures that completely describes the dynamics of this model giving a Morse decomposition for the skew-product attractor. The complexity of the isolated invariant sets in the global attractor of the associated skew-product semigroup is associated to the complexity of the attractor of the associated driving semigroup. In particular, if beta is asymptotically almost periodic, the isolated invariant sets will be almost periodic hyperbolic global solutions of an associated globally defined problem.