In this work we will study the structure of the skew-product attractor for a planar diffusively coupled ordinary differential equation, given by $\dot{x}= k(y-x)+x-\beta(t)x^3$ and $\dot{y}= k(x-y)+y-\beta(t)y^3$, $t\geq 0$. 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.
