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

Abstract

<p style='text-indent:20px;'>In this work we will study the structure of the skew-product attractor for a planar diffusively coupled ordinary differential equation, given by <inline-formula><tex-math id="M1">\begin{document}$ \dot{x} = k(y-x)+x-\beta(t)x^3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \dot{y} = k(x-y)+y-\beta(t)y^3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ t\geq 0 $\end{document}</tex-math></inline-formula>. We identify the non-autonomous structures that completely describes the dynamics of this model giving a Morse decomposition for the skew-product attractor. The complexity of the isolated invariant sets in the global attractor of the associated skew-product semigroup is associated to the complexity of the attractor of the associated driving semigroup. In particular, if <inline-formula><tex-math id="M4">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> is asymptotically almost periodic, the isolated invariant sets will be almost periodic hyperbolic global solutions of an associated globally defined problem.</p>