The Morse Smale property for time-periodic scalar reaction-diffusion equation on the circle

TL;DR

This paper investigates the Morse-Smale property for a class of time-periodic scalar reaction-diffusion equations on the circle, establishing conditions for hyperbolicity and transversality of invariant manifolds.

Contribution

It proves the Morse-Smale property for the equation under conditions on hyperbolic fixed points and their invariant manifolds, extending understanding of dynamics in periodic reaction-diffusion systems.

Findings

Transversality of stable and unstable manifolds for heteroclinic hyperbolic fixed points.

Exclusion of homoclinic connections for hyperbolic fixed points.

Morse-Smale property holds when all omega-limit sets are hyperbolic.

Abstract

\begin{abstract} We study the Morse-Smale property for the following scalar semilinear parabolic equation on the circle $S^1$, \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2\pi \mathbb{Z}, \end{equation*} where $f$ is a $C^2$ function and $T$-periodic in $t$. Assume that the equation admits a compact global attractor $\mathcal{A}$ and let $P$ be the Poincar\'{e} map of this equation. We exclude homoclinic connection for hyperbolic fixed points of $P$ and prove that stable and unstable manifolds for any two heteroclinic hyperbolic fixed points of $P$ intersect transversely. Further, this equation admits the Morse-Smale property provided that all $\omega$-limit sets (in the case $f(t,u,u_x)=f(t,u,-u_x)$, the $\omega$-limit set is just a fixed point) of the corresponding Poincar\'{e} map are hyperbolic. \end{abstract}