Generic Morse-Smale property for the parabolic equation on the circle

TL;DR

This paper proves that for scalar reaction-diffusion equations on the circle, the Morse-Smale property is generic with respect to the non-linearity, ensuring transversality of connecting orbits and a simplified structure of the non-wandering set.

Contribution

It completes previous results by showing that all connecting orbits between hyperbolic equilibria or periodic orbits are transverse, and that connections between equilibria with the same Morse index are generically absent.

Findings

Generic transversality of connecting orbits between hyperbolic states.

Non-existence of connections between equilibria with same Morse index.

Finite non-wandering set consisting of hyperbolic equilibria and periodic orbits.

Abstract

In this paper, we show that, for scalar reaction-diffusion equations $u_t=u_{xx}+f(x,u,u_x)$ on the circle $S^1$, the Morse-Smale property is generic with respect to the non-linearity $f$. In \cite{CR}, Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown that, generically with respect to the non-linearity $f$, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showing that any connecting orbit between two hyperbolic equilibria with distinct Morse indices or between a hyperbolic equilibrium and a hyperbolic periodic orbit is automatically transverse. We also show that, generically with respect to $f$, there does not exist any connection between equilibria with the same Morse index. The above properties, together with the existence of a compact global attractor and the Poincar\'e-Bendixson property, allow us to deduce that, generically with respect to $f$, the non-wandering set consists in a finite number of hyperbolic equilibria and periodic orbits . The main tools in the proofs include the lap number property, exponential dichotomies and the Sard-Smale theorem. The proofs also require a careful analysis of the asymptotic behavior of solutions of the linearized equations along the connecting orbits.