Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle

TL;DR

This paper proves that for scalar reaction-diffusion equations on the circle, hyperbolicity of all equilibria and periodic orbits is a generic property in the space of nonlinearities, using lap number and Sard-Smale theorem.

Contribution

It establishes the generic hyperbolicity of equilibria and periodic orbits for scalar reaction-diffusion equations on the circle, extending understanding of their stability properties.

Findings

Hyperbolicity is generic among nonlinearities in the considered equations.

The set of nonlinearities with all hyperbolic equilibria and orbits is a countable intersection of open dense sets.

The proof utilizes lap number properties and Sard-Smale theorem.

Abstract

In this paper, we show that, for scalar reaction-diffusion equations on the circle S1, the property of hyperbolicity of all equilibria and periodic orbits is generic with respect to the non-linearity . In other words, we prove that in an appropriate functional space of nonlinear terms in the equation, the set of functions, for which all equilibria and periodic orbits are hyperbolic, is a countable intersection of open dense sets. The main tools in the proof are the property of the lap number and the Sard-Smale theorem.