# Generic transversality of heteroclinic and homoclinic orbits for scalar parabolic equations

## TL;DR

This paper proves that heteroclinic and homoclinic orbits in scalar reaction-diffusion equations are generically transverse, advancing understanding of the structural stability of these dynamical systems.

## Contribution

It establishes the generic transversality of heteroclinic and homoclinic orbits for scalar parabolic equations, a key step towards proving the Kupka-Smale property.

## Key findings

- Heteroclinic and homoclinic orbits are transverse for generic reaction-diffusion equations.
- The study includes an analysis of the singular nodal set of solutions of linear parabolic equations.
- Results contribute to the understanding of the generic hyperbolicity of orbits in these systems.

## Abstract

In this paper, we consider the scalar reaction-diffusion equations $\partial_t u=\Delta u + f(x,u,\nabla u)$ on a bounded domain $\Omega\subset\mathbb{R}^d$ of class $C^2$. We show that the heteroclinic and homoclinic orbits connecting hyperbolic equilibria and hyperbolic periodic orbits are transverse, generically with respect to f. One of the main ingredients of the proof is an accurate study of the singular nodal set of solutions of linear parabolic equations. Our main result is a first step for proving the genericity of Kupka-Smale property, the generic hyperbolicity of periodic orbits remaining unproved.