Generic Transversality of Heteroclinic and Homoclinic Orbits for Scalar Parabolic Equations

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 $$\mathcal {C}^{2,\gamma }$$. 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.