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

Romain Joly (1) , Geneviève Raugel (2)

1 IF - Institut Fourier
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: Hyperbolicity; genericity; periodic orbits; equilibria; Sard-Smale; lap number

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Romain Joly, Geneviève Raugel. Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle. Transactions of the American Mathematical Society, 2010, 362, p. 5189-5211. ⟨hal-00288326v2⟩