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

Joly, Romain; Raugel, Geneviève

doi:10.1090/S0002-9947-2010-04890-1

Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle
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by Romain Joly and Geneviève Raugel PDF

Trans. Amer. Math. Soc. 362 (2010), 5189-5211 Request permission

Abstract:

In this paper, we show that, for scalar reaction-diffusion equations on the circle $S^1$, 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 non-linear 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.

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