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


Authors: Romain Joly and Geneviève Raugel
Journal: Trans. Amer. Math. Soc. 362 (2010), 5189-5211
MSC (2010): Primary 35B10, 35B30, 35K57, 37D05, 37D15, 37L45; Secondary 35B40
DOI: https://doi.org/10.1090/S0002-9947-2010-04890-1
Published electronically: May 20, 2010
MathSciNet review: 2657677
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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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Additional Information

Romain Joly
Affiliation: Institut Fourier, UMR CNRS 5582, Université de Grenoble I, B.P. 74, 38402 Saint-Martin-d’Hères, France
Email: Romain.Joly@ujf-grenoble.fr

Geneviève Raugel
Affiliation: CNRS, Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, F-91405 Orsay Cedex, France
Email: Genevieve.Raugel@math.u-psud.fr

DOI: https://doi.org/10.1090/S0002-9947-2010-04890-1
Keywords: Hyperbolicity, genericity, periodic orbits, equilibria, Sard-Smale, lap number
Received by editor(s): May 20, 2008
Published electronically: May 20, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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