Comptes Rendus
no. 8
Dynamical Systems/Ordinary Differential Equations
Birth control in generalized Hopf bifurcations
[Contrôle des naissances dans les bifurcations de Hopf généralisées]

Présenté par : by Étienne Ghys

Marc Chaperon1
1 Institut de mathématiques de Jussieu & Université Paris 7, UFR de mathématiques, case 7012, 2, place Jussieu, 75251 Paris cedex 05, France
Comptes Rendus. Mathématique, Volume 345 (2007) no. 8, pp. 453-458.

Nous énonçons un lemme très général garantissant, aux points stationnaires partiellement elliptiques de familles de champs de vecteurs ou de transformations, la naissance de variétés compactes invariantes normalement hyperboliques. Quelques exemples suivent.

We state a very general lemma ensuring, at partially elliptic rest points of families of vector fields or transformations, the birth of normally hyperbolic invariant compact manifolds. A few examples follow.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.09.016
Marc Chaperon 1

1 Institut de mathématiques de Jussieu & Université Paris 7, UFR de mathématiques, case 7012, 2, place Jussieu, 75251 Paris cedex 05, France 
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     title = {Birth control in generalized {Hopf} bifurcations},
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     language = {en},
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Marc Chaperon. Birth control in generalized Hopf bifurcations. Comptes Rendus. Mathématique, Volume 345 (2007) no. 8, pp. 453-458. doi : 10.1016/j.crma.2007.09.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.09.016/

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[2] M. Chaperon, S. López de Medrano, Generalized Hopf bifurcations, in press

[3] M. Chaperon; M. Kammerer-Colin de Verdière; S. López de Medrano More compact invariant manifolds appearing in the non-linear coupling of oscillators, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006), pp. 301-305

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[5] M.W. Hirsch Systems of differential equations that are competitive or cooperative. III: Competing species, Nonlinearity, Volume 1 (1988), pp. 51-71

[6] M.W. Hirsch; C.C. Pugh; M. Shub Invariant Manifolds, Lecture Notes in Mathematics, vol. 583, Springer-Verlag, 1977

[7] M. Kammerer-Colin de Verdière Stable products of spheres in the non-linear coupling of oscillators or quasi-periodic motions, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2004), pp. 625-629

[8] M. Kammerer-Colin de Verdière, Bifurcations de variétés invariante, Thèse, Université de Bourgogne, décembre 2006

[9] S. López de Medrano The space of Siegel leaves of a holomorphic vector field, Dynamical Systems, Lecture Notes in Mathematics, vol. 1345, Springer-Verlag, 1988, pp. 233-245

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