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On the saddle loop bifurcation

Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1455)

Abstract

It is shown that the set of C (generic) saddle loop bifurcations has a unique modulus of stability γ ≥]0, 1[∪]1, ∞[ for (C0, Cr)-equivalence, with 1≤r≤∞. We mean for an equivalence (x,μ) ↦ (h(x,μ), ϕ(μ)) with h continuous and ϕ of class Cr. The modulus γ is the ratio of hyperbolicity at the saddle point of the connection. Already asking ϕ to be a lipeomorphism forces two saddle loop bifurcations to have the same modulus, while two such bifurcations with the same modulus are (C0,±Identity)-equivalent.

A side result states that the Poincaré map of the connection is C1-conjugate to the mapping x↦xγ.

In the last part of the paper is shown how to finish the proof that the Bogdanov-Takens bifurcation has exactly two models for (C0,C)-equivalence.

Keywords

Vector Field Saddle Point Bifurcation Diagram Unstable Manifold Stable Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  1. 1.Limburgs Universitair CentrumDiepenbeekBelgium
  2. 2.Département de MathématiquesUniversité de Bourgogne, UFR de Sciences et Techniques, Laboratoire de TopologieDijon CedexFrance

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