On the saddle loop bifurcation

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


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.


Vector Field Saddle Point Bifurcation Diagram Unstable Manifold Stable Manifold 
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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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