On the saddle loop bifurcation
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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.
KeywordsVector Field Saddle Point Bifurcation Diagram Unstable Manifold Stable Manifold
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- [A.L]A. Andronov, E. Leontovich, et al. Theory of Bifurcations of Dynamical Systems on a Plane I.P.S.T., Jerusalem, 1971.Google Scholar
- [A.A.D.]H. Annabi, M.L. Annabi, F. Dumortier. Continuous dependence on parameters in the Bogdanov-Takens bifurcation. To appear in the proceedings of the workshop on Chaotic Dynamics and Bifurcations, Longman Research Notes.Google Scholar
- [M.P.]I.P. Malta, J. Palis Families of vector fields with finite modulus of stability. Lecture Notes in Mathematics 898, Dyn. Systems and Turbulence, Warwick 1980, Springer-Verlag, 212–229, 1981.Google Scholar
- [T1]F. Takens. Forced oscillations and bifurcations. Applications of Global Analysis 1, Communications of Math. Inst. Rijksuniv. Utrecht, 3, 1974.Google Scholar