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.
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References
A. Andronov, E. Leontovich, et al. Theory of Bifurcations of Dynamical Systems on a Plane I.P.S.T., Jerusalem, 1971.
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.
R.I. Bogdanov. Versal deformation of a singularity of a vector field on the Plane in the Case of Zero Eigenvalues (R) Seminar Petrovski, 1976, (E) Selecta Mathematica Sovietica, Vol. 1, 4, 389–421, 1981.
M.H. Dulac. Sur les cycles limites. Bull. Soc. Math. France 51, 45–188, 1923.
F. Dumortier, R. Roussaire, J. Sotomayor. Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case. Erg. Theor. and Dyn. Sys. 7, 375–413, 1987.
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.
F. Takens. Forced oscillations and bifurcations. Applications of Global Analysis 1, Communications of Math. Inst. Rijksuniv. Utrecht, 3, 1974.
F. Takens. Unfoldings of Certain Singularities of Vector fields-Generalized Hopf Bifurcations. Journal of Diff. Equations 14, 476–493, 1973.
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© 1990 Springer-Verlag
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Dumortier, F., Roussarie, R. (1990). On the saddle loop bifurcation. In: Françoise, JP., Roussarie, R. (eds) Bifurcations of Planar Vector Fields. Lecture Notes in Mathematics, vol 1455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085390
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DOI: https://doi.org/10.1007/BFb0085390
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