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Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms

Published online by Cambridge University Press:  14 October 2010

Henk Broer ,
Robert Roussarie  and
Carles Simó
Henk Broer
Affiliation:
Afdeling Wiskunde en Informatica, Rijksuniversiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands
Robert Roussarie
Affiliation:
Laboratoire de Topologie, CNRS, UA755, Université de Bourgogne, BP 138, 21004 Dijon, France
Carles Simó
Affiliation:
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
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Abstract

We study a generic, real analytic unfolding of a planar diffeomorphism having a fixed point with unipotent linear part. In the analogue for vector fields an open parameter domain is known to exist, with a unique limit cycle. This domain is bounded by curves corresponding to a Hopf bifurcation and to a homoclinic connection. In the present case of analytic diffeomorphisms, a similar domain is shown to exist, with a normally hyperbolic invariant circle. It follows that all the ‘interesting’ dynamics, concerning the destruction of the invariant circle and the transition to trivial dynamics by the creation and death of homoclinic points, takes place in an exponentially small part of the parameter-plane. Partial results were stated in [5]. Related numerical results appeared in [16].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 6 , December 1996 , pp. 1147 - 1172
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[1] Arnol'd, V. I.. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, 1983.CrossRefGoogle Scholar
[2] Baesens, C. and Nicolis, G.. Complex bifurcation phenomena in a periodically forced normal form. Z. Phys. 52B (1983), 345–354.Google Scholar
[3] Baesens, C.. Complex bifurcation phenomena in a periodically forced normal form. Led. Notes Phys. 179 (1983), 223–224.CrossRefGoogle Scholar
[4] Bogdanov, R. I.. Versal deformation of a singularity of a vector field in the plane in case of zero eigenvalues (Seminar Petrovski, 1976). Selecta Matheinatica Sovietica. 1, 4 (1981), 389–421. (In Russian.)Google Scholar
[5] Broer, H. W., Roussarie, R. and Simó, C.. On the Bogdanov-Takens bifurcation for planar diffeomorphisms. International Conference on Differential Equations (Equadiff 91, Barcelona). Eds Perelló, C.et al. World Scientific, 1993, pp. 81–92.Google Scholar
[6] Broer, H. W., and Tokens, F.. Formally symmetric normal forms and genericity. Dynamics Reported 2 (1989), 39–59.CrossRefGoogle Scholar
[7] Broer, H. W. and Tangerman, F. M.. From a differentiable to a real analytic perturbation theory. Applications to the Kupka-Smale Theorem. Ergod. Th. & Dynam. Sys. 6 (1986), 345–362.CrossRefGoogle Scholar
[8] Broer, H. W. and Vegter, G.. Bifurcational aspects of parametric resonance. Dynamics Reported (New Series) 1 (1992), 1–51.CrossRefGoogle Scholar
[9] Dumortier, F., Roussarie, R. and Sotomayor, J.. Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case. Ergod. Th. & Dynam. Sys. 7 (1987), 375–413.CrossRefGoogle Scholar
[10] Fontich, E. and Simó, C.. The splitting of separatrices for analytic diffeomorphisms. Ergod. Th. & Dynam. Sys. 10 (1990), 295–318.CrossRefGoogle Scholar
[11] Fontich, E. and Simó, C.. Invariant manifolds for near the identity differentiate maps and splitting of separatrices. Ergod. Th. & Dynam. Sys. 10 (1990), 319–346.CrossRefGoogle Scholar
[12] Gelfreich, V. G.. Separatrices splitting for polynomial area-preserving maps. Topics in Mathematical Physics, vol. 13. Ed. Birman, M. Sh.. Leningrad State University, 1991, pp. 108–116 (in Russian).Google Scholar
[13] Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics 583). Springer, 1977.CrossRefGoogle Scholar
[14] Neishtadt, A. I.. The separation of motions in systems with rapidly rotating phase. J. Appl. Math. Mech. 48 (1984), 133–139.CrossRefGoogle Scholar
[15] Roussarie, R.. On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol. Soc. Bras. Mat. 17 (1986), 67–101.CrossRefGoogle Scholar
[16] Simó, C.Broer, H. W. and Roussarie, R.. A numerical survey on the Takens-Bogdanov bifurcation for diffeomorphisms. European Conference on Iteration Theory (EOT 89). Eds. Mira, C.et al. World Scientific, 1991, pp. 320–334.Google Scholar
[17] Simó, C.. Averaging under fast quasiperiodic motion. Integrable and Chaotic Behaviour in Hamiltonian Systems (Torun, Poland, 1993). Ed. Seimenis, I.. Plenum, 1994, pp. 13–34.Google Scholar
[18] Takens, F.. Forced oscillations and bifurcations. Applications of Global Analysis I, Comm. Math. Inst. Univ. of Utrecht. 3 (1974), 1–59.Google Scholar
[19] Takens, F.. Singularities of vector fields, Publ. Math. I.H.E.S. 43 (1974), 48–100.CrossRefGoogle Scholar