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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

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