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The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit

Published online by Cambridge University Press:  19 September 2008

Ale Jan Homburg ,
Hiroshi Kokubu  and
Martin Krupa
Ale Jan Homburg
Affiliation:
Department of Mathematics, Rijksuniversiteit Groningen, PB 800, 9700 AV Groningen, The Netherlands
Hiroshi Kokubu
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606, Japan
Martin Krupa
Affiliation:
Department of Mathematics, Rijksuniversiteit Groningen, PB 800, 9700 AV Groningen, The Netherlands
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Abstract

Deng has demonstrated a mechanism through which a perturbation of a vector field having an inclination-flip homoclinic orbit would have a Smale horseshoe. In this article we prove that if the eigenvalues of the saddle to which the homoclinic orbit is asymptotic satisfy the condition 2λu > min{−λs, λuu} then there are arbitrarily small perturbations of the vector field which possess a Smale horseshoe. Moreover we analyze a sequence of bifurcations leading to the annihilation of the horseshoe. This sequence contains, in particular, the points of existence of n-homoclinic orbits with arbitrary n.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 4 , December 1994 , pp. 667 - 693
Copyright
Copyright © Cambridge University Press 1994

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