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Strange attractor in the unfolding of an inclination-flip homoclinic orbit

Published online by Cambridge University Press:  14 October 2010

Vincent Naudot
Vincent Naudot
Affiliation:
Université de Bourgogne, Département de mathématiques, Laboratoire de topologie, U.R.A 755 21004 Dijon Cedex, France
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Abstract

We study the unfolding of a smooth vector-field X on ℝ3 having a homoclinic orbit to a hyperbolic equilibrium point with three real eigenvalues satisfying − λss < λs < 0 < λu We say that Γ is an inclination-flip homoclinic orbit if the extended unstable manifold at the equilibrium point is, along Γ, non-transverse to the stable manifold and that Γ is of weak type if the unstable manifold has a non-trivial intersection with a special C2 weak stable manifold of dimension one. In this paper, we show the existence of a strange attractor in the unfolding of an inclination-flip homoclinic orbit (of weak type) in the case where the divergence at the equilibrium point is negative. The crucial idea is to compare the Poincaré return map with the Hénon family: being close to 0.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 5 , October 1996 , pp. 1071 - 1086
Copyright
Copyright © Cambridge University Press 1996

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