Ergodic Theory and Dynamical Systems

Research Article

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

Ale Jan Homburga1, Hiroshi Kokubua2 and Martin Krupaa1

a1 Department of Mathematics, Rijksuniversiteit Groningen, PB 800, 9700 AV Groningen, The Netherlands

a2 Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606, Japan

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.

(Received April 16 1993)