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  • Ergodic Theory and Dynamical Systems, Volume 14, Issue 4
  • December 1994, pp. 667-693

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

  • Ale Jan Homburg (a1), Hiroshi Kokubu (a2) and Martin Krupa (a1)
  • DOI:
  • Published online: 01 September 2008

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.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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