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  • Cited by 26
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    Terada, K. Yoshizawa, S. and Nishimura, C. 1999. IEEE SMC'99 Conference Proceedings. 1999 IEEE International Conference on Systems, Man, and Cybernetics (Cat. No.99CH37028). Vol. 2, Issue. , p. 371.

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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: http://dx.doi.org/10.1017/S0143385700008117
  • Published online: 01 September 2008
Abstract
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.

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[Deng93]B. Deng . Homoclinic twisting bifurcations and cusp horseshoe maps. J. Dyn. Diff. Eq. 5 (1993), 417467.

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[Yan87]E. Yanagida . Branching of double pulse solutions from single pulse solutions in nerve axon equations. J. Diff. Eq. 66 (1987), 243262.

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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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