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A non-transverse homoclinic orbit to a saddle-node equilibrium

Published online by Cambridge University Press:  19 September 2008

Alan R. Champneys,
Jörg Härterich and
Björn Sandstede
Alan R. Champneys
Affiliation:
Department of Engineering Mathematics, University of Bristol, Queen's Building, University Walk, Bristol BS8 1TR, UK
Jörg Härterich
Affiliation:
Institut für Mathematik I, Freie Universität Berlin, Arnimallee 2–6, 14195 Berlin, Germany
Björn Sandstede
Affiliation:
Weierstraβ-Institut für Angewandte Analysis und Stochastik, Mohrenstraβe 39, 10117 Berlin, Germany
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Abstract

A homoclinic orbit is considered for which the center-stable and center-unstable manifolds of a saddle-node equilibrium have a quadratic tangency. This bifurcation is of codimension two and leads generically to the creation of a bifurcation curve defining two independent transverse homoclinic orbits to a saddle-node. This latter case was shown by Shilnikov to imply shift dynamics. It is proved here that in a large open parameter region of the codimension-two singularity, the dynamics are completely described by a perturbation of the Hénon-map giving strange attractors, Newhouse sinks and the creation of the shift dynamics. In addition, an example system admitting this bifurcation is constructed and numerical computations are performed on it.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 3 , June 1996, pp. 431 - 450
Copyright
Copyright © Cambridge University Press 1996

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References

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