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

  • Alan R. Champneys (a1), Jörg Härterich (a2) and Björn Sandstede (a3)

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.

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

  • Alan R. Champneys (a1), Jörg Härterich (a2) and Björn Sandstede (a3)

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