Homoclinic Snaking near a Heteroclinic Cycle in Reversible Systems

Knobloch, J. and Wagenknecht, T. (2005) Homoclinic Snaking near a Heteroclinic Cycle in Reversible Systems. Physica D, 206 (1-2). pp. 82-93. ISSN 0167-2789

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Abstract

Snaking curves of homoclinic orbits have been found numerically in a number of ODE models from water wave theory and structural mechanics. Along such a curve infinitely many fold bifurcation of homoclinic orbits occur. Thereby the corresponding solutions spread out and develop more and more bumps (oscillations) about their own centre. A common feature of the examples is that the systems under consideration are reversible. In this paper it is shown that such a homoclinic snaking can be caused by a heteroclinic cycle between two equilibria, one of which is a bi-focus. Using Lin’s method a snaking of 1-homoclinic orbits is proved to occur in an unfolding of such a cycle. Further dynamical consequences are discussed. As an application a system of Boussinesq equations is considered, where numerically a homoclinic snaking curve is detected and it is shown that the homoclinic orbits accumulate along a heteroclinic cycle between a real saddle and a bi-focus equilibrium.

Item Type:	Article
Uncontrolled Keywords:	Bifurcation; Heteroclinic cycle; Homoclinic snaking; Lin’s method; Boussinesq system
Subjects:	PACS 2010, the AIP's Physics and Astronomy Classification Scheme > 00 GENERAL PHYSICS > 02 Mathematical methods in physics PACS 2010, the AIP's Physics and Astronomy Classification Scheme > 00 GENERAL PHYSICS > 05 Statistical physics, thermodynamics, and nonlinear dynamical systems
Depositing User:	Thomas Wagenknecht
Date Deposited:	07 Dec 2006
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/663

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