2006.413: Homoclinic Snaking near a Heteroclinic Cycle in Reversible Systems

2006.413: J. Knobloch and T. Wagenknecht (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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DOI: 10.1016/j.physd.2005.04.018

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.

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