Homoclinic snaking near a heteroclinic cycle in reversible systems

PHYSICA D-NONLINEAR PHENOMENA

卷:206

Article

Knobloch, J ; Wagenknecht, T

关键词: bifurcation;    heteroclinic cycle;    homoclinic snaking;    Lin's method;    Boussinesq system;   

DOI  :  10.1016/j.physd.2005.04.018

来源: Elsevier

【 摘 要 】

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. (c) 2005 Elsevier B.V. All rights reserved.

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