Homoclinic snaking near a heteroclinic cycle in reversible systems

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.