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Homoclinic snaking near a heteroclinic cycle in reversible systems. (English) Zbl 1084.34049

Snaking curves of homoclinic orbits have been found numerically in a number of ODE models ranging from structural mechanics to water wave theory and nonlinear optics. Along such a curve infinitely many fold bifurcations of homoclinic orbits occur. Thereby, the corresponding solutions spread out and develop more and more bumps (oscillations) about their own center. A common feature of the examples is that the systems under consideration are reversible.

In the 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.

MSC:
34C37Homoclinic and heteroclinic solutions of ODE
34C23Bifurcation (ODE)
34E05Asymptotic expansions (ODE)