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.

Coexistence of stable dark- and bright-soliton Kerr combs in normal-dispersion resonators

“…These dark solitons are connected by unstable solution branches that serve to add additional spatial oscillations in their profiles, leading to the broadening of the dark states. This type of bifurcation structure is called collapsed snaking [16,17,41,42], which is significantly different from the homoclinic snaking appearing for dissipative solitons associated with subcritical patterns. Such homoclinic snaking, where many solutions coexist over a fixed parameter range around the Maxwell point, is probably better known and has been widely studied in physics [43,44] and optics [13,[45][46][47].…”

Nonsnaking doubly diffusive convectons and the twist instability

“…States of this type expand abruptly in the extended direction near a special point in parameter space corresponding to the formation of a pair of heteroclinic connections between two different fixed points in a spatial dynamics view of the system. 26 This point, variously referred to as the nonsnaking 35 or protosnaking 36 point plays the role of a Maxwell point in systems, like the Swift-Hohenberg equation (1), with gradient dynamics. If the spatial eigenvalues of one of the fixed points are complex the resulting behavior may be termed collapsed snaking.…”

Snaking of Multiple Homoclinic Orbits in Reversible Systems

“…In the analysis in Section 4, we will particularly focus on the case where ω is the first passage past p 2 , and we will discuss other possibilities only briefly. This paper can be seen as a follow-up to [10], where we discussed one-homoclinic orbits near an EE cycle. There it has also be shown that p 2 has to be of saddle-focus type, in order to find snaking behaviour of one-homoclinic orbits.…”

“…This feature allows one to study the scenario using a local bifurcation analysis. In an earlier paper [10] it has been shown rigorously by the authors that heteroclinic cycles between symmetric equilibria of saddle focus type generate a snaking behaviour. In contrast to that behaviour the snaking related to an EP cycle is generically a global phenomenon.…”

“…Similar to the procedure in [10] we will use Lin's method [8,13] to derive bifurcation equations for N -homoclinic orbits near the cycle. The general setup will be introduced in Section 3.…”