Snaking bifurcations of localized patterns on ring lattices

Abstract

We study the structure of stationary patterns in bistable lattice dynamical systems posed on rings with a symmetric coupling structure in the regime of small coupling strength. We show that sparse coupling (for instance, nearest-neighbour or next-nearest-neighbour coupling) and all-to-all coupling lead to significantly different solution branches. In particular, sparse coupling leads to snaking branches with many saddle-node bifurcations, whilst all-to-all coupling leads to branches with six saddle nodes, regardless of the size of the number of nodes in the graph.

Snakes on Lieb lattice

“…The pinning region in the discrete case was first approximated analytically by Matthews and Susanto [48] and Dean et al [23]. Some of the present authors have also studied snaking in higher-dimensional discrete systems [40] where details of the bifurcation diagram are rather more involved (see also [9,65]). The complexity and width of the snaking diagrams depend on the number of "patch interfaces" admitted by the lattice patterns.…”