Snaking bifurcations of localized patterns on ring lattices

TL;DR

This paper investigates how different coupling structures in ring lattice systems influence the formation and bifurcation patterns of stationary localized solutions, revealing distinct snaking behaviors for sparse versus all-to-all coupling.

Contribution

It provides a detailed analysis of how coupling topology affects bifurcation structures in bistable ring lattices, highlighting differences between sparse and all-to-all coupling.

Findings

Sparse coupling results in snaking branches with many saddle-node bifurcations.

All-to-all coupling produces solution branches with exactly six saddle nodes.

Coupling structure significantly influences the bifurcation landscape of localized patterns.

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.