Localized patterns in planar bistable weakly coupled lattice systems

TL;DR

This paper analyzes localized patterns in two-dimensional bistable lattice systems, explaining the snaking bifurcation structure near the anti-continuum limit and examining stability and bifurcations through analytical and numerical methods.

Contribution

It provides the first analytical explanation of snaking in planar bistable lattices near the anti-continuum limit using Lyapunov--Schmidt reduction.

Findings

Snaking curves are characterized near the anti-continuum limit.

Stability of localized patterns is established.

Shape of snaking curves changes with spatial coupling strength.

Abstract

Localized planar patterns in spatially extended bistable systems are known to exist along intricate bifurcation diagrams, which are commonly referred to as snaking curves. Their analysis is challenging as techniques such as spatial dynamics that have been used to explain snaking in one space dimension no longer work in the planar case. Here, we consider bistable systems posed on square lattices and provide an analytical explanation of snaking near the anti-continuum limit using Lyapunov--Schmidt reduction. We also establish stability results for localized patterns, discuss bifurcations to asymmetric states, and provide further numerical evidence that the shape of snaking curves changes drastically as the coefficient that reflects the strength of the spatial coupling crosses a finite threshold.