Spatially Localized Structures in Lattice Dynamical Systems

TL;DR

This paper studies localized stationary patterns in bistable lattice systems, revealing their bifurcation structures such as isolas and snaking curves, with theoretical proofs and numerical evidence.

Contribution

It provides the first rigorous bifurcation analysis of localized patterns in bistable lattice dynamical systems, especially near the anti-continuum limit.

Findings

Existence of isolas and snaking bifurcation curves for localized patterns.

Numerical evidence of snaking diagrams on square and hexagonal lattices.

Bifurcation theory applied to prove pattern existence in a discrete Ginzburg--Landau model.

Abstract

We investigate stationary, spatially localized patterns in lattice dynamical systems that exhibit bistability. The profiles associated with these patterns have a long plateau where the pattern resembles one of the bistable states, while the profile is close to the second bistable state outside this plateau. We show that the existence branches of such patterns generically form either an infinite stack of closed loops (isolas) or intertwined s-shaped curves (snaking). We then use bifurcation theory near the anti-continuum limit, where the coupling between edges in the lattice vanishes, to prove existence of isolas and snaking in a bistable discrete real Ginzburg--Landau equation. We also provide numerical evidence for the existence of snaking diagrams for planar localized patches on square and hexagonal lattices and outline a strategy to analyse them rigorously.