Spatially Localized Structures in Lattice Dynamical Systems

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.