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.

arXiv commentThis paper was published in the Journal of Nonlinear Science volume 30 in 2020. We wanted to make a preprint available as well

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Cited by in corpus (10)

• Existence and spectral stability of multi-pulses in discrete Hamiltonian lattice systems
• Saddle Transport and Chaos in the Double Pendulum
• Localized patterns in planar bistable weakly coupled lattice systems
• Pattern Formation in Random Networks Using Graphons
• Isolas of multi-pulse solutions to lattice dynamical systems
• Snaking bifurcations of localized patterns on ring lattices
• Complex localization mechanisms in networks of coupled oscillators: two case studies
• Approximate localised dihedral patterns near a Turing instability
• From Snaking to Isolas: A One-Active-Site Approximation in Discrete Optical Cavities
• Dynamics of two languages competing on a network: a case study