Localized synchronous patterns in weakly coupled bistable oscillators

TL;DR

This paper investigates localized synchronous patterns in weakly coupled bistable oscillators, revealing their existence on specific solution branches depending on coupling strength, with implications for understanding complex oscillatory systems.

Contribution

It introduces the existence of localized time-periodic solutions in coupled oscillator chains with Ginzburg--Landau equations, highlighting their branch structures under weak coupling.

Findings

Localized synchrony patterns exist in weakly coupled oscillators.

Patterns are on isola or snaking branches depending on coupling.

Results extend understanding of localized solutions in nonlinear oscillator systems.

Abstract

Motivated by numerical continuation studies of coupled mechanical oscillators, we investigate branches of localized time-periodic solutions of one-dimensional chains of coupled oscillators. We focus on Ginzburg--Landau equations with nonlinearities of Lambda-Omega type and establish the existence of localized synchrony patterns in the case of weak coupling and weak-amplitude dependence of the oscillator periods. Depending on the coupling, localized synchrony patterns lie on a discrete stack of isola branches or on a single connected snaking branch.