Complex localization mechanisms in networks of coupled oscillators: two case studies

TL;DR

This paper investigates complex localization phenomena in coupled oscillator networks, revealing new bifurcation routes and the influence of symmetries on localized states through detailed numerical analysis.

Contribution

It provides the first detailed numerical continuation and bifurcation analysis of localized time-periodic states in coupled oscillator systems, uncovering novel localization mechanisms.

Findings

Identification of bifurcations of heteroclinic cycles in Janus oscillators

Discovery of strange bifurcation diagrams resembling chaotic tangles

Highlighting the role of discrete symmetries in localization phenomena

Abstract

Localized phenomena abound in nature and throughout the physical sciences. Some universal mechanisms for localization have been characterized, such as in the snaking bifurcations of localized steady states in pattern-forming partial differential equations. While much of this understanding has been targeted at steady states, recent studies have noted complex dynamical localization phenomena in systems of coupled oscillators. These localized states can come in the form of symmetry-breaking chimera patterns that exhibit a coexistence of coherence and incoherence in symmetric networks of coupled oscillators and gap solitons emerging in the band gap of parametrically driven networks of oscillators. Here, we report detailed numerical continuations of localized time-periodic states in systems of coupled oscillators, while also documenting the numerous bifurcations they give way to. We find novel routes to localization involving bifurcations of heteroclinic cycles in networks of Janus oscillators and strange bifurcation diagrams resembling chaotic tangles in a parametrically driven array of coupled pendula. We highlight the important role of discrete symmetries and the symmetric branch points that emerge in symmetric models.