Topological synchronization of coupled nonlinear oscillators

TL;DR

This paper introduces the concept of topological synchronization in coupled nonlinear oscillators, where edge oscillators synchronize while bulk oscillators remain chaotic, leveraging topological properties for robust control.

Contribution

It proposes a novel nonlinear topological phenomenon, demonstrating edge-only synchronization and the emergence of unconventional boundary modes in topological nonlinear systems.

Findings

Edge oscillators synchronize while bulk oscillators are chaotic

Presence of positive Lyapunov exponents localized at edges

Emergence of unconventional topological boundary modes

Abstract

Synchronization of coupled oscillators is a ubiquitous phenomenon found throughout nature. Its robust realization is crucial to our understanding of various nonlinear systems, ranging from biological functions to electrical engineering. On another front, in condensed matter physics, topology is utilized to realize robust properties like topological edge modes, as demonstrated by celebrated topological insulators. Here, we integrate these two research avenues and propose a nonlinear topological phenomenon, namely topological synchronization, where only the edge oscillators synchronize while the bulk ones exhibit chaotic dynamics. We analyze concrete prototypical models to demonstrate the presence of positive Lyapunov exponents and Lyapunov vectors localized along the edge. As a unique characteristic of topology in nonlinear systems, we find that unconventional extra topological boundary modes appear at emerging effective boundaries. Furthermore, our proposal shows promise for spatially controlling synchronization, such as on-demand pattern designing and defect detection. The topological synchronization can ubiquitously appear in topological nonlinear oscillators and thus can provide a guiding principle to realize synchronization in a robust, geometrical, and flexible way.