Topological synchronization of coupled nonlinear oscillators

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.

Automated summary

This article combines the synchronization phenomenon of coupled oscillators with the field of condensed matter physics, proposing a nonlinear topological phenomenon called topological synchronization. In this phenomenon, only the edge oscillators synchronize while the bulk oscillators exhibit chaotic dynamics. The article provides analysis and models to demonstrate the existence of this phenomenon and discovers the emergence of unconventional extra topological boundary modes at effective boundaries. Furthermore, the proposal shows promise for spatially controlling synchronization, such as on-demand pattern designing and defect detection.