The symmetry of the Kuramoto system and the essence of the cluster synchronization

TL;DR

This paper introduces a symmetry transformation approach to explain cluster synchronization in the higher harmonic Kuramoto system, providing a mathematical foundation and numerical validation for understanding CS.

Contribution

It presents a novel symmetry-based method that offers a clear mathematical and physical explanation of cluster synchronization in the Kuramoto model.

Findings

Numerical results confirm the theoretical predictions of the symmetry approach.

The method clarifies the physical meaning of cluster synchronization.

The approach enhances understanding of CS in various systems.

Abstract

The cluster synchronization (CS) is a very important characteristic for the higher harmonic cou- pling Kuramoto system. A novel method from the symmetry transformation is provided, and it gives CS a profoundly mathematical explanation and clear physical annotation. Detailed numerical studies for the order parameters in various conditions confirm the theoretical predictions from this new view of the symmetry transformation. The work is very beneficial to the further study on CS in various systems.