Hidden symmetry and Collective behavior

TL;DR

This paper investigates the relationship between coupling structure and partial synchronization in dynamical systems, revealing that topological symmetry is sufficient but not necessary, and provides a method to identify all partial synchronous solutions.

Contribution

It challenges the common belief by showing symmetry is not necessary for partial synchronization and introduces a comprehensive method to find all such solutions.

Findings

Topological symmetry is sufficient but not necessary for partial synchronization.

A new method to identify all existing partially synchronous solutions.

Established necessary and sufficient conditions for partial synchronization.

Abstract

We study the relationship between the partially synchronous state and the coupling structure in general dynamical systems. Our results show that, on the contrary to the widely accepted concept, topological symmetry in a coupling structure is the sufficient condition but not the necessary condition. Furthermore, we find the necessary and sufficient condition for the existence of the partial synchronization and develop a method to obtain all of the existing partially synchronous solutions for all nonspecific dynamics from a very large number of possible candidates.