Topological Symmetry And Existence of Partial Synchronization

TL;DR

This paper investigates the conditions under which partial synchronization occurs in dynamical systems, revealing that symmetry is sufficient but not necessary, and that equal-degree random structures can support PaS states.

Contribution

It establishes the necessary and sufficient conditions for partial synchronization, clarifying the role of coupling structure symmetry and introducing equal-degree random structures as viable configurations.

Findings

Symmetry of coupling structure is sufficient but not necessary for PaS.

Equal-degree random structures can support partial synchronization.

Provides an exact proof of the conditions for PaS existence.

Abstract

We study the relationship between the partial synchronous (PaS) state and the coupling structure in general dynamical systems. By the exact proof, we find the sufficient and necessary condition of the existence of PaS state for the coupling structure. Our result shows that the symmetry of the coupling structure is not the equivalent condition which is supposed before but only the sufficient condition. Furthermore, for the existence of the PaS state, the general structure is the equal-degree random.