Phase Synchronization of non-Abelian Oscillators on Small-World Networks

TL;DR

This paper extends phase synchronization concepts to non-Abelian oscillators on small-world networks, analyzing various Lie group systems and revealing that network disorder enhances synchronizability, with high-dimensional systems showing unexpected synchronization in regular networks.

Contribution

It introduces a generalized phase synchronization framework for non-Abelian oscillators on complex networks, including high-dimensional Lie group systems, and uncovers novel synchronization phenomena.

Findings

Greater network disorder improves synchronizability.

Small-world networks enable global synchronization at sufficient coupling.

High-dimensional systems can synchronize in regular networks, unlike low-dimensional systems.

Abstract

In this paper, by extending the concept of Kuramoto oscillator to the left-invariant flow on general Lie group, we investigate the generalized phase synchronization on networks. The analyses and simulations of some typical dynamical systems on Watts-Strogatz networks are given, including the $n$-dimensional torus, the identity component of 3-dimensional general linear group, the special unitary group, and the special orthogonal group. In all cases, the greater disorder of networks will predict better synchronizability, and the small-world effect ensures the global synchronization for sufficiently large coupling strength. The collective synchronized behaviors of many dynamical systems, such as the integrable systems, the two-state quantum systems and the top systems, can be described by the present phase synchronization frame. In addition, it is intuitive that the low-dimensional systems are more easily to synchronize, however, to our surprise, we found that the high-dimensional systems display obviously synchronized behaviors in regular networks, while these phenomena can not be observed in low-dimensional systems.