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

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.

5 pages, and 3 figures

References in corpus (4)

• Synchronization on small-world networks
• Ising model in small-world networks
• Comment on "Ising model on a small world network"
• Scaling in directed dynamical small-world networks with random responses

Cited by in corpus (5)

• Continuous versus Discontinuous Transitions in the $D$-Dimensional Generalized Kuramoto Model: Odd $D$ is Different
• Solvable model of the collective motion of heterogeneous particles interacting on a sphere
• Synchrony for weak coupling in the complexified Kuramoto model
• On synchronization in Kuramoto models on spheres
• Ott-Antonsen reduction for the non-Abelian Kuramoto model on the 3-sphere