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

Zhi-Ming Gu·Ming Zhao·Tao Zhou·Chen-Ping Zhu·Bing-Hong Wang
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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.

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Zhi-Ming Gu, Ming Zhao, Tao Zhou, Chen-Ping Zhu, Bing-Hong Wang. 2006-07-05. Phase Synchronization of non-Abelian Oscillators on Small-World Networks. https://doi.org/10.1016/j.physleta.2006.10.010

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