Center for Control, Dynamical Systems, and Computation, University of California, Santa Barbara, CA 93106, USA.
Proc Natl Acad Sci U S A. 2013 Feb 5;110(6):2005-10. doi: 10.1073/pnas.1212134110. Epub 2013 Jan 14.
The emergence of synchronization in a network of coupled oscillators is a fascinating topic in various scientific disciplines. A widely adopted model of a coupled oscillator network is characterized by a population of heterogeneous phase oscillators, a graph describing the interaction among them, and diffusive and sinusoidal coupling. It is known that a strongly coupled and sufficiently homogeneous network synchronizes, but the exact threshold from incoherence to synchrony is unknown. Here, we present a unique, concise, and closed-form condition for synchronization of the fully nonlinear, nonequilibrium, and dynamic network. Our synchronization condition can be stated elegantly in terms of the network topology and parameters or equivalently in terms of an intuitive, linear, and static auxiliary system. Our results significantly improve upon the existing conditions advocated thus far, they are provably exact for various interesting network topologies and parameters; they are statistically correct for almost all networks; and they can be applied equally to synchronization phenomena arising in physics and biology as well as in engineered oscillator networks, such as electrical power networks. We illustrate the validity, the accuracy, and the practical applicability of our results in complex network scenarios and in smart grid applications.
在耦合振荡器网络中出现同步是各个科学领域中一个引人入胜的话题。耦合振荡器网络的一个广泛采用的模型的特点是具有异质相振荡器的群体、描述它们之间相互作用的图以及扩散和正弦耦合。已知强耦合且足够均匀的网络会同步，但从非同步到同步的确切阈值是未知的。在这里，我们提出了一种独特、简洁且闭式的全非线性、非平衡和动态网络同步条件。我们的同步条件可以用网络拓扑和参数来优雅地表示，或者等效地用直观、线性和静态辅助系统来表示。我们的结果大大改进了迄今为止提出的条件，它们对于各种有趣的网络拓扑和参数是精确的；对于几乎所有网络，它们在统计学上都是正确的；并且它们可以同样应用于物理学和生物学中以及工程振荡器网络中出现的同步现象，例如电力网络。我们在复杂网络场景和智能电网应用中说明了我们的结果的有效性、准确性和实际适用性。
