Synchronization in Oscillatory Networks

Theformationofcollective behaviorinlargeensemblesornetworksofcoupled oscillatory elements is one of the oldest problem in the study of dyna- cal systems. Nevertheless, it is an actually challenging ?eld for a theoretical understanding as well as for applications in various disciplines, ranging from physics, chemistry, earth sciences via biology and neuroscience to engineering, business and social sciences. Due to the large number of e?ective degrees of freedominspatiallyextendedsystems,arichvarietyofspatiotemporalregimes is observed. Three main types of collective behavior are distinguished (1) a fully incoherent state or highly developed spatiotemporal disorder; (2) p- tially coherent states, where some of the participants in the network behave in some common rhythm, forming clusters; (3) a fully coherent state or a regime of globally synchronized elements. The basic phenomenon of these structure formations is synchronization, i. e. regime of coherent activity, which is u- versal in many dynamical systems and can be understood from the analysis of common models of oscillatory networks. Cooperative phenomena in ensembles of globally (mean-?eld) coupled phase equations were studied ?rst by Winfree and Kuramoto. They showed that if the coupling is strong enough mutual synchronization emerges. In contrast to the mean-?eld Winfree and Kuramoto models, Ermentrout and Kopell’s classic works deal with chains of phase equations. Besides cluster and global synchronization e?ects, the main results there involve travelling waves. A main part of this book presents di?erent aspects of synchronization in chains and lattices of locally interconnected nonidentical nonlinear oscil- tory elements.