Synchronization in complex networks of phase oscillators: A survey

Abstract

The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in real-world synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology. We review the history and the countless applications of this model throughout science and engineering. We justify the importance of the widespread coupled oscillator model as a locally canonical model and describe some selected applications relevant to control scientists, including vehicle coordination, electric power networks, and clock synchronization. We introduce the reader to several synchronization notions and performance estimates. We propose analysis approaches to phase and frequency synchronization, phase balancing, pattern formation, and partial synchronization. We present the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. We conclude by summarizing the limitations of existing analysis methods and by highlighting some directions for future research.

Transition to synchrony in degree-frequency correlated Sakaguchi-Kuramoto model

“…mainly to investigate rhythmicity and synchronization (see the review [10,14] and references therein for details). On the other hand, theoretical investigations of * Electronic address: pinaki.pal@maths.nitdgp.ac.in the Kuramoto model and its generalizations largely focus on the transition to synchronization in coupled networks [10].…”

“…Examples include flashing of fireflies [5], applauding persons [6], systems describing circadian rhythms in animals [7], Josephson junction arrays [8], moving pedestrians on footbridges [9] and many more [10]. Mathematical modeling of these complex systems often involves the important steps of capturing dynamics of the interacting units by nonlinear oscillators and interaction by suitable coupling functions [11].…”

Event-based exponential synchronization of complex networks

“…The previous most common models are investigated Dörfler and Bullo 2014;Yu et al 2009aYu et al , 2013aYu et al , 2009bZhang et al 2013d;, however, their insurmountable deficiency is very obvious: each node's controller implementation must realize its all neighbors, current states at every time instant, that means continuous information communication between autonomous nodes is needed, which promotes nodes have to be equipped with high performance processors and communication facilities. Therefore, these traditional synchronization strategies undoubtedly will consume much more energy and be largely restricted in practical applications with limited computing resources and network bandwidth.…”

“…It is noteworthy that the communications between nodes in Zhang et al (c, d 2015a, b), Reyes and Tanner (2015), Olfati-Saber and Murray (2004), Das and Ghose (2015), An et al (2014), Dörfler and Bullo (2014), Yu et al (2009aYu et al ( , b, 2013a; and are continuous. The continuous communications have an obvious insurmountable deficiency: the designed control law requires real-time updates, which promotes network nodes to be equipped with high performance processors and high-speed communication channels.…”

“…A complex network is a large set of interconnected nodes, where the nodes and connections can be anything, examples are internet, transportation networks, coupled biological and chemical engineering systems, neural networks in human brains and so on. The synchronization is one of the most important dynamical properties of complex networks (Zhang et al 2011(Zhang et al , 2013a(Zhang et al , b, c, d, 2014a(Zhang et al , 2015aBalasubramaniam and Jarina 2014;Alofi et al 2015;Reyes and Tanner 2015;Das and Ghose 2015;Olfati-Saber and Murray 2004;An et al 2014;Dörfler and Bullo 2014;Yu et al 2009aYu et al , b, 2013a. In essence, synchronization is a form of self-organization.…”