Synchronization in complex networks of phase oscillators: A survey <s> Synchronization in Infinite-Dimensional Networks <s> Improved mobility control in secondary and tertiary recovery processes is obtained by injecting at least two slugs into the reservoir, one slug designed to effect a permeability reduction and the other slug containing an agent which more effectively increases the fluid viscosity. For example, an aqueous slug containing a partially hydrolyzed, high molecular weight polyacrylamide can be injected (this slug efficiently reduces the relative permeability to the flow of the aqueous slug through the reservoir), followed by an aqueous slug containing a biopolymer, e.g., a linear polysaccharide having a molecular weight of about one million (this agent more effectively increases the viscosity of the water and has a lesser effect on permeability reduction). This combination of slugs gives more efficient and economical mobility control in flooding subterranean oil-bearing formations to recover oil therefrom. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Synchronization in Infinite-Dimensional Networks <s> The Kuramoto model describes a large population of coupled limit-cycle oscillators whose natural frequencies are drawn from some prescribed distribution. If the coupling strength exceeds a certain threshold, the system exhibits a phase transition: some of the oscillators spontaneously synchronize, while others remain incoherent. The mathematical analysis of this bifurcation has proved both problematic and fascinating. We review 25 years of research on the Kuramoto model, highlighting the false turns as well as the successes, but mainly following the trail leading from Kuramoto’s work to Crawford’s recent contributions. It is a lovely winding road, with excursions through mathematical biology, statistical physics, kinetic theory, bifurcation theory, and plasma physics. © 2000 Elsevier Science B.V. All rights reserved. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Synchronization in Infinite-Dimensional Networks <s> The purpose of this paper is to understand phase transition in noncooperative dynamic games with a large number of agents. Applications are found in neuroscience, biology, economics, as well as traditional engineering applications. The focus of analysis is a variation of the large population LQG model of Huang et. al. 2007 [6], comprised here of a controlled nonlinear N-dimensional stochastic differential equation model, coupled only through a nonlinear cost function. The states are interpreted as the phase angle for a collection of non-homogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. A deterministic PDE model is proposed, which is shown to approximate the stochastic system as the population size approaches infinity. Key to the analysis of the PDE model is the existence of a particular Nash equilibrium in which the agents ‘opt out’ of the game, setting their controls to zero, resulting in the ‘incoherence’ equilibrium. Methods from dynamical systems theory are used in a bifurcation analysis, based on a linearization of the PDE model about the incoherence equilibrium. A critical value of the control cost parameter is identified: Above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Synchronization in Infinite-Dimensional Networks <s> The aim of this paper is to describe a coupled oscillator model for Bayesian inference. The coupled oscillator model comprises of a large number of oscillators with mean-field coupling. The collective dynamics of the oscillators are used to solve an inference problem: the empirical distribution of the population encodes a ‘belief state' (posterior distribution) that is continuously updated based on noisy measurements. In effect, the coupled oscillator model works as a particle filter. The framework is described here with the aid of a model problem involving estimation of a walking gait cycle. For this problem, the coupled oscillator particle filter is developed, and demonstrated on experimental data taken from an Ankle-foot Orthosis (AFO) device. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Synchronization in Infinite-Dimensional Networks <s> A novel and simple network model that is capable to reproduce quasi-stationary behavior and propagation phenomena in electric power grid dynamics is introduced. A new Kuramoto approximation to distributed generator dynamics is obtained from combining a continuous spatial interaction function with a discrete lattice model representing generator positions and network structure over a continuous spatial domain. At hand of model properties and a numerical study quasi-stationarity of electric power grid dynamics results, being related to non-vanishing fluctuation and oscillations in stationary state. <s> BIB005
For a complete coupling graph with uniform weights a ij = K/n, where K > 0, the coupled oscillator model (1) reduces to the celebrated Kuramoto model given in (3). By means of the order parameter re iψ defined in equation (16), the Kuramoto model (3) can be rewritten in the insightful forṁ Equation (17) gives the intuition that the oscillators synchronize because of their coupling to a mean field represented by the order parameter re iψ , which itself is a function of θ(t). Intuitively, for small coupling strength K each oscillator rotates with its distinct natural frequency ω i , whereas for large coupling strength K all angles θ i (t) will entrain to the mean field re iψ , and the oscillators synchronize. The transition from incoherence to synchronization occurs at a critical threshold value of the coupling strength, denoted by K critical . The analysis of this phase transition based on a meanfield and statistical mechanics viewpoint has been the subject of numerous investigations, starting with Kuramoto's own ingenious analysis in BIB001  . As neatly described by BIB002 , Kuramoto assumed the a priori existence of solutions to (17) which feature a stationary order parameter r(t)e iψ(t) = constant. Following this assumption and his intuition, Kuramoto derived a set of self-consistency equations. A rigorous mathematical underpinning to Kuramoto's mean-field approach can be established by a time-scale separation or in the continuum limit as the number of oscillators tends to infinity, and the natural frequencies ω are sampled from a distribution function g : R → R ≥0 . The continuum-limit model has enjoyed a considerable amount of attention by the physics and dynamics communities. Related controltheoretical applications of the continuum-limit model are estimation of gait cycles BIB004 , spatial power grid modeling and analysis BIB005 , and game theoretic approaches BIB003 .