Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included. CONTENTS

A model for synchronization of globally coupled phase oscillators including "inertial" effects is analyzed. In such a model, both oscillator frequencies and phases evolve in time. Stationary solutions include incoherent (unsynchronized) and synchronized states of the oscillator population. Assuming a Lorentzian distribution of oscillator natural frequencies, g(Ω), both larger inertia or larger frequency spread stabilize the incoherent solution, thereby making harder to synchronize the population. In the limiting case g(Ω) = δ(Ω), the critical coupling becomes independent of inertia. A richer phenomenology is found for bimodal distributions. For instance, inertial effects may destabilize incoherence, giving rise to bifurcating synchronized standing wave states. Inertia tends to harden the bifurcation from incoherence to synchronized states: at zero inertia, this bifurcation is supercritical (soft), but it tends to become subcritical (hard) as inertia increases. Nonlinear stability is investigated in the limit of high natural frequencies.

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We study the noisy FitzHugh-Nagumo model, representative of the dynamics of excitable neural elements, and derive a Fokker-Planck equation for both a single element and for a network of globally coupled elements. We introduce an efficient way to numerically solve this Fokker-Planck equation, especially for large noise levels. We show that, contrary to the single element, the network can undergo a Hopf bifurcation as the coupling strength is increased. Furthermore, we show that an external sinusoidal driving force leads to a classical resonance when its frequency matches the underlying system frequency. This resonance is also investigated analytically by exploiting the different time scales in the problem.

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It is well known that overdamped unforced dynamical systems do not oscillate. However, well-designed coupling schemes, together with the appropriate choice of initial conditions, can induce oscillations when a control parameter exceeds a threshold value. In a recent publication [Phys. Rev. E 68, 045102 (2003)]], we demonstrated this behavior in a specific prototype system, a soft-potential mean-field description of the dynamics in a hysteretic "single-domain" ferromagnetic sample. The previous analysis of this work showed that N (odd) unidirectionally coupled elements with cyclic boundary conditions would, in fact, oscillate when a control parameter-in this case the coupling strength-exceeded a critical value. These oscillations are now finding utility in the detection of very weak "target" signals, via their effect on the oscillation characteristics, e.g., the frequency and asymmetry of the oscillation wave forms. In this paper we explore the underlying dynamics of this system. Scaling laws that govern the oscillation frequency in the vicinity of the critical point, as well as the zero-crossing intervals in the presence of a symmetry-breaking target dc signal, are derived; these quantities are germane to signal detection and analysis.