We provide a complete analysis of the Kuramoto model of coupled nonlinear oscillators with uncertain natural frequencies and arbitrary interconnection topology. Our work extends and supersedes existing, partial results for the case of an all-to-all connected network. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value all the oscillators synchronize, resulting in convergence of all phase di#erences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity.

We solve a longstanding stability problem for the Kuramoto model of coupled oscilla-tors. This system has attracted mathematical attention, in part because of its applications in fields ranging from neuroscience to condensed-matter physics, and also because it pro-vides a beautiful connection between nonlinear dynamics and statistical mechanics. The model consists of a large population of phase oscillators with all-to-all sinusoidal coupling. The oscillators ’ intrinsic frequencies are randomly distributed across the population ac-cording to a prescribed probability density, here taken to be unimodal and symmetric about its mean. As the coupling between the oscillators is increased, the system sponta-neously synchronizes: the oscillators near the center of the frequency distribution lock their phases together and run at the same frequency, while those in the tails remain unlocked and drift at different frequencies. Although this “partially locked ” state has been observed in simulations for decades, its stability has never been analyzed mathematically. Part of the difficulty is in formulating a reasonable infinite-N limit of the model. Here we describe such a continuum limit, and prove that the corresponding partially locked state is, in fact, neutrally stable, contrary to what one might have expected. The possible implications of this result are discussed.

frequencies

Abstract not found

We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for all-to-all networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value, the synchronized state is locally asymptotically stable, resulting in convergence of all phase differences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity. Over the past decade, considerable attention has been devoted to the problem of coordinated motion of multiple autonomous agents. A variety of disciplines (as diverse as ecology, the social sciences, statistical physics, computer graphics and, indeed, systems and control theory) are developing an understanding of how a group of moving objects (such as flocks of birds, schools of fish, crowds of people [11], [20], or collections of autonomous robots or unmanned