Analysis of a power grid using the Kuramoto-like model

We show that there is a link between the Kuramoto paradigm and another system of synchronized oscillators, namely an electrical power distribution grid of generators and consumers. The purpose of this work is to show both the formal analogy and some practical consequences. The mapping can be made quantitative, and under some necessary approximations a class of Kuramoto-like models, those with bimodal distribution of the frequencies, is most appropriate for the power-grid. In fact in the power-grid there are two kinds of oscillators: the 'sources' delivering power to the 'consumers'.Comment: 24 pages, including 7 figures. To appear on Eur. Phys. J.

The Spectrum of the Partially Locked State for the Kuramoto Model

We solve a longstanding stability problem for the Kuramoto model of coupled oscillators. 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 provides 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 according 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 spontaneously 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

Pinned states in Josephson arrays: A general stability theorem

Using the lumped circuit equations, we derive a stability criterion for superconducting pinned states in two-dimensional arrays of Josephson junctions. The analysis neglects quantum, thermal, and inductive effects, but allows disordered junctions, arbitrary network connectivity, and arbitrary spatial patterns of applied magnetic flux and DC current injection. We prove that a pinned state is linearly stable if and only if its corresponding stiffness matrix is positive definite. This algebraic condition can be used to predict the critical current and frustration at which depinning occurs.Comment: To appear in Phys. Rev.

Time-periodic phases in populations of nonlinearly coupled oscillators with bimodal frequency distributions

The mean field Kuramoto model describing the synchronization of a population of phase oscillators with a bimodal frequency distribution is analyzed (by the method of multiple scales) near regions in its phase diagram corresponding to synchronization to phases with a time periodic order parameter. The richest behavior is found near the tricritical point were the incoherent, stationarily synchronized, ``traveling wave'' and ``standing wave'' phases coexist. The behavior near the tricritical point can be extrapolated to the rest of the phase diagram. Direct Brownian simulation of the model confirms our findings.Comment: Revtex,16 pag.,10 fig., submitted to Physica

Asymptotic description of transients and synchronized states of globally coupled oscillators

A two-time scale asymptotic method has been introduced to analyze the multimodal mean-field Kuramoto-Sakaguchi model of oscillator synchronization in the high-frequency limit. The method allows to uncouple the probability density in different components corresponding to the different peaks of the oscillator frequency distribution. Each component evolves toward a stationary state in a comoving frame and the overall order parameter can be reconstructed by combining them. Synchronized phases are a combination of traveling waves and incoherent solutions depending on parameter values. Our results agree very well with direct numerical simulations of the nonlinear Fokker-Planck equation for the probability density. Numerical results have been obtained by finite differences and a spectral method in the particular case of bimodal (symmetric and asymmetric) frequency distribution with or without external field. We also recover in a very easy and intuitive way the only other known analytical results: those corresponding to reflection-symmetric bimodal frequency distributions near bifurcation points.Comment: Revtex,12 pag.,9 fig.;submitted to Physica

Effects of the network structural properties on its controllability

In a recent paper, it has been suggested that the controllability of a diffusively coupled complex network, subject to localized feedback loops at some of its vertices, can be assessed by means of a Master Stability Function approach, where the network controllability is defined in terms of the spectral properties of an appropriate Laplacian matrix. Following that approach, a comparison study is reported here among different network topologies in terms of their controllability. The effects of heterogeneity in the degree distribution, as well as of degree correlation and community structure, are discussed.Comment: Also available online at: http://link.aip.org/link/?CHA/17/03310

Nonlinear Dynamics of the Rock-Paper-Scissors Game with Mutations

We analyze the replicator-mutator equations for the Rock-Paper-Scissors game. Various graph-theoretic patterns of mutation are considered, ranging from a single unidirectional mutation pathway between two of the species, to global bidirectional mutation among all the species. Our main result is that the coexistence state, in which all three species exist in equilibrium, can be destabilized by arbitrarily small mutation rates. After it loses stability, the coexistence state gives birth to a stable limit cycle solution created in a supercritical Hopf bifurcation. This attracting periodic solution exists for all the mutation patterns considered, and persists arbitrarily close to the limit of zero mutation rate and a zero-sum game.Comment: 6 pages, 5 figure