4,396 research outputs found
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
Experimental evidence for mixed reality states
Recently researchers at the University of Illinois coupled a real pendulum to
its virtual counterpart. They observed that the two pendulums suddenly start to
move in synchrony if their lengths are sufficiently close. In this synchronized
state, the boundary between the real system and the virtual system is blurred,
that is, the pendulums are in a mixed reality state. An instantaneous,
bidirectional coupling is a prerequisite for mixed reality states. In this
article we explore the implications of mixed reality states in the context of
controlling real-world systems.Comment: 2 pages, 2 figure
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
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.
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
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