2,887 research outputs found
Critical Coupling and Synchronized Clusters in Arbitrary Networks of Kuramoto Oscillators
abstract: The Kuramoto model is an archetypal model for studying synchronization in groups
of nonidentical oscillators where oscillators are imbued with their own frequency and
coupled with other oscillators though a network of interactions. As the coupling
strength increases, there is a bifurcation to complete synchronization where all oscillators
move with the same frequency and show a collective rhythm. Kuramoto-like
dynamics are considered a relevant model for instabilities of the AC-power grid which
operates in synchrony under standard conditions but exhibits, in a state of failure,
segmentation of the grid into desynchronized clusters.
In this dissertation the minimum coupling strength required to ensure total frequency
synchronization in a Kuramoto system, called the critical coupling, is investigated.
For coupling strength below the critical coupling, clusters of oscillators form
where oscillators within a cluster are on average oscillating with the same long-term
frequency. A unified order parameter based approach is developed to create approximations
of the critical coupling. Some of the new approximations provide strict lower
bounds for the critical coupling. In addition, these approximations allow for predictions
of the partially synchronized clusters that emerge in the bifurcation from the
synchronized state.
Merging the order parameter approach with graph theoretical concepts leads to a
characterization of this bifurcation as a weighted graph partitioning problem on an
arbitrary networks which then leads to an optimization problem that can efficiently
estimate the partially synchronized clusters. Numerical experiments on random Kuramoto
systems show the high accuracy of these methods. An interpretation of the
methods in the context of power systems is provided.Dissertation/ThesisDoctoral Dissertation Applied Mathematics 201
Synchronisation Properties of Trees in the Kuramoto Model
We consider the Kuramoto model of coupled oscillators, specifically the case
of tree networks, for which we prove a simple closed-form expression for the
critical coupling. For several classes of tree, and for both uniform and
Gaussian vertex frequency distributions, we provide tight closed form bounds
and empirical expressions for the expected value of the critical coupling. We
also provide several bounds on the expected value of the critical coupling for
all trees. Finally, we show that for a given set of vertex frequencies, there
is a rearrangement of oscillator frequencies for which the critical coupling is
bounded by the spread of frequencies.Comment: 21 pages, 19 Figure
Effect of time delay on the onset of synchronization of the stochastic Kuramoto model
We consider the Kuramoto model of globally coupled phase oscillators with
time-delayed interactions, that is subject to the Ornstein-Uhlenbeck (Gaussian)
colored or the non-Gaussian colored noise. We investigate numerically the
interplay between the influences of the finite correlation time of noise
and the time delay on the onset of the synchronization process. Both
cases for identical and nonidentical oscillators had been considered. Among the
obtained results for identical oscillators is a large increase of the
synchronization threshold as a function of time delay for the colored
non-Gaussian noise compared to the case of the colored Gaussian noise at low
noise correlation time . However, the difference reduces remarkably for
large noise correlation times. For the case of nonidentical oscillators, the
incoherent state may become unstable around the maximum value of the threshold
(as a function of time delay) even at lower coupling strength values in the
presence of colored noise as compared to the noiseless case. We had studied the
dependence of the critical value of the coupling strength (the threshold of
synchronization) on given parameters of the stochastic Kuramoto model in great
details and presented results for possible cases of colored Gaussian and
non-Gaussian noises.Comment: 19 pages with 7 figure
The emergence of coherence in complex networks of heterogeneous dynamical systems
We present a general theory for the onset of coherence in collections of
heterogeneous maps interacting via a complex connection network. Our method
allows the dynamics of the individual uncoupled systems to be either chaotic or
periodic, and applies generally to networks for which the number of connections
per node is large. We find that the critical coupling strength at which a
transition to synchrony takes place depends separately on the dynamics of the
individual uncoupled systems and on the largest eigenvalue of the adjacency
matrix of the coupling network. Our theory directly generalizes the Kuramoto
model of equal strength, all-to-all coupled phase oscillators to the case of
oscillators with more realistic dynamics coupled via a large heterogeneous
network.Comment: 4 pages, 1 figure. Published versio
A power-law distribution of phase-locking intervals does not imply critical interaction
Neural synchronisation plays a critical role in information processing,
storage and transmission. Characterising the pattern of synchronisation is
therefore of great interest. It has recently been suggested that the brain
displays broadband criticality based on two measures of synchronisation - phase
locking intervals and global lability of synchronisation - showing power law
statistics at the critical threshold in a classical model of synchronisation.
In this paper, we provide evidence that, within the limits of the model
selection approach used to ascertain the presence of power law statistics, the
pooling of pairwise phase-locking intervals from a non-critically interacting
system can produce a distribution that is similarly assessed as being power
law. In contrast, the global lability of synchronisation measure is shown to
better discriminate critical from non critical interaction.Comment: (v3) Fixed error in Figure 1; (v2) Added references. Minor edits
throughout. Clarified relationship between theoretical critical coupling for
infinite size system and 'effective' critical coupling system for finite size
system. Improved presentation and discussion of results; results unchanged.
Revised Figure 1 to include error bars on r and N; results unchanged; (v1) 11
pages, 7 figure
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