1,666 research outputs found
Global phase-locking in finite populations of phase-coupled oscillators
We present new necessary and sufficient conditions for the existence of fixed
points in a finite system of coupled phase oscillators on a complete graph. We
use these conditions to derive bounds on the critical coupling.Comment: 31 pages; to appear in SIAM journal of dynamical systems (SIADS
Synchronization of oscillators with long range interaction: phase transition and anomalous finite size effects
Synchronization in a lattice of a finite population of phase oscillators with
algebraically decaying, non-normalized coupling is studied by numerical
simulations. A critical level of decay is found, below which full locking takes
place if the population contains a sufficiently large number of elements. For
large number of oscillators and small coupling constant, numerical simulations
and analytical arguments indicate that a phase transition separating
synchronization from incoherence appears at a decay exponent value equal to the
number of dimensions of the lattice. In contrast with earlier results on
similar systems with normalized coupling, we have indication that for the decay
exponent less than the dimensions of the lattice and for large populations,
synchronization is possible even if the coupling is arbitarily weak. This
finding suggests that in organisms interacting through slowly decaying signals
like light or sound, collective oscillations can always be established if the
population is sufficiently large.Comment: 15 pages, 12 figures, submitted to Phys. Rev. E; Text slightly
changed; References added; Fig. 9 update
Time Delay Effects on Coupled Limit Cycle Oscillators at Hopf Bifurcation
We present a detailed study of the effect of time delay on the collective
dynamics of coupled limit cycle oscillators at Hopf bifurcation. For a simple
model consisting of just two oscillators with a time delayed coupling, the
bifurcation diagram obtained by numerical and analytical solutions shows
significant changes in the stability boundaries of the amplitude death, phase
locked and incoherent regions. A novel result is the occurrence of amplitude
death even in the absence of a frequency mismatch between the two oscillators.
Similar results are obtained for an array of N oscillators with a delayed mean
field coupling and the regions of such amplitude death in the parameter space
of the coupling strength and time delay are quantified. Some general analytic
results for the N tending to infinity (thermodynamic) limit are also obtained
and the implications of the time delay effects for physical applications are
discussed.Comment: 20 aps formatted revtex pages (including 13 PS figures); Minor
changes over the previous version; To be published in Physica
The Kuramoto model: A simple paradigm for synchronization phenomena
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
Dispersal and noise: Various modes of synchrony in\ud ecological oscillators
We use the theory of noise-induced phase synchronization to analyze the effects of dispersal on the synchronization of a pair of predator-prey systems within a fluctuating environment (Moran effect). Assuming that each isolated local population acts as a limit cycle oscillator in the deterministic limit, we use phase reduction and averaging methods to derive a Fokker–Planck equation describing the evolution of the probability density for pairwise phase differences between the oscillators. In the case of common environmental noise, the oscillators ultimately synchronize. However the approach to synchrony depends on whether or not dispersal in the absence of noise supports any stable asynchronous states. We also show how the combination of correlated (shared) and uncorrelated (unshared) noise with dispersal can lead to a multistable\ud
steady-state probability density
Synchronization of Integrate and Fire oscillators with global coupling
In this article we study the behavior of globally coupled assemblies of a
large number of Integrate and Fire oscillators with excitatory pulse-like
interactions. On some simple models we show that the additive effects of pulses
on the state of Integrate and Fire oscillators are sufficient for the
synchronization of the relaxations of all the oscillators. This synchronization
occurs in two forms depending on the system: either the oscillators evolve ``en
bloc'' at the same phase and therefore relax together or the oscillators do not
remain in phase but their relaxations occur always in stable avalanches. We
prove that synchronization can occur independently of the convexity or
concavity of the oscillators evolution function. Furthermore the presence of
disorder, up to some level, is not only compatible with synchronization, but
removes some possible degeneracy of identical systems and allows new mechanisms
towards this state.Comment: 37 pages, 19 postscript figures, Latex 2
Dynamics of fully coupled rotators with unimodal and bimodal frequency distribution
We analyze the synchronization transition of a globally coupled network of N
phase oscillators with inertia (rotators) whose natural frequencies are
unimodally or bimodally distributed. In the unimodal case, the system exhibits
a discontinuous hysteretic transition from an incoherent to a partially
synchronized (PS) state. For sufficiently large inertia, the system reveals the
coexistence of a PS state and of a standing wave (SW) solution. In the bimodal
case, the hysteretic synchronization transition involves several states.
Namely, the system becomes coherent passing through traveling waves (TWs), SWs
and finally arriving to a PS regime. The transition to the PS state from the SW
occurs always at the same coupling, independently of the system size, while its
value increases linearly with the inertia. On the other hand the critical
coupling required to observe TWs and SWs increases with N suggesting that in
the thermodynamic limit the transition from incoherence to PS will occur
without any intermediate states. Finally a linear stability analysis reveals
that the system is hysteretic not only at the level of macroscopic indicators,
but also microscopically as verified by measuring the maximal Lyapunov
exponent.Comment: 22 pages, 11 figures, contribution for the book: Control of
Self-Organizing Nonlinear Systems, Springer Series in Energetics, eds E.
Schoell, S.H.L. Klapp, P. Hoeve
Emergent dynamics of the Kuramoto ensemble under the effect of inertia
We study the emergent collective behaviors for an ensemble of identical
Kuramoto oscillators under the effect of inertia. In the absence of inertial
effects, it is well known that the generic initial Kuramoto ensemble relaxes to
the phase-locked states asymptotically (emergence of complete synchronization)
in a large coupling regime. Similarly, even for the presence of inertial
effects, similar collective behaviors are observed numerically for generic
initial configurations in a large coupling strength regime. However, this
phenomenon has not been verified analytically in full generality yet, although
there are several partial results in some restricted set of initial
configurations. In this paper, we present several improved complete
synchronization estimates for the Kuramoto ensemble with inertia in two
frameworks for a finite system. Our improved frameworks describe the emergence
of phase-locked states and its structure. Additionally, we show that as the
number of oscillators tends to infinity, the Kuramoto ensemble with infinite
size can be approximated by the corresponding kinetic mean-field model
uniformly in time. Moreover, we also establish the global existence of
measure-valued solutions for the Kuramoto equation and its large-time
asymptotics
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