173 research outputs found
Comment on "Asymptotic Phase for Stochastic Oscillators"
Definition of the phase of oscillations is straightforward for deterministic
periodic processes but nontrivial for stochastic ones. Recently, Thomas and
Lindner in [Phys. Rev. Lett., v. 113, 254101 (2014)] suggested to use the
argument of the complex eigenfunction of the backward density evolution
operator with the smallest real part of the eigenvalue, as an asymptotic phase
of stochastic oscillations. Here I show that this definition does not generally
provide a correct asymptotic phase
Maximizing coherence of oscillations by external locking
We study how the coherence of noisy oscillations can be optimally enhanced by
external locking. Basing on the condition of minimizing the phase diffusion
constant, we find the optimal forcing explicitly in the limits of small and
large noise, in dependence of phase sensitivity of the oscillator. We show that
the form of the optimal force bifurcates with the noise intensity. In the limit
of small noise, the results are compared with purely deterministic conditions
of optimal locking
Transition to Collective Oscillations in Finite Kuramoto Ensembles
We present an alternative approach to finite-size effects around the
synchronization transition in the standard Kuramoto model. Our main focus lies
on the conditions under which a collective oscillatory mode is well defined.
For this purpose, the minimal value of the amplitude of the complex Kuramoto
order parameter appears as a proper indicator. The dependence of this minimum
on coupling strength varies due to sampling variations and correlates with the
sample kurtosis of the natural frequency distribution. The skewness of the
frequency sample determines the frequency of the resulting collective mode. The
effects of kurtosis and skewness hold in the thermodynamic limit of infinite
ensembles. We prove this by integrating a self-consistency equation for the
complex Kuramoto order parameter for two families of distributions with
controlled kurtosis and skewness, respectively.Comment: 11 pages, 8 figures, Editors' Suggestion PR
Phase demodulation with iterative Hilbert transform embeddings
We propose an efficient method for demodulation of phase modulated signals
via iterated Hilbert transform embeddings. We show that while a usual approach
based on one application of the Hilbert transform provides only an
approximation to a proper phase, with iterations the accuracy is essentially
improved, up to precision limited mainly by the discretization effects. We
demonstrate that the method is applicable to arbitrarily complex waveforms, and
to modulations fast compared to the basic frequency. Furthermore, we develop a
perturbative theory applicable to simple cosine waveforms, showing convergence
of the technique.Comment: 20 pages, 7 figure
Partially integrable dynamics of ensembles of nonidentical oscillators
We consider ensembles of sine-coupled phase oscillators consisting of
subpopulations of identical units, with a general heterogeneous coupling
between subpopulations. Using the Watanabe-Strogatz ansatz we reduce the
dynamics of the ensemble to a relatively small number of dynamical variables
plus microscopic constants of motion. This reduction is independent of the
sizes of subpopulations and remains valid in the thermodynamic limits, where
these sizes or/and the number of subpopulations are infinite. We demonstrate
that the approach to the dynamics of such systems, recently proposed by Ott and
Antonsen, corresponds to a particular choice of microscopic constants of
motion. The theory is applied to the standard Kuramoto model and to the
description of two interacting subpopulations, exhibiting a chimera state.
Furthermore, we analyze the dynamics of the extension of the Kuramoto model for
the case of nonlinear coupling and demonstrate the multistability of
synchronous states.Comment: 13 figure
Nonreciprocal wave scattering on nonlinear string-coupled oscillators
We study scattering of a periodic wave in a string on two lumped oscillators
attached to it. The equations can be represented as a driven (by the incident
wave) dissipative (due to radiation losses) system of delay differential
equations of neutral type. Nonlinearity of oscillators makes the scattering
non-reciprocal: the same wave is transmitted differently in two directions.
Periodic regimes of scattering are analysed approximately, using amplitude
equation approach. We show that this setup can act as a nonreciprocal modulator
via Hopf bifurcations of the steady solutions. Numerical simulations of the
full system reveal nontrivial regimes of quasiperiodic and chaotic scattering.
Moreover, a regime of a "chaotic diode", where transmission is periodic in one
direction and chaotic in the opposite one, is reported.Comment: Version accepted for publicatio
Finite-size-induced transitions to synchrony in oscillator ensembles with nonlinear global coupling
We report on finite-sized-induced transitions to synchrony in a population of
phase oscillators coupled via a nonlinear mean field, which microscopically is
equivalent to a hypernetwork organization of interactions. Using a
self-consistent approach and direct numerical simulations, we argue that a
transition to synchrony occurs only for finite-size ensembles, and disappears
in the thermodynamic limit. For all considered setups, that include purely
deterministic oscillators with or without heterogeneity in natural oscillatory
frequencies, and an ensemble of noise-driven identical oscillators, we
establish scaling relations describing the order parameter as a function of the
coupling constant and the system size
Numerical phase reduction beyond the first order approximation
We develop a numerical approach to reconstruct the phase dynamics of driven
or coupled self-sustained oscillators. Employing a simple algorithm for
computation of the phase of a perturbed system, we construct numerically the
equation for the evolution of the phase. Our simulations demonstrate that the
description of the dynamics solely by phase variables can be valid for rather
strong coupling strengths and large deviations from the limit cycle. Coupling
functions depend crucially on the coupling and are generally non-decomposable
in phase response and forcing terms. We also discuss limitations of the
approach.Comment: 6 pages, 7 figure
Stochastic bursting in unidirectionally delay-coupled noisy excitable systems
We show that \emph{stochastic bursting} is observed in a ring of
unidirectional delay-coupled noisy excitable systems, thanks to the
combinational action of time-delayed coupling and noise. Under the
approximation of timescale separation, i.e., when the time delays in each
connection are much larger than the characteristic duration of the spikes, the
observed rather coherent spike pattern can be described by idealized coupled
point processes with a leader-follower relationship. We derive analytically the
statistics of the spikes in each unit, pairwise correlations between any two
units, and the spectrum of the total output from the network. Theory is in a
good agreement with the simulations with a network of theta-neurons.Comment: accepted in Chao
Inter-community resonances in multifrequency ensembles of coupled oscillators
We generalize the Kuramoto model of globally coupled oscillators to
multifrequency communities. A situation when mean frequencies of two
subpopulations are close to resonance 2:1 is considered in detail. We derive
uniformly rotating solutions describing synchronization inside communities and
between them. Remarkably, cross-coupling between the frequency scales can
promote synchrony even when ensembles are separately asynchronous. We also show
that the transition to synchrony due to the cross-coupling is accompanied by a
huge multiplicity of distinct synchronous solutions what is directly related to
a multi-branch entrainment. On the other hand, for synchronous populations, the
cross-frequency coupling can destroy a phase-locking and lead to chaos of mean
fields
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