Refine
Has Fulltext
- no (181)
Year of publication
Document Type
- Article (181) (remove)
Language
- English (181) (remove)
Is part of the Bibliography
- yes (181)
Keywords
- Kuramoto model (6)
- synchronization (4)
- Coupled oscillators (3)
- Lyapunov exponent (3)
- Hilbert transform (2)
- Network reconstruction (2)
- Nonlinear coupling (2)
- Oscillator ensembles (2)
- Synchronization (2)
- chaos (2)
Institute
We consider chimera states in a one-dimensional medium of nonlinear nonlocally coupled phase oscillators. In terms of a local coarse-grained complex order parameter, the problem of finding stationary rotating nonhomogeneous solutions reduces to a third-order ordinary differential equation. This allows finding chimera-type and other inhomogeneous states as periodic orbits of this equation. Stability calculations reveal that only some of these states are stable. We demonstrate that an oscillatory instability leads to a breathing chimera, for which the synchronous domain splits into subdomains with different mean frequencies. Further development of instability leads to turbulent chimeras. Published by AIP Publishing.
Objective: Several different measures of heart rate variability, and particularly of respiratory sinus arrhythmia, are widely used in research and clinical applications. For many purposes it is important to know which features of heart rate variability are directly related to respiration and which are caused by other aspects of cardiac dynamics. Approach: Inspired by ideas from the theory of coupled oscillators, we use simultaneous measurements of respiratory and cardiac activity to perform a nonlinear disentanglement of the heart rate variability into the respiratory-related component and the rest. Main results: The theoretical consideration is illustrated by the analysis of 25 data sets from healthy subjects. In all cases we show how the disentanglement is manifested in the different measures of heart rate variability. Significance: The suggested technique can be exploited as a universal preprocessing tool, both for the analysis of respiratory influence on the heart rate and in cases when effects of other factors on the heart rate variability are in focus.
We develop an approach for the description of the dynamics of large populations of phase oscillators based on "circular cumulants" instead of the Kuramoto-Daido order parameters. In the thermodynamic limit, these variables yield a simple representation of the Ott-Antonsen invariant solution [E. Ott and T. M. Antonsen, Chaos 18, 037113 (2008)] and appear appropriate for constructing perturbation theory on top of the Ott-Antonsen ansatz. We employ this approach to study the impact of small intrinsic noise on the dynamics. As a result, a closed system of equations for the two leading cumulants, describing the dynamics of noisy ensembles, is derived. We exemplify the general theory by presenting the effect of noise on the Kuramoto system and on a chimera state in two symmetrically coupled populations.
We study populations of globally coupled noisy rotators (oscillators with inertia) allowing a nonequilibrium transition from a desynchronized state to a synchronous one (with the nonvanishing order parameter). The newly developed analytical approaches resulted in solutions describing the synchronous state with constant order parameter for weakly inertial rotators, including the case of zero inertia, when the model is reduced to the Kuramoto model of coupled noise oscillators. These approaches provide also analytical criteria distinguishing supercritical and subcritical transitions to the desynchronized state and indicate the universality of such transitions in rotator ensembles. All the obtained analytical results are confirmed by the numerical ones, both by direct simulations of the large ensembles and by solution of the associated Fokker-Planck equation. We also propose generalizations of the developed approaches for setups where different rotators parameters (natural frequencies, masses, noise intensities, strengths and phase shifts in coupling) are dispersed.
Efficient determination of synchronization domains from observations of asynchronous dynamics
(2018)
We develop an approach for a fast experimental inference of synchronization properties of an oscillator. While the standard technique for determination of synchronization domains implies that the oscillator under study is forced with many different frequencies and amplitudes, our approach requires only several observations of a driven system. Reconstructing the phase dynamics from data, we successfully determine synchronization domains of noisy and chaotic oscillators. Our technique is especially important for experiments with living systems where an external action can be harmful and shall be minimized. Published by AIP Publishing.
Phase response curve is an important tool in the studies of stable self-sustained oscillations; it describes a phase shift under action of an external perturbation. We consider multistable oscillators with several stable limit cycles. Under a perturbation, transitions from one oscillating mode to another one may occur. We define phase transfer curves to describe the phase shifts at such transitions. This allows for a construction of one-dimensional maps that characterize periodically kicked multistable oscillators. We show that these maps are good approximations of the full dynamics for large periods of forcing. Published by AIP Publishing.
We analyze the accuracy of different low-dimensional reductions of the collective dynamics in large populations of coupled phase oscillators with intrinsic noise. Three approximations are considered: (i) the Ott-Antonsen ansatz, (ii) the Gaussian ansatz, and (iii) a two-cumulant truncation of the circular cumulant representation of the original system’s dynamics. For the latter, we suggest a closure, which makes the truncation, for small noise, a rigorous first-order correction to the Ott-Antonsen ansatz, and simultaneously is a generalization of the Gaussian ansatz. The Kuramoto model with intrinsic noise and the population of identical noisy active rotators in excitable states with the Kuramoto-type coupling are considered as examples to test the validity of these approximations. For all considered cases, the Gaussian ansatz is found to be more accurate than the Ott-Antonsen one for high-synchrony states only. The two-cumulant approximation is always superior to both other approximations.
Synchrony of large ensembles of coupled elements can be characterised by the order parameters—the mean fields. Quite often, the evolution of these collective variables is surprisingly simple, which makes a description with only a few order parameters feasible. Thus, one tries to construct accurate closed low-dimensional mathematical models for the dynamics of the first few order parameters. These models represent useful tools for gaining insight into the underlaying mechanisms of some more sophisticated collective phenomena: for example, one describes coupled populations by virtue of coupled equations for the relevant order parameters. A regular approach to the construction of closed low-dimensional systems is also beneficial for dealing with phenomena, which are beyond the applicability scope of these models; for instance, with such an approach, one can determine constraints on clustering in populations. There are two prominent types of situations, where the low-dimensional models can be constructed: (i) for a certain class of ideal paradigmatic systems of coupled phase oscillators, the Ott-Antonsen ansatz yields an exact equation for the main order parameter and (ii) the Gaussian approximation for the probability density of the phases, also yielding a low-dimensional closure, is frequently quite accurate. In this paper, we compare applications of these two model reductions for situations, where neither of them is perfectly accurate. Furthermore, we construct a new reduction approach which practically works as a first-order correction to the best of the two basic approximations.
Low-dimensional dynamics for higher-order harmonic, globally coupled phase-oscillator ensembles
(2019)
The Kuramoto model, despite its popularity as a mean-field theory for many synchronization phenomenon of oscillatory systems, is limited to a first-order harmonic coupling of phases. For higher-order coupling, there only exists a low-dimensional theory in the thermodynamic limit. In this paper, we extend the formulation used by Watanabe and Strogatz to obtain a low-dimensional description of a system of arbitrary size of identical oscillators coupled all-to-all via their higher-order modes. To demonstrate an application of the formulation, we use a second harmonic globally coupled model, with a mean-field equal to the square of the Kuramoto mean-field. This model is known to exhibit asymmetrical clustering in previous numerical studies. We try to explain the phenomenon of asymmetrical clustering using the analytical theory developed here, as well as discuss certain phenomena not observed at the level of first-order harmonic coupling.
We show that a combined action of noise and delayed feedback on an excitable theta-neuron leads to rather coherent stochastic bursting. An idealized point process, valid if the characteristic timescales in the problem are well separated, is used to describe statistical properties such as the power spectral density and the interspike interval distribution. We show how the main parameters of the point process, the spontaneous excitation rate, and the probability to induce a spike during the delay action can be calculated from the solutions of a stationary and a forced Fokker-Planck equation.
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 the limitations of the approach. Published under license by AIP Publishing.
We consider the dynamics of the Kuramoto ensemble oscillators not included in a common synchronized cluster, where the mean field is subject to fluctuations. The fluctuations can be either related to the finite size of the ensemble or superimposed on the mean field in the form of common noise due to the constructive features of the system. It is shown that the states of such oscillators with close natural frequencies appear correlated with each other, since the mean-field fluctuations act as common noise. We quantify the effect with the synchronization index of two oscillators, which is calculated numerically and analytically as a function of the frequency difference and noise intensity. The results are rigorous for large ensembles with additional noise superimposed on the mean field and are qualitatively true for the systems where the mean-field fluctuations are due to the finite size of the ensemble. In the latter case, the effect is found to be independent of the number of oscillators in the ensemble.
We have developed a method for deriving systems of closed equations for the dynamics of order parameters in the ensembles of phase oscillators. The Ott-Antonsen equation for the complex order parameter is a particular case of such equations. The simplest nontrivial extension of the Ott-Antonsen equation corresponds to two-bunch states of the ensemble. Based on the equations obtained, we study the dynamics of multi-bunch chimera states in coupled Kuramoto-Sakaguchi ensembles. We show an increase in the dimensionality of the system dynamics for two-bunch chimeras in the case of identical phase elements and a transition to one-bunch "Abrams chimeras" for imperfect identity (in the latter case, the one-bunch chimeras become attractive).
We consider the Kuramoto-Sakaguchi model of identical coupled phase oscillators with a common noisy forcing. While common noise always tends to synchronize the oscillators, a strong repulsive coupling prevents the fully synchronous state and leads to a nontrivial distribution of oscillator phases. In previous numerical simulations, the formation of stable multicluster states has been observed in this regime. However, we argue here that because identical phase oscillators in the Kuramoto-Sakaguchi model form a partially integrable system according to the Watanabe-Strogatz theory, the formation of clusters is impossible. Integrating with various time steps reveals that clustering is a numerical artifact, explained by the existence of higher order Fourier terms in the errors of the employed numerical integration schemes. By monitoring the induced change in certain integrals of motion, we quantify these errors. We support these observations by showing, on the basis of the analysis of the corresponding Fokker-Planck equation, that two-cluster states are non-attractive. On the other hand, in ensembles of general limit cycle oscillators, such as Van der Pol oscillators, due to an anharmonic phase response function as well as additional amplitude dynamics, multiclusters can occur naturally. Published under license by AIP Publishing.
We show that "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 an idealized coupled point processwith a leader-follower relationship. We derive analytically the statistics of the spikes in each unit, the pairwise correlations between any two units, and the spectrum of the total output from the network. Theory is in good agreement with the simulations with a network of theta-neurons. Published under license by AIP Publishing.
We consider collective dynamics in the ensemble of serially connected spin-torque oscillators governed by the Landau-Lifshitz-Gilbert-Slonczewski magnetization equation. Proximity to homoclinicity hampers synchronization of spin-torque oscillators: when the synchronous ensemble experiences the homoclinic bifurcation, the growth rate per oscillation of small deviations from the ensemble mean diverges. Depending on the configuration of the contour, sufficiently strong common noise, exemplified by stochastic oscillations of the current through the circuit, may suppress precession of the magnetic field for all oscillators. We derive the explicit expression for the threshold amplitude of noise, enabling this suppression.
In globally coupled ensembles of identical oscillators so-called chimera states can be observed. The chimera state is a symmetry-broken regime, where a subset of oscillators forms a cluster, a synchronized population, while the rest of the system remains a collection of nonsynchronized, scattered units. We describe here a blinking chimera regime in an ensemble of seven globally coupled rotators (Kuramoto oscillators with inertia). It is characterized by a death-birth process, where a long-term stable cluster of four oscillators suddenly dissolves and is very quickly reborn with a new reshuffled configuration. We identify three different kinds of rare blinking events and give a quantitative characterization by applying stability analysis to the long-lived chaotic state and to the short-lived regular regimes that arise when the cluster dissolves.
Supercritical Kuramoto oscillators with distributed frequencies can be separated into two disjoint groups: an ordered one locked to the mean field, and a disordered one consisting of effectively decoupled oscillators-at least so in the thermodynamic limit. In finite ensembles, in contrast, such clear separation fails: The mean field fluctuates due to finite-size effects and thereby induces order in the disordered group. This publication demonstrates this effect, similar to noise-induced synchronization, in a purely deterministic system. We start by modeling the situation as a stationary mean field with additional white noise acting on a pair of unlocked Kuramoto oscillators. An analytical expression shows that the cross-correlation between the two increases with decreasing ratio of natural frequency difference and noise intensity. In a deterministic finite Kuramoto model, the strength of the mean-field fluctuations is inextricably linked to the typical natural frequency difference. Therefore, we let a fluctuating mean field, generated by a finite ensemble of active oscillators, act on pairs of passive oscillators with a microscopic natural frequency difference between which we then measure the cross-correlation, at both super- and subcritical coupling.
We study the dynamics of the ring of identical phase oscillators with nonlinear nonlocal coupling. Using the Ott - Antonsen approach, the problem is formulated as a system of partial derivative equations for the local complex order parameter. In this framework, we investigate the existence and stability of twisted states. Both fully coherent and partially coherent stable twisted states were found (the latter ones for the first time for identical oscillators). We show that twisted states can be stable starting from a certain critical value of the medium length, or on a length segment. The analytical results are confirmed with direct numerical simulations in finite ensembles.
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 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 a simple cosine waveform, showing convergence of the technique.
Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here, we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the periodically forced Stuart-Landau oscillator as the paradigmatic model, we determine and numerically analyse the coupling functions up to the fourth order in the force strength. We show that the found nonlinear phase coupling functions can be used for predicting synchronization regions of the forced oscillator.