Phase synchronization between collective rhythms of globally coupled oscillator groups: noisy identical case

We theoretically investigate the collective phase synchronization between interacting groups of globally coupled noisy identical phase oscillators exhibiting macroscopic rhythms. Using the phase reduction method, we derive coupled collective phase equations describing the macroscopic rhythms of the groups from microscopic Langevin phase equations of the individual oscillators via nonlinear Fokker-Planck equations. For sinusoidal microscopic coupling, we determine the type of the collective phase coupling function, i.e., whether the groups exhibit in-phase or antiphase synchronization. We show that the macroscopic rhythms can exhibit effective antiphase synchronization even if the microscopic phase coupling between the groups is in-phase, and vice versa. Moreover, near the onset of collective oscillations, we analytically obtain the collective phase coupling function using center-manifold and phase reductions of the nonlinear Fokker-Planck equations.     

Phase synchronization between collective rhythms of globally coupled oscillator groups: noiseless nonidentical case

We theoretically study the synchronization between collective oscillations exhibited by two weakly interacting groups of nonidentical phase oscillators with internal and external global sinusoidal couplings of the groups. Coupled amplitude equations describing the collective oscillations of the oscillator groups are obtained by using the Ott-Antonsen ansatz, and then coupled phase equations for the collective oscillations are derived by phase reduction of the amplitude equations. The collective phase coupling function, which determines the dynamics of macroscopic phase differences between the groups, is calculated analytically. We demonstrate that the groups can exhibit effective antiphase collective synchronization even if the microscopic external coupling between individual oscillator pairs belonging to different groups is in-phase, and similarly effective in-phase collective synchronization in spite of microscopic antiphase external coupling between the groups.     

Synchronization and oscillator death in oscillatory media with stirring

The effect of stirring in an inhomogeneous oscillatory medium is investigated. We show that the stirring rate can control the macroscopic behavior of the system producing collective oscillations (synchronization) or complete quenching of the oscillations (oscillator death). We interpret the homogenization rate due to mixing as a measure of global coupling and compare the phase diagrams of stirred oscillatory media and of populations of globally coupled oscillators.     

Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities

A model dynamical system with a great many degrees of freedom is proposed for which the critical condition for the onset of collective oscillations, the evolution of a suitably defined order parameter, and its fluctuations around steady states can be studied analytically. This is a rotator model appropriate for a large population of limit cycle oscillators. It is assumed that the natural frequencies of the oscillators are distributed and that each oscillator interacts with all the others uniformly. An exact self-consistent equation for the stationary amplitude of the collective oscillation is derived and is extended to a dynamical form. This dynamical extension is carried out near the transition point where the characteristic time scales of the order parameter and of the individual oscillators become well separated from each other. The macroscopic evolution equation thus obtained generally involves a fluctuating term whose irregular temporal variation comes from a deterministic torus motion of a subpopulation. The analysis of this equation reveals order parameter behavior qualitatively different from that in thermodynamic phase transitions, especially in that the critical fluctuations in the present system are extremely small.Dedicated to Ilya Prigogine on the occasion of his 70th birthday.     

Collective behavior of coupled nonuniform stochastic oscillators

Theoretical studies of synchronization are usually based on models of coupled phase oscillators which, when isolated, have constant angular frequency. Stochastic discrete versions of these uniform oscillators have also appeared in the literature, with equal transition rates among the states. Here we start from the model recently introduced by Wood et al. [K. Wood, C. Van den Broeck, R. Kawai, K. Lindenberg, Universality of synchrony: critical behavior in a discrete model of stochastic phase-coupled oscillators, Phys. Rev. Lett. 96 (2006) 145701], which has a collectively synchronized phase, and parametrically modify the phase-coupled oscillators to render them (stochastically) nonuniform. We show that, depending on the nonuniformity parameter 0≤α≤1, a mean field analysis predicts the occurrence of several phase transitions. In particular, the phase with collective oscillations is stable for the complete graph only for α≤α^′<1. At α=1 the oscillators become excitable elements and the system has an absorbing state. In the excitable regime, no collective oscillations were found in the model.     

Analysis of regions of chaotic oscillations in coupled phase systems

We discuss the problems of generation of chaotically modulated oscillations in small ensembles of coupled self-excited oscillators with phase control. Special attention is paid to analyzing the regions of generation of chaotic oscillations in parameter space. It is shown that transition to collective dynamics allows us to efficiently solve the problem of generation of chaotically modulated oscillations in a sufficiently wide parameter-space region. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 5, pp. 448–457, May 2006.     

Collective dynamics of chaotic chemical oscillators and the law of large numbers

Experiments on the nontrivial collective dynamics and phase synchronization of populations of nonidentical chaotic electrochemical oscillators are presented. Without added coupling no deviation from the law of large numbers is observed. Deviations do arise with weak global or short-range coupling; large, irregular, and periodic mean field oscillations occur along with (partial) phase synchronization.     

耦合相振子系统同步的序参量理论

节律行为,即系统行为呈现随时间的周期变化,在我们的周围随处可见.不同节律之间可以通过相互影响、相互作用产生自组织,其中同步是最典型、最直接的有序行为,它也是非线性波、斑图、集群行为等的物理内在机制.不同的节律可以用具有不同频率的振子(极限环)来刻画,它们之间的同步可以用耦合极限环系统的动力学来加以研究.微观动力学表明,随着耦合强度增强,振子同步伴随着动力学状态空间降维到一个低维子空间,该空间由序参量来描述.序参量的涌现及其所描述的宏观动力学行为可借助于协同学与流形理论等降维思想来进行.本文从统计物理学的角度讨论了耦合振子系统序参量涌现的几种降维方案,并对它们进行了对比分析.序参量理论可有效应用于耦合振子系统的同步自组织与相变现象的分析,通过进一步研究序参量的动力学及其分岔行为,可以对复杂系统的涌现动力学有更为深刻的理解.     

Evidence for quantized displacement in macroscopic nanomechanical oscillators

We report the observation of discrete displacement of nanomechanical oscillators with gigahertz-range resonance frequencies at millikelvin temperatures. The oscillators are nanomachined single-crystal structures of silicon, designed to provide two distinct sets of coupled elements with very low and very high frequencies. With this novel design, femtometer-level displacement of the frequency-determining element is amplified into collective motion of the entire micron-sized structure. The observed discrete response possibly results from energy quantization at the onset of the quantum regime in these macroscopic nanomechanical oscillators.     

Persistent spin oscillations in a spin-orbit-coupled superconductor

Quasi-two-dimensional superconductors with tunable spin-orbit coupling are very interesting systems with properties that are also potentially useful for applications. In this Letter we demonstrate that these systems exhibit undamped collective spin oscillations that can be excited by the application of a supercurrent. We propose to use these collective excitations to realize persistent spin oscillators operating in the frequency range of 10 GHz-1 THz.     

A model for current oscillations in the Si-HF system based on a quantitative analysis of current transients

The physical character of local oscillators on a nm scale will be outlined. Next-neighbor coupling leads to percolation areas of about 100 nm as result of Monte Carlo simulations. The percolation provides an intrinsic synchronization mechanism leading to macroscopic oscillations. Received: 24 September 1997/Accepted: 22 April 1998     

Order parameter analysis of synchronization transitions on star networks

The collective behaviors of populations of coupled oscillators have attracted significant attention in recent years. In this paper, an order parameter approach is proposed to study the low-dimensional dynamical mechanism of collective synchronizations, by adopting the star-topology of coupled oscillators as a prototype system. The order parameter equation of star-linked phase oscillators can be obtained in terms of the Watanabe–Strogatz transformation, Ott–Antonsen ansatz, and the ensemble order parameter approach. Different solutions of the order parameter equation correspond to the diverse collective states, and different bifurcations reveal various transitions among these collective states. The properties of various transitions in the star-network model are revealed by using tools of nonlinear dynamics such as time reversibility analysis and linear stability analysis.     

Waves and patterns in ring lattices with delays

We investigate the existence and the stability of waves and phase locked states in rings of coupled oscillators with delayed interactions. Using center manifold reduction and the normal form method, we reduce the equation governing the dynamics of the whole network to an amplitude-phase model (i.e. a set of coupled ordinary differential equations describing the evolution of both the amplitudes and the phases of the oscillators). Then we prove the existence of traveling waves, in-phase and anti-phase locked oscillations, in both one-dimensional and two-dimensional lattices. The influence of the interaction strength and the number of oscillators is investigated, and the possible coexistence of waves and phase locked oscillations is shown.     

Sequential activity and multistability in an ensemble of coupled Van der Pol oscillators

In this paper collective dynamics of an ensemble of inhibitory coupled Van der Pol oscillators are studied. It was found that a stable heteroclinic contour and a stable heteroclinic channel between saddle cycles exist. These heteroclinic structures are responsible for the sequential activity of different oscillations. The corresponding bifurcations leading to the appearance of heteroclinic trajectories are analyzed.     

Travelling waves in chains of pulse-coupled integrate-and-fire oscillators with distributed delays

We analyse travelling waves in a chain of pulse-coupled integrate-and-fire oscillators with nearest-neighbour coupling and delayed interactions. This is achieved by approximating the equations for phase-locking in terms of a singularly perturbed two-point (continuum) boundary value problem. The latter has a solution provided that a self-consistent value for the collective frequency of oscillations can be found. We investigate how the qualitative behaviour of travelling waves depends on the distribution of natural frequencies across the chain and the form of delayed interactions. A linear stability analysis of phase-locked solutions is carried out in terms of perturbations of the firing times of the oscillators. It is shown how travelling waves destabilize when the detuning between oscillators or the strength of the coupling becomes too large.     

Controlling synchronization in an ensemble of globally coupled oscillators

We propose a technique to control coherent collective oscillations in ensembles of globally coupled units (self-sustained oscillators or maps). We demonstrate numerically and theoretically that a time delayed feedback in the mean field can, depending on the parameters, enhance or suppress the self-synchronization in the population. We discuss possible applications of the technique.     

Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators

Synaptically coupled neurons show in-phase or antiphase synchrony depending on the chemical and dynamical nature of the synapse. Deterministic theory helps predict the phase differences between two phase-locked oscillators when the coupling is weak. In the presence of noise, however, deterministic theory faces difficulty when the coexistence of multiple stable oscillatory solutions occurs. We analyze the solution structure of two coupled neuronal oscillators for parameter values between a subcritical Hopf bifurcation point and a saddle node point of the periodic branch that bifurcates from the Hopf point, where a rich variety of coexisting solutions including asymmetric localized oscillations occurs. We construct these solutions via a multiscale analysis and explore the general bifurcation scenario using the lambda-omega model. We show for both excitatory and inhibitory synapses that noise causes important changes in the phase and amplitude dynamics of such coupled neuronal oscillators when multiple oscillatory solutions coexist. Mixed-mode oscillations occur when distinct bistable solutions are randomly visited. The phase difference between the coupled oscillators in the localized solution, coexisting with in-phase or antiphase solutions, is clearly represented in the stochastic phase dynamics.     

Noise enhanced phase synchronization and coherence resonance in sets of chaotic oscillators with weak global coupling

The effect of noise on phase synchronization in small sets and larger populations of weakly coupled chaotic oscillators is explored. Both independent and correlated noise are found to enhance phase synchronization of two coupled chaotic oscillators below the synchronization threshold; this is in contrast to the behavior of two coupled periodic oscillators. This constructive effect of noise results from the interplay between noise and the locking features of unstable periodic orbits. We show that in a population of nonidentical chaotic oscillators, correlated noise enhances synchronization in the weak coupling region. The interplay between noise and weak coupling induces a collective motion in which the coherence is maximal at an optimal noise intensity. Both the noise-enhanced phase synchronization and the coherence resonance numerically observed in coupled chaotic R?ssler oscillators are verified experimentally with an array of chaotic electrochemical oscillators.     

Forced Synchronization of Electroconvective Roll Oscillations in Nematic Liquid Crystals

The forced phase locking in a system of the oscillating electroconvective rolls that form in a nematic liquid crystal layer in a dc electric field is studied. As a result of the action of an additive ac electric field with a small amplitude, the system of oscillators is found to be divided into clusters, where oscillations are fully phase locked. The electroconvective rolls in neighboring clusters oscillate in antiphase and the clusters are separated by Ising walls. The phase locking is shown to be maximal at the forcing frequency that is close to the double frequency of the natural oscillations of rolls. A model is proposed to describe spatially distributed phase oscillators, and it takes into account the symmetries of a system of electroconvective rolls and external forcing. The results of numerical simulations agree well with the experimental data.