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.     

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.     

An Approach to Analyse Phase Synchronization in Oscillator Networks with Weak Coupling

We study phase synchronization in oscillator networks through phase reduced method. The dynamics of networks is reduced to phase equations by this method. Analysing the phase equations through the master stability function method, one obtains that the oscillators with identical frequency can be in-phase synchronized by weak balanced coupling. Similarly, the problem of frequency synchronization of oscillators with different frequencies is transformed to the existence of a locally asymptotically stable equilibrium of the phase error system.     

Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators

Experimental observations of time-delay-induced amplitude death in two coupled nonlinear electronic circuits that are individually capable of exhibiting limit-cycle oscillations are described. The existence of multiply connected death islands in the parameter space of coupling strength and time delay for coupled identical oscillators is established. The existence of such regions was predicted earlier on theoretical grounds [Phys. Rev. Lett. 80, 5109 (1998); Physica (Amsterdam) 129D, 15 (1999)]. The experiments also reveal the occurrence of multiple frequency states, frequency suppression of oscillations with increased time delay, and the onset of both in-phase and antiphase collective oscillations.     

Synchronization phenomena in nephron-nephron interaction

Experimental data for tubular pressure oscillations in rat kidneys are analyzed in order to examine the different types of synchronization that can arise between neighboring functional units. For rats with normal blood pressure, the individual unit (the nephron) typically exhibits regular oscillations in its tubular pressure and flow variations. For such rats, both in-phase and antiphase synchronization can be demonstrated in the experimental data. For spontaneously hypertensive rats, where the pressure variations in the individual nephrons are highly irregular, signs of chaotic phase and frequency synchronization can be observed. Accounting for a hemodynamic as well as for a vascular coupling between nephrons that share a common interlobular artery, we develop a mathematical model of the pressure and flow regulation in a pair of adjacent nephrons. We show that this model, for appropriate values of the parameters, can reproduce the different types of experimentally observed synchronization. (c) 2001 American Institute of Physics.     

Amplitude and phase effects on the synchronization of delay-coupled oscillators

We consider the behavior of Stuart-Landau oscillators as generic limit-cycle oscillators when they are interacting with delay. We investigate the role of amplitude and phase instabilities in producing symmetry-breaking/restoring transitions. Using analytical and numerical methods we compare the dynamics of one oscillator with delayed feedback, two oscillators mutually coupled with delay, and two delay-coupled elements with self-feedback. Taking only the phase dynamics into account, no chaotic dynamics is observed, and the stability of the identical synchronization solution is the same in each of the three studied networks of delay-coupled elements. When allowing for a variable oscillation amplitude, the delay can induce amplitude instabilities. We provide analytical proof that, in case of two mutually coupled elements, the onset of an amplitude instability always results in antiphase oscillations, leading to a leader-laggard behavior in the chaotic regime. Adding self-feedback with the same strength and delay as the coupling stabilizes the system in the transverse direction and, thus, promotes the onset of identically synchronized behavior.     

Collective phase sensitivity

The collective phase response to a macroscopic external perturbation of a population of interacting nonlinear elements exhibiting collective oscillations is formulated for the case of globally coupled oscillators. The macroscopic phase sensitivity is derived from the microscopic phase sensitivity of the constituent oscillators by a two-step phase reduction. We apply this result to quantify the stability of the macroscopic common-noise-induced synchronization of two uncoupled populations of oscillators undergoing coherent collective oscillations.     

Time delay effect in a living coupled oscillator system with the plasmodium of Physarum polycephalum

A living coupled oscillator system was constructed by a cell patterning method with a plasmodial slime mold, in which parameters such as coupling strength and distance between the oscillators can be systematically controlled. Rich oscillation phenomena between the two-coupled oscillators, namely, desynchronizing and antiphase/in-phase synchronization were observed according to these parameters. Both experimental and theoretical approaches showed that these phenomena are closely related to the time delay effect in interactions between the oscillators.     

From low-dimensional synchronous chaos to high-dimensional desynchronous spatiotemporal chaos in coupled systems

The dynamic behavior of coupled chaotic oscillators is investigated. For small coupling, chaotic state undergoes a transition from a spatially disordered phase to an ordered phase with an orientation symmetry breaking. For large coupling, a transition from full synchronization to partial synchronization with translation symmetry breaking is observed. Two bifurcation branches, one in-phase branch starting from synchronous chaos and the other antiphase branch bifurcated from spatially random chaos, are identified by varying coupling strength epsilon. Hysteresis, bistability, and first-order transitions between these two branches are observed.     

Experimental evidence of anomalous phase synchronization in two diffusively coupled Chua oscillators

We study the transition to phase synchronization in two diffusively coupled, nonidentical Chua oscillators. In the experiments, depending on the used parameterization, we observe several distinct routes to phase synchronization, including states of either in-phase, out-of-phase, or antiphase synchronization, which may be intersected by an intermediate desynchronization regime with large fluctuations of the frequency difference. Furthermore, we report the first experimental evidence of an anomalous transition to phase synchronization, which is characterized by an initial enlargement of the natural frequency difference with coupling strength. This results in a maximal frequency disorder at intermediate coupling levels, whereas usual phase synchronization via monotonic decrease in frequency difference sets in only for larger coupling values. All experimental results are supported by numerical simulations of two coupled Chua models.     

Phase synchronization and collective interaction of multiple flamelets in a laboratory scale spray combustor

In this paper, we investigate the coupled behvior of the acoustic field in the confinement and the unsteady flame dynamics in a laboratory scale spray combustor. We study this interaction during the intermittency route to thermoacoustic instability when the location of the flame is varied inside the combustor. As the flame location is changed, the synchronization properties of the coupled acoustic pressure and heat release rate signals change from desynchronized aperiodicity (combustion noise) to phase synchronized periodicity (thermoacoustic instability) through intermittent phase synchronization (intermittency). We also characterize the collective interaction between the multiple flamelets anchored at the flame holder and the acoustic field in the system, during different dynamical states observed in the combustor operation. When the signals are desynchronized, we notice that the flamelets exhibit a steady combustion without the exhibition of a prominent feedback with the acoustic field. In a state of intermittent phase synchronization, we observe the existence of a short-term coupling between the heat release rate and the acoustic field. We notice that the onset of collective synchronization in the oscillations of multiple flamelets and the acoustic field leads to the simultaneous emergence of periodicity in the global dynamics of the system. This collective periodicity in both the subsystems causes enhancement of oscillations during epochs of amplitude growth in the intermittency signal. On the contrary, the weakening of the coupling induces suppression of periodic oscillations during epochs of amplitude decay in the intermittency signal. During phase synchronization, we notice a sustained synchronized movement of all flamelets with the periodicity of the acoustic cycle in the system.     

Synchronization of self-sustained oscillators inertially coupled through common damped system

We study the dynamics of two self-oscillating systems inertially coupled to a linear oscillator. This interaction mechanism results in various types of synchronous motions such as in-phase, anti-phase and phase synchronization. We demonstrate the existence of mono-stable regimes and multi-stable behavior with two or more coexisting attractors. We present the bifurcational analysis revealing transition mechanisms between these regimes. In the multi-stable case, we examine the role of coupling parameter and shape of oscillations (the parameter indicating nonlinearity and strength of the damping) in various structure formations of attraction basins.     

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.     

Why two clocks synchronize: energy balance of the synchronized clocks

We consider the synchronization of two clocks which are accurate (show the same time) but have pendulums with different masses. We show that such clocks hanging on the same beam beside the complete (in-phase) and antiphase synchronizations perform the third type of synchronization in which the difference of the pendulums' displacements is a periodic function of time. We identify this period to be a few times larger than the period of pendulums' oscillations in the case when the beam is at rest. Our approximate analytical analysis allows to derive the synchronizations conditions, explains the observed types of synchronizations, and gives the approximate formula for both the pendulums' amplitudes and the phase shift between them. We consider the energy balance in the system and show how the energy is transferred between pendulums via oscillating beam allowing pendulums' synchronization.     

Synchronization by dynamical relaying in electronic circuit arrays

We experimentally study the synchronization of two chaotic electronic circuits whose dynamics is relayed by a third parameter-matched circuit, to which they are coupled bidirectionally in a linear chain configuration. In a wide range of operating parameters, this setup leads to synchronization between the outer circuits, while the relaying element remains unsynchronized. The specifics of the synchronization differ with the coupling level: for low couplings a state of intermittent synchronization between the outer circuits coexists with one of antiphase synchronization. Synchronization becomes in phase for moderate couplings, and for strong coupling identical synchronization is observed between the outer elements, which are themselves synchronized in a generalized way with the relaying element. In the latter situation, the middle element displays a triple-scroll attractor that is not possible to obtain when the chaotic oscillator is isolated.     

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.     

Intensity coupling and synchronization of chaotic lasers

The synchronization of chaotic lasers was studied considering the coupling scheme with the intensity of light pulses from one laser saturating the other one. The experimental synchronization of two single mode CO2 lasers with saturable absorber is reviewed. Numerical simulations are presented for diode lasers made chaotic by optical feedback. Optical intensity saturation coupling was introduced in Lang-Kobayashi equations. Parameters were chosen such that both lasers were uncorrelated and chaotic before coupling and partially synchronize after coupling. Correlation functions for the numerical solutions were used to characterize the properties of the synchronism. They show partial in-phase and antiphase synchronism for the pulse intensities, depending on the coupling mechanism. These synchronisms are independent of optical interferometric effects and so, relevant for communication applications.     

振幅耦合动态网络中相邻结点间的相同步

研究用适当的量化指标来刻画动态网络的相同步,为此定义了新的量化指标:相邻结点的网络平均锁相值和网络平均相频差.动态网络结点选择的是多旋转中心的Lorenz混沌振子,对Lorenz系统进行柱面坐标变换,用振幅耦合方法构造动态网络.分别对星形网络和小世界网络进行了仿真计算,结果表明随着耦合强度的增大,网络中相邻结点的两个系统之间存在相同步现象,而且相同步行为与定义的量化指标之间存在较准确的对应关系.     

Synchronization of phase oscillators with heterogeneous coupling: A solvable case

We consider an extension of Kuramoto’s model of coupled phase oscillators where oscillator pairs interact with different strengths. When the coupling coefficient of each pair can be separated into two different factors, each one associated to an oscillator, Kuramoto’s theory for the transition to synchronization can be explicitly generalized, and the effects of coupling heterogeneity on synchronized states can be analytically studied. The two factors are respectively interpreted as the weight of the contribution of each oscillator to the mean field, and the coupling of each oscillator to that field. We explicitly analyze the effects of correlations between those weights and couplings, and show that synchronization can be completely inhibited when they are strongly anti-correlated. Numerical results validate the theory, but suggest that finite-size effect are relevant to the collective dynamics close to the synchronization transition, where oscillators become entrained in synchronized frequency clusters.     

Synthetic gene network for entraining and amplifying cellular oscillations

We present a model for a synthetic gene oscillator and consider the coupling of the oscillator to a periodic process that is intrinsic to the cell. We investigate the synchronization properties of the coupled system, and show how the oscillator can be constructed to yield a significant amplification of cellular oscillations. We reduce the driven oscillator equations to a normal form, and analytically determine the amplification as a function of the strength of the cellular oscillations. The ability to couple naturally occurring genetic oscillations to a synthetically designed network could lead to possible strategies for entraining and/or amplifying oscillations in cellular protein levels.