Phase Synchronization Phenomena Research Articles

Detecting synchronization in coupled stochastic ecosystem networks

Instantaneous phase difference, synchronization index and mutual information are considered in order to detect phase transitions, collective behaviours and synchronization phenomena that emerge for different levels of diffusive and reactive activity in stochastic networks. The network under investigation is a spatial 2D lattice which serves as a substrate for Lotka–Volterra dynamics with 3rd order nonlinearities. Kinetic Monte Carlo simulations demonstrate that the system spontaneously organizes into a number of asynchronous local oscillators, when only nearest neighbour interactions are considered. In contrast, the oscillators can be correlated, phase synchronized and completely synchronized when introducing different interactivity rules (diffusive or reactive) for nearby and distant species. The quantitative measures of synchronization show that long distance diffusion coupling induces phase synchronization after a well defined transition point, while long distance reaction coupling induces smeared phase synchronization.

Phase multistability and phase synchronization in an array of locally coupled period-doubling oscillators

We consider phase multistability and phase synchronization phenomena in a chain of period-doubling oscillators. The synchronization in arrays of diffusively coupled self-sustained oscillators manifests itself as rotating wave regimes, which are characterized by equal amplitudes and phases in every site which are shifted by a constant value. The value of the phase shift is preserved while the shape of motion becomes more complex through a period-doubling cascade. The number of coexisting attractors increases drastically after the transition from period-one to period-two oscillations and then after every following period-doubling bifurcation. In the chaotic region, we observe a number of phase-synchronized modes with instantaneous phases locked in different values. The loss of phase synchronization with decreasing coupling is accompanied by intermittency between several synchronous regimes.

Phase synchronization in tilted inertial ratchets as chaotic rotators

The phenomenon of phase synchronization for a particle in a periodic ratchet potential is studied. We consider the deterministic dynamics in the underdamped case where the inertia plays an important role since the dynamics can become chaotic. The ratchet potential is tilted due to a constant external force and is rocking by an external periodic forcing. This potential has to be tilted in order to obtain a rotator or self-sustained nonlinear oscillator in the absence of the external periodic forcing; this oscillator then acquires an intrinsic frequency that can be locked with the frequency of the external driving. We introduced an instantaneous linear phase, using a set of discrete time markers, and the associated average frequency, and show that this frequency can be synchronized with the frequency of the driving. We calculate Arnold tongues in a two-dimensional parameter space and discuss their implications for the chaotic transport in ratchets. We show that the local maxima in the current correspond to the borders of these Arnold tongues; in this way we established a link between optimal transport in ratchets and phase synchronization.

Network mutual information and synchronization under time transformations

We investigate the effect of general time transformations on the phase synchronization (PS) phenomenon and the mutual information rate (MIR) between pairs of nodes in dynamical networks. We demonstrate two important results concerning the invariance of both PS and the MIR. Under time transformations PS can neither be introduced nor destroyed and the MIR cannot be raised from zero. On the other hand, for proper time transformations the timing between the cycles of the coupled oscillators can be largely improved. Finally, we discuss the relevance of our findings for communication in dynamical networks.

PHASE AND LAG SYNCHRONIZATION IN COUPLED FRACTIONAL ORDER CHAOTIC OSCILLATORS

Chaotic dynamics of fractional (non-integer) order systems have begun to attract much attention in recent years. In this paper, we study the phase and lag synchronization in coupled two fractional order chaotic oscillators. It is shown that with the increase of coupling strength, the system first undergoes a transition to phase synchronization, and when the coupling strength is increased further, another transition to lag synchronization occurs. It is further shown that the system with two chaotic oscillators of different orders can also exhibit phase synchronization phenomena, although the identical synchronization is impossible in this case.

Phase and average period of chaotic oscillators

Recently, there has been a great effort to extract the phase of chaotic attractors and complex oscillators. As a consequence many phases have been introduced, as example the standard phase θ based on the rotation of the vector position, and the phase ϕ based on the rotation of the tangent vector. Despite of the large interest in the phase dynamics of coupled oscillators there is still a lack of approaches that analyze whether these phase are equivalent and on what conditions these phases work. In this work, we show that the phase ϕ generalizes the standard phase θ, and it is equal to the length of the Gauss map, the generator of the curvature in differential geometry. Furthermore, we demonstrate, for a broad class of attractors, that the phase synchronization phenomenon between two coherent chaotic oscillators is invariant under the phase definition. Moreover, we discuss to which classes of oscillators the defined phases can be used to calculate quantities as the average frequency and the average period of oscillators. Finally, we generalize the phase ϕ which allows its use also to homoclinic attractors.

Geometric framework for phase synchronization in coupled noisy nonlinear systems

A geometric approach is introduced for understanding the phenomenon of phase synchronization in coupled nonlinear systems in the presence of additive noise. We show that the emergence of cooperative behavior through a change of stability via a Hopf bifurcation entails the spontaneous appearance of a gauge structure in the system, arising from the evolution of the slow dynamics, but induced by the fast variables. The conditions for the oscillators to be synchronised in phase are obtained. The role of weak noise appears to be to drive the system towards a more synchronized behavior. Our analysis provides a framework to explain recent experimental observations on noise-induced phase synchronization in coupled nonlinear systems.

Phase synchronization of chaotic lasers

We experimentally investigate phase synchronization between two electronically coupled diode laser pumped Nd:YAG lasers. As the coupling strength increases, the phase of the two chaotic laser outputs develops from a nonsynchronous state to a phase synchronous one through a phase jump state. We find that there are 2pi phase jumps and a pi/2 phase shift between the two laser outputs unlike in optically coupled Nd:YAG lasers. To clarify the transition to phase synchronization with a pi/2 phase shift, we analyze the phenomenon of phase synchronization by using a phase portrait, phase difference dynamics, and frequency variation depending on the coupling strength and obtain the scaling rule of the average phase locking time in the intermittent phase jump state.

Mapping Model of Chaotic Phase Synchronization

A coupled map model for the chaotic phase synchronization and its desynchronization phenomenon is proposed. The model is constructed by integrating the coupled kicked oscillator system, kicking strength depending on the complex state variables. It is shown that the proposed model clearly exhibits the chaotic phase synchronization phenomenon. Furthermore, we numerically prove that in the region where the phase synchronization is weakly broken, the anomalous scaling of the phase difference rotation number is observed. This proves that the present model belongs to the same universality class found by Pikovsky et al.. Furthermore, the phase diffusion coefficient in the de-synchronization state is analyzed.

Quantification of phase synchronization phenomena and their importance for verbal memory processes

In the past years, interest in brain oscillations and their possible role in perceptual and cognitive processes has greatly increased. The two oscillations that have received the most attention are the theta and the gamma rhythm. In this study, the functioning and properties of phase synchronization parameters for these two frequency bands estimated by means of Gabor expansion were demonstrated with simulations for the phase-locking index (PLI) and the 1:1 as well as n:m phase synchronization indices. In order to demonstrate the importance of phase synchronization phenomena for memory performance, power, PLI and the 1:1 as well as n:m phase synchronization indices were calculated for EEG data on verbal memory encoding. These parameters showed various dissociations for recalled versus not-recalled nouns. In particular, the calculation of phase synchronization among different frequencies either at the same electrode or at different electrodes provided a completely new picture of dynamic neuronal interaction accompanying memory processing.

Theory of frequency and phase synchronization in a rocked bistable stochastic system

We investigate the role of noise in the phenomenon of stochastic synchronization of switching events in a rocked, overdamped bistable potential driven by white Gaussian noise, the archetype description of stochastic resonance. We present an approach to the stochastic counting process of noise-induced switching events: starting from the Markovian dynamics of the nonstationary, continuous particle dynamics, one finds upon contraction onto two states a non-Markovian renewal dynamics. A proper definition of an output discrete phase is given, and the time rate of change of its noise average determines the corresponding output frequency. The phenomenon of noise-assisted phase synchronization is investigated in terms of an effective, instantaneous phase diffusion. The theory is applied to rectangular-shaped rocking signals versus increasing input-noise strengths. In this case, for an appropriate choice of the parameter values, the system exhibits a noise-induced frequency locking accompanied by a very pronounced suppression of the phase diffusion of the output signal. Precise numerical simulations corroborate very favorably our analytical results. The novel theoretical findings are also compared with prior ones.

Predicting phase synchronization in a spiking chaoticCO2laser

An approach is presented for the reconstruction of phase synchronization phenomena in a chaotic CO2 laser from experimental data. We analyze this laser system in a regime able to phase synchronize with a weak sinusoidal forcing. Our technique recovers the synchronization diagram of the experimental system from only few measurement data sets, thus allowing the prediction of the regime of phase synchronization as well as nonsynchronization in a broad parameter space of forcing frequency and amplitude without further experiments.

AN APPROACH TO MULTIVARIATE PHASE SYNCHRONIZATION ANALYSIS AND ITS APPLICATION TO EVENT-RELATED POTENTIALS

A method for the multivariate analysis of statistical phase synchronization phenomena in empirical data is presented. A first statistical approach is complemented by a stochastic dynamic model, to result in a data analysis algorithm which can in a specific sense be shown to be a generic multivariate statistical phase synchronization analysis. The method is applied to EEG data from a psychological experiment, obtaining results which indicate the relevance of this method in the context of cognitive science as well as in other fields.

Anomalous phase synchronization in populations of nonidentical oscillators.

We report the phenomenon of anomalous phase synchronization in interacting oscillator systems with randomly distributed parameters. We show that coupling is first able to enlarge the frequency disorder leading to maximal decoherence for intermediate levels of coupling strength before reaching synchronization. Anomalous synchronization arises when the natural frequency covaries with nonisochronicity and allows for synchronization control by adjustment of system parameters.

Frequency and phase synchronization in stochastic systems.

The phenomenon of frequency and phase synchronization in stochastic systems requires a revision of concepts originally phrased in the context of purely deterministic systems. Various definitions of an instantaneous phase are presented and compared with each other with special attention paid to their robustness with respect to noise. We review the results of an analytic approach describing noise-induced phase synchronization in a thermal two-state system. In this context exact expressions for the mean frequency and the phase diffusivity are obtained that together determine the average length of locking episodes. A recently proposed method to quantify frequency synchronization in noisy potential systems is presented and exemplified by applying it to the periodically driven noisy harmonic oscillator. Since this method is based on a threshold crossing rate pioneered by Rice the related phase velocity is termed the Rice frequency. Finally, we discuss the relation between the phenomenon of stochastic resonance and noise-enhanced phase coherence by applying the developed concepts to the periodically driven bistable Kramers oscillator.

Oscillatory systems driven by noise: frequency and phase synchronization.

The phenomenon of effective phase synchronization in stochastic oscillatory systems can be quantified by an average frequency and a phase diffusion coefficient. A different approach to compute the noise-averaged frequency is put forward. The method is based on a threshold crossing rate pioneered by Rice. After the introduction of the Rice frequency for noisy systems we compare this quantifier with those obtained in the context of other phase concepts, such as the natural and the Hilbert phase, respectively. It is demonstrated that the average Rice frequency <omega>R typically supersedes the Hilbert frequency <omega>H, i.e. <omega>R > or = <omega>H. We investigate next the Rice frequency for the harmonic and the damped, bistable Kramers oscillator, both without and with external periodic driving. Exact and approximative analytic results are corroborated by numerical simulation results. Our results complement and extend previous findings for the case of noise-driven inertial systems.

PHASE SYNCHRONIZATION IN REGULAR AND CHAOTIC SYSTEMS

In this contribution we present a brief introduction to the theory of synchronization of self-sustained oscillators. Classical results for synchronization of periodic motions and effects of noise on this process are reviewed and compared with recently found phase synchronization phenomena in chaotic oscillators. The basic notions of phase and frequency locking are reconsidered within a common framework. The application of phase synchronization to data analysis is discussed.

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Phase synchronization phenomena of neuronal networks are one of many features depicted by real networks that can be studied using computational models. Here, we proceed with numerical simulations of a globally connected network composed of non-identical (distinct) Izhikevich neuron model to study clustered phase synchronization. We investigate the case in which, once coupled, there exist two main neuron clusters in the network: one of them is bi-stable, depicting phase-synchronized or unsynchronized states, depending on the initial conditions; and the second one shows just an unsynchronized state. For the set of initial conditions that lead the first cluster to the synchronized regime, we observe a chimera-like pattern of the network. For small networks, the dynamics can also present intermittent chimera-like scenarios. In this context, the mechanism for intermittent chimera states is based on two features: the coexistence of a synchronized cluster with an unsynchronized one; and the capability of one cluster to display bi-stability depending on the signal trait by which it is forced. We conclude with an understanding of intermittent chimera-like dynamics as the limit case where bi-stability is not maintained, which occurs due to the loss of uniformity in the neuron input synaptic currents.