Phase synchronization of chaotic oscillators

Learning phase synchronization from nonsynchronized chaotic regimes

We present a novel modeling approach for reconstruction of the global behavior of coupled chaotic systems from bivariate time series. We analyze two coupled chaotic oscillators, which are able to phase synchronize due to coupling. It is shown that our technique enables the recovery of the synchronization diagram from only three data sets. In particular, this allows the estimate of the relative strength of the coupling and the parameter mismatch of both subsystems. The method is most efficient if only data from the nonsynchronized regime are used for the model learning. We also apply this approach to experimental data of a paced plasma tube.

Does synchronization reflect a true interaction in the cardiorespiratory system?

Cardiorespiratory synchronization, studied within the framework of phase synchronization, has recently raised interest as one of the interactions in the cardiorespiratory system. In this work, we present a quantitative approach to the analysis of this nonlinear phenomenon. Our primary aim is to determine whether synchronization between HR and respiration rate is a real phenomenon or a random one. First, we developed an algorithm, which detects epochs of synchronization automatically and objectively. The algorithm was applied to recordings of respiration and HR obtained from 13 normal subjects and 13 heart transplant patients. Surrogate data sets were constructed from the original recordings, specifically lacking the coupling between HR and respiration. The statistical properties of synchronization in the two data sets and in their surrogates were compared. Synchronization was observed in all groups: in normal subjects, in the heart transplant patients and in the surrogates. Interestingly, synchronization was less abundant in normal subjects than in the transplant patients, indicating that the unique physiological condition of the latter promote cardiorespiratory synchronization. The duration of synchronization epochs was significantly shorter in the surrogate data of both data sets, suggesting that at least some of the synchronization epochs are real. In view of those results, cardiorespiratory synchronization, although not a major feature of cardiorespiratory interaction, seems to be a real phenomenon rather than an artifact.

Regular and chaotic phase synchronization of coupled circle maps

We study the effects of regular and chaotic phase synchronization in ensembles of coupled nonidentical circle maps (CMs) and find phase-locking regions for both types of synchronization. We show that synchronization of chaotic CMs is crucially influenced by the three quantities: (i) rotation number difference, (ii) variance of the phase evolution, and (iii)relative duration of intervals of phase increase respect decrease. In the case of regular CMs, only variance and rotation number difference are important. It is demonstrated that with increase of noncoherence of phase evolutionsin the regular and chaotic regime, the regions of the main (1:1) synchronization are usually decreased. We present a chaotic synchronization in the systems of coupled nonidentical circle maps where phase entrainment occurs and it is not accompanied by bifurcations of the chaotic set. For ensembles (chains) of coupled CMs with linear and random distributions of the individual frequencies soft and hard transitions to global synchronization are found.

Unexpected coherence and conservation

The effects of migration in a network of patch populations, or metapopulation, are extremely important for predicting the possibility of extinctions both at a local and a global scale. Migration between patches synchronizes local populations and bestows upon them identical dynamics (coherent or synchronous oscillations), a feature that is understood to enhance the risk of global extinctions. This is one of the central theoretical arguments in the literature associated with conservation ecology. Here, rather than restricting ourselves to the study of coherent oscillations, we examine other types of synchronization phenomena that we consider to be equally important. Intermittent and out-of-phase synchronization are but two examples that force us to reinterpret some classical results of the metapopulation theory. In addition, we discuss how asynchronous processes (for example, random timing of dispersal) can paradoxically generate metapopulation synchronization, another non-intuitive result that cannot easily be explained by the standard theory.

Phase locking in on-off intermittency

Dynamical behavior of on-off intermittency around chaos synchronization-desynchronization bifurcation parameter line is investigated in coupled identical chaotic oscillators. Along this parameter line, we find that on-off intermittency can transit from phase-unlocking status to phase-locking one in the phase space of variable differences, which can be regarded as a codimension-two bifurcation, i.e., combinative bifurcations of desynchronization and phase locking. In the phase-locking case, the motions of all oscillators are chaotic and they show on-off intermittency with respect to the synchronous manifold, however, spatial phase order of variable differences is clearly established.

Comment on "Intermittency in chaotic rotations"

Lai et al. [Phys. Rev. E 62, R29 (2000)] claim that the angular velocity of the phase point moving along the chaotic trajectory in a properly chosen projection (the instantaneous frequency) is intermittent. Using the same examples, namely the Rössler and the Lorenz systems, we show the absence of intermittency in the dynamics of the instantaneous frequency. This is confirmed by demonstrating that the phase dynamics exhibits normal diffusion. We argue that the nonintermittent behavior is generic.

Phase synchronization of chaotic systems with small phase diffusion

The geometric theory of phase locking between periodic oscillators is extended to phase coherent chaotic systems. This approach explains the qualitative features of phase locked chaotic systems and provides an analytical tool for a quantitative description of the phase locked states. Moreover, this geometric viewpoint allows us to identify obstructions to phase locking even in systems with negligible phase diffusion, and to provide sufficient conditions for phase locking to occur. We apply these techniques to the Rössler system and a phase coherent electronic circuit and find that numerical results and experiments agree well with theoretical predictions.

Comparison of Hilbert transform and wavelet methods for the analysis of neuronal synchrony

The quantification of phase synchrony between neuronal signals is of crucial importance for the study of large-scale interactions in the brain. Two methods have been used to date in neuroscience, based on two distinct approaches which permit a direct estimation of the instantaneous phase of a signal [Phys. Rev. Lett. 81 (1998) 3291; Human Brain Mapping 8 (1999) 194]. The phase is either estimated by using the analytic concept of Hilbert transform or, alternatively, by convolution with a complex wavelet. In both methods the stability of the instantaneous phase over a window of time requires quantification by means of various statistical dependence parameters (standard deviation, Shannon entropy or mutual information). The purpose of this paper is to conduct a direct comparison between these two methods on three signal sets: (1) neural models; (2) intracranial signals from epileptic patients; and (3) scalp EEG recordings. Levels of synchrony that can be considered as reliable are estimated by using the technique of surrogate data. Our results demonstrate that the differences between the methods are minor, and we conclude that they are fundamentally equivalent for the study of neuroelectrical signals. This offers a common language and framework that can be used for future research in the area of synchronization.

Detecting direction of coupling in interacting oscillators

We propose a method for experimental detection of directionality of weak coupling between two self-sustained oscillators from bivariate data. The technique is applicable to both noisy and chaotic systems that can be nonidentical or even structurally different. We introduce an index that quantifies the asymmetry in coupling.

Phase synchronization and suppression of chaos through intermittency in forcing of an electrochemical oscillator

External periodic forcing was applied to a chaotic chemical oscillator in experiments on the electrodissolution of Ni in sulfuric acid solution. The amplitude and the frequency (Omega) of the forcing signal were varied in a region around Omega=omega(0), where omega(0) is the frequency of the unforced signal. Phase synchronization occurred with increase in the amplitude of the forcing. For Omega/omega(0) near 1 the signal remained chaotic after the transition to the phase-locked state; for Omega/omega(0) somewhat farther from 1 the transition was to a periodic state via intermittency. The experimental results are supported by numerical simulations using a general model for electrochemical oscillations.

Synchronization of hyperexcitable systems with phase-repulsive coupling

We study two-dimensional arrays of FitzHugh-Nagumo elements with nearest-neighbor coupling from the viewpoint of synchronization. The elements are diffusively coupled. By varying the diffusion coefficient from positive to negative values, interesting synchronization patterns are observed. The results of the simulations resemble the intracellular oscillation patterns observed in cultured human epileptic astrocytes. Three measures are proposed to determine the degree of synchronization (or coupling) in both the simulated and the experimental system.

Experimental observation of generalized time-lagged chaotic synchronization

We investigate, experimentally, synchronization in coupled chaotic oscillators in the presence of large parameter mismatches and identify a different phenomenon: generalized time-lagged synchronization. Specifically, we find that there can be a functional relation between time-lagged dynamical variables of the coupled oscillators in wide parameter regimes.

Synchronization effects in a dual-wavelength class-B laser with modulated losses

Different types of synchronization: in-phase, antiphase, phase, and lag synchronization, as well as amplitude death have been found theoretically in a dual-wavelength class-B laser with modulated losses in one of the channels. Depending on the laser parameters, oscillations in master and slave channels can be either completely or partially synchronized. The conditions for the dual-wavelength regime have been established. The analysis has been performed on the basis of transfer functions of the master and slave channels.

Spatiotemporal variation of metabolism in a plant circadian rhythm: the biological clock as an assembly of coupled individual oscillators

The complex dynamic properties of biological timing in organisms remain a central enigma in biology despite the increasingly precise genetic characterization of oscillating units and their components. Although attempts to obtain the time constants from oscillations of gene activity and biochemical units have led to substantial progress, we are still far from a full molecular understanding of endogenous rhythmicity and the physiological manifestations of biological clocks. Applications of nonlinear dynamics have revolutionized thinking in physics and in biomedical and life sciences research, and spatiotemporal considerations are now advancing our understanding of development and rhythmicity. Here we show that the well known circadian rhythm of a metabolic cycle in a higher plant, namely the crassulacean acid metabolism mode of photosynthesis, is expressed as dynamic patterns of independently initiated variations in photosynthetic efficiency (phi(PSII)) over a single leaf. Noninvasive highly sensitive chlorophyll fluorescence imaging reveals randomly initiated patches of varying phi(PSII) that are propagated within minutes to hours in wave fronts, forming dynamically expanding and contracting clusters and clearly dephased regions of phi(PSII). Thus, this biological clock is a spatiotemporal product of many weakly coupled individual oscillators, defined by the metabolic constraints of crassulacean acid metabolism. The oscillators operate independently in space and time as a consequence of the dynamics of metabolic pools and limitations of CO(2) diffusion between tightly packed cells.

EEG gamma-band phase synchronization between posterior and frontal cortex during mental rotation in humans

The main purpose of the present paper was: (1) to study the phase synchronization pattern in the gamma-band while performing the classical Shepard-Metzler task of mental rotation; (2) to investigate the role of musical training; and (3) to study hemispheric differences in the degree of synchronization during mental rotation. Multivariate electroencephalograph signals from 20 male subjects (ten musicians and ten non-musicians) were recorded while performing the mental rotation task and also at resting condition. Phase synchronization was measured by a recent index, mean phase coherence. It was found that synchronization between frontal cortex and right parietal cortex was significantly increased during mental rotation with respect to rest, whereby musicians showed significantly higher degrees of synchronization than non-musicians. Left hemispheric dominance in the degree of phase synchronization, stronger in the posterior right parietal and occipital regions, was observed in musicians. Right hemispheric dominance was generally observed in non-musicians.

Phase synchronization in coupled Nd:YAG lasers

We investigate experimentally the transition to phase synchronization in coupled Nd:YAG lasers. As the coupling strength increases, the phase difference of two chaotic laser outputs develops from nonsynchronous to a phase-synchronous state via +/-2pi phase jumps. We analyze this transition.

How does a periodic rotating wave emerge from high-dimensional chaos in a ring of coupled chaotic oscillators?

A route to typical rotating waves from high-dimensional chaos is investigated in diffusively coupled chaotic Rössler oscillators. By increasing the coupling from zero, a high-dimensional spatiotemporal chaos changes into a coherent state, which is periodic in time and well ordered in space, through consecutive transitions. A crossover transition from spatially random chaos to spatially ordered chaos with phase locking and orientational equality (for two directions) breaking is a crucial step for establishing the typical spatial order of the rotating wave.

Analytical and numerical studies of noise-induced synchronization of chaotic systems

We study the effect that the injection of a common source of noise has on the trajectories of chaotic systems, addressing some contradictory results present in the literature. We present particular examples of one-dimensional maps and the Lorenz system, both in the chaotic region, and give numerical evidence showing that the addition of a common noise to different trajectories, which start from different initial conditions, leads eventually to their perfect synchronization. When synchronization occurs, the largest Lyapunov exponent becomes negative. For a simple map we are able to show this phenomenon analytically. Finally, we analyze the structural stability of the phenomenon. (c) 2001 American Institute of Physics.

Array-enhanced coherence resonance: nontrivial effects of heterogeneity and spatial independence of noise

We demonstrate the effect of coherence resonance in a heterogeneous array of coupled Fitz Hugh-Nagumo neurons. It is shown that coupling of such elements leads to a significantly stronger coherence compared to that of a single element. We report nontrivial effects of parameter heterogeneity and spatial independence of noise on array-enhanced coherence resonance; especially, we find that (i) the coherence increases as spatial correlation of the noise decreases, and (ii) inhomogeneity in the parameters of the array enhances the coherence. Our results have the implication that generic heterogeneity and background noise can play a constructive role to enhance the time precision of firing in neural systems.

Phase synchronization of diffusively coupled Rössler oscillators with funnel attractors

Recently, an antiphase phase-synchronized state in a system of diffusively coupled Rössler oscillators has been reported [Gang Hu et al., Phys. Rev. Lett. 85, 3377 (2000)]. In the current paper this antiphase state is explored in detail. Our interests are concentrated on the comparison with the normal in-phase phase-synchronized state for phase-coherent oscillators and the effect of the lattice size. Our main results are that (i) this antiphase synchronization is only for funnel Rössler attractors and cannot be observed in a system of coupled phase-coherent oscillators; (ii) it can be observed only for intermediate values of the lattice size while it disappears for quite low or large values of the lattice size; and (iii) it is different from the in-phase phase-synchronized state of phase-coherent oscillators in many respects.

Modelling the dynamics of angles of human R-R intervals

Heart rate variability (HRV) data from young healthy humans is expanded into two components, namely, the angles and radii of a map of R-R intervals. It is shown that. for most subjects at rest breathing spontaneously, the map of successive angles reveals a highly deterministic structure after the frequency range below approximately 0.05 Hz has been filtered out. However, no obvious low-dimensional structure is found in the map of successive radii. A recently proposed model describing the map of angles for a periodic self-oscillator under external periodic and quasiperiodic forcing is successfully applied to model the dynamics of such angles.

Effect of noise on the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors

We study the influence of external noise on the relaxation to an invariant probability measure for two types of nonhyperbolic chaotic attractors, a spiral (or coherent) and a noncoherent one. We find that for the coherent attractor the rate of mixing changes under the influence of noise, although the largest Lyapunov exponent remains almost unchanged. A mechanism of the noise influence on mixing is presented which is associated with the dynamics of the instantaneous phase of chaotic trajectories. This also explains why the noncoherent regime is robust against the presence of external noise.

Detecting phase synchronization in a chaotic laser array

Detection of phase synchronization of coupled chaotic oscillators is examined experimentally for the case of a linear laser array. Phase variables are computed by applying a Gaussian filter, peaked at a positive frequency, to the signal obtained from the intensity time series of the individual lasers. Relationships between different frequency components of the oscillator dynamics that are not otherwise apparent are unambiguously detected.

Spatial-temporal structures of human alpha rhythms: theory, microcurrent sources, multiscale measurements, and global binding of local networks

A theoretical framework supporting experimental measures of dynamic properties of human EEG is proposed with emphasis on distinct alpha rhythms. Robust relationships between measured dynamics and cognitive or behavioral conditions are reviewed, and proposed physiological bases for EEG at cellular levels are considered. Classical EEG data are interpreted in the context of a conceptual framework that distinguishes between locally and globally dominated dynamic processes, as estimated with coherence or other measures of phase synchronization. Macroscopic (scalp) potentials generated by cortical current sources are described at three spatial scales, taking advantage of the columnar structure of neocortex. New EEG data demonstrate that both globally coherent and locally dominated behavior can occur within the alpha band, depending on narrow band frequency, spatial measurement scale, and brain state. Quasi-stable alpha phase structures consistent with global standing waves are observed. At the same time, alpha and theta phase locking between cortical regions during mental calculations is demonstrated, consistent with neural network formation. The brain-binding problem is considered in the context of EEG dynamic behavior that generally exhibits both of these local and global aspects. But specific experimental designs and data analysis methods may severely bias physiological interpretations in either local or global directions.

Intermittent phase synchronization of coupled spatiotemporal chaotic systems

Phase synchronization is studied with a discrete system formed by two coupled map lattices, in which phases are measured in two-dimensional vectors. Simulation results show that by imposing external coupling between the two lattices, phase synchronization can be found in all two-dimensional phase planes between them. When the system is approaching the phase synchronizing state, unstable phase synchronization is observed. This is referred to as intermittent phase synchronization that appears when the trajectories on two interacting phase planes have opposite directions of rotation but with only a small phase difference. The intermittent phase synchronization could also be observed in coupled autonomous systems with diffusive attractors although their phase concepts are inconsistent. Our results show that the intermittent phase synchronization of both discrete and autonomous systems relates to the diffusion or the complexity of the attractors.

Enhanced phase synchrony in the electroencephalograph gamma band for musicians while listening to music

Multichannel electroencephalograph signals from two broad groups, 10 musicians and 10 nonmusicians, recorded in different states (in resting states or no task condition, with eyes opened and eyes closed, and with two musical tasks, listening to two different pieces of music) were studied. Degrees of phase synchrony in various frequency bands were assessed. No differences in the degree of synchronization in any frequency band were found between the two groups in resting conditions. Yet, while listening to music, significant increases of synchronization were found only in the gamma-frequency range (>30 Hz) over large cortical areas for the group of musicians. This high degree of synchronization elicited by music in the group of musicians might be due to their ability to host long-term memory representations of music and mediate access to these stored representations.

Unifying framework for synchronization of coupled dynamical systems

A definition of synchronization of coupled dynamical systems is provided. We discuss how such a definition allows one to identify a unifying framework for synchronization of dynamical systems, and show how to encompass some of the different phenomena described so far in the context of synchronization of chaotic systems.

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.

Influence of low intensity noise on assemblies of diffusively coupled chaotic cells

The effect of time-correlated and white Gaussian noises of low intensity on one-dimensional arrays consisting of diffusively coupled chaotic cells is analyzed. An improvement or worsening of the synchronization between cells of the array driven by low-intensity colored noise is observed for a resonant interval of time correlation values. A comparison between colored and white noise and additive and multiplicative contribution has been carried out investigating the nonlinear cooperative effects of noise strength, correlation time, and coupling strength to control spatiotemporal chaos in coupled arrays of chaotic cells. The possibility to distinguish highly correlated areas of a diffusively coupled network of cells by using low-intensity time correlated noise is discussed. (c) 2001 American Institute of Physics.

Characterizing neurodynamic changes before seizures

The study of dynamic changes in neural activity preceding epileptic seizure allows the characterization of a preictal state several minutes before seizure onset. This opens up new perspectives for studying the mechanisms of epileptogenesis as well as for possible therapeutic interventions, which represent a major breakthrough. In this review the authors present and discuss the results from their group in this domain using nonlinear analysis of brain signals, as well as the limitations of this topic and current questions.

Comment on "Phase synchronization in discrete chaotic systems"

Chen et al. [Phys. Rev. E 61, 2559 (2000)] recently proposed an extension of the concept of phase for discrete chaotic systems. Using the newly introduced definition of phase they studied the dynamics of coupled map lattices and compared these dynamics with phase synchronization of coupled continuous-time chaotic systems. In this paper we illustrate by two simple counterexamples that the angle variable introduced by Chen et al. fails to satisfy the basic requirements to the proper phase. Furthermore, we argue that an extension of the notion of phase synchronization to generic discrete maps is doubtful.

Early seizure detection

For patients with medically intractable epilepsy, there have been few effective alternatives to resective surgery, a destructive, irreversible treatment. A strategy receiving increased attention is using interictal spike patterns and continuous EEG measurements from epileptic patients to predict and ultimately control seizure activity via chemical or electrical control systems. This work compares results of seven linear and nonlinear methods (analysis of power spectra, cross-correlation, principal components, phase, wavelets, correlation integral, and mutual prediction) in detecting the earliest dynamical changes preceding 12 intracranially-recorded seizures from 4 patients. A method of counting standard deviations was used to compare across methods, and the earliest departures from thresholds determined from non-seizure EEG were compared to a neurologist's judgement. For these data, the nonlinear methods offered no predictive advantage over the linear methods. All the methods described here were successful in detecting changes leading to a seizure between one and two minutes before the first changes noted by the neurologist, although analysis of phase correlation proved the most robust. The success of phase analysis may be due in part to its complete insensitivity to amplitude, which may provide a significant source of error.

Multistability formation and synchronization loss in coupled Hénon maps: two sides of the single bifurcational mechanism

We investigate phenomena of multistability and complete chaos synchronization in coupled Hénon maps, which is an invertible system. Multiparametric analysis of a selected family of periodic orbits for coupled Henon maps shows that a single bifurcational mechanism describes both a loss of chaos synchronization and multistability formation. The process of bubbling transition and riddle basins, and the multistability formation in invertible systems are described in detail.

Phase clustering and transition to phase synchronization in a large number of coupled nonlinear oscillators

The transition to phase synchronization in systems consisting of a large number (N) of coupled nonlinear oscillators via the route of phase clustering (phase synchronization among subsets of oscillators) is investigated. We elucidate the mechanism for the merger of phase clusters and find an algebraic scaling between the critical coupling parameter required for phase synchronization and N. Our result implies that, in realistic situations, phase clustering may be more prevalent than full phase synchronization.

Large-scale synchrony in weakly interacting automata

We study the behavior of two spatially distributed (sandpile) models which are weakly linked with one another. Using a Monte Carlo implementation of the renormalization-group and algebraic methods, we describe how large-scale correlations emerge between the two systems, leading to synchronized behavior.

Experimental observation of superpersistent chaotic transients

We present the first experimental observation of superpersistent chaotic transients. In particular, we investigate the effect of noise on phase synchronization in coupled chaotic electronic circuits and obtain the scaling relation that is characteristic of those extremely long chaotic transients.

Synchronization of single-side locally averaged adaptive coupling and its application to shock capturing

We propose a single-sided locally averaged adaptive coupling scheme for the synchronization of spatially extended systems. Coupling and synchronization are analyzed from the viewpoint of image filter construction and numerical dissipation. Single-sided locally averaged coupling is introduced based on the resolution argument of control process. Control sensors are adaptively selected and automatically adjusted according to the magnitude of local oscillations. We demonstrate that the present scheme can effectively suppress and control spatiotemporal oscillations and, thus, provide a powerful approach for shock capturing. Both the Navier-Stokes equation and Burgers' equation are used to illustrate the idea.

Parameter-adaptive identical synchronization disclosing Lorenz chaotic masking

A parameter-adaptive rule that globally synchronizes oscillatory Lorenz chaotic systems with initially different parameter values is reported. In principle, the adaptive rule requires access to the three state variables of the drive system but it has been readapted to work with the exclusive knowledge of only one variable, a potential message carrier. The rule is very robust and can be used to trace parameter modulation conveying hidden messages. The driven system is defined according to a drive-driven type of coupling that guarantees synchronization if parameters are identical. From any arbitrary initial state, the parameters of the driven system are dynamically adapted to reach convergence to the drive parameter values. At this point, synchronization mismatch or parameter tracing is used to unmask any potential hidden message.

Synchronization experiments with an atmospheric global circulation model

Synchronization in a chaotic system with many degrees of freedom is investigated by coupling two identical global atmospheric circulation models. Starting from different initial conditions, the two submodels show complete synchronization as well as noncomplete synchronization depending on the coupling strength. The relatively low value of the coupling strength threshold for complete synchronization indicates the potential importance of synchronization mechanisms involved in climate variability. In addition, the results suggest synchronization experiments as a valuable additional method to analyze complex dynamical models, e.g., to estimate the largest Lyapunov exponent. (c) 2001 American Institute of Physics.

Phase signal coupling induced n:m phase synchronization in drive-response oscillators

We have studied phase synchronization between two identical Rössler oscillators connected in the drive-response configuration by a single phase signal. Before the transition to phase synchronization, the distribution of the time interval between consecutive 2pi jumps shows several sharp peaks. With a strong phase signal coupling, the n:m phase synchronization between the oscillators can be achieved. For the n (not equal) m phase synchronizing state, some values of coupling strength result in a phenomenon characterized by a reduction in the mean amplitude of the response termed amplitude reduction. In these regions, the mean rotation speed of the response remains approximately constant while the locking ratio n:m varies.

Transcritical riddling in a system of coupled maps

The transition from fully synchronized behavior to two-cluster dynamics is investigated for a system of N globally coupled chaotic oscillators by means of a model of two coupled logistic maps. An uneven distribution of oscillators between the two clusters causes an asymmetry to arise in the coupling of the model system. While the transverse period-doubling bifurcation remains essentially unaffected by this asymmetry, the transverse pitchfork bifurcation is turned into a saddle-node bifurcation followed by a transcritical riddling bifurcation in which a periodic orbit embedded in the synchronized chaotic state loses its transverse stability. We show that the transcritical riddling transition is always hard. For this, we study the sequence of bifurcations that the asynchronous point cycles produced in the saddle-node bifurcation undergo, and show how the manifolds of these cycles control the magnitude of asynchronous bursts. In the case where the system involves two subpopulations of oscillators with a small mismatch of the parameters, the transcritical riddling will be replaced by two subsequent saddle-node bifurcations, or the saddle cycle involved in the transverse destabilization of the synchronized chaotic state may smoothly shift away from the synchronization manifold. In this way, the transcritical riddling bifurcation is substituted by a symmetry-breaking bifurcation, which is accompanied by the destruction of a thin invariant region around the symmetrical chaotic state.

Phase-frequency synchronization in a chain of periodic oscillators in the presence of noise and harmonic forcings

We study numerically the effects of noise and periodic forcings on cluster synchronization in a chain of Van der Pol oscillators. We generalize the notion of effective synchronization to the case of a spatially extended system. It is shown that the structure of synchronized clusters can be effectively controlled by applying local external forcings. The effect of amplitude relations on the phase dynamics is also explored.

Two-dimensional type-I intermittency

The general structure of two-dimensional intermittency is discussed. The structure of channel and the trajectory in the return map are compared with those of one-dimensional intermittency and the scaling relations are obtained according to the trajectory. We illustrate the temporal behavior and scaling relations in a coupled map. The numerical results agree well with the theoretical predication of <l> approximately equal 1/sqrt[epsilon].

Phase synchronization between several interacting processes from univariate data

A novel approach is suggested for detecting the presence or absence of synchronization between two or three interacting processes with different time scales in univariate data. It is based on an angle-of-return-time map. A model is derived to describe analytically the behavior of angles for a periodic oscillator under weak periodic and quasiperiodic forcing. An explicit connection is demonstrated between the return angle and the phase of the external periodic forcing. The technique is tested on simulated nonstationary data and applied to human heart rate variability data.

Generalized splay state in coupled chaotic oscillators induced by weak mutual resonant interactions

Dynamic behavior of coupled chaotic oscillators is investigated. A transition from high-dimensional hyperchaos to a generalized periodic splay state is found for extremely weak coupling. Chaotic nature of a single oscillator and mutual resonant interactions are regarded to be responsible for this self-organized ordering. The functional phase distribution of the generalized splay state, which is essentially different from the equal-phase-separation distribution of the conventional splay states, can be well predicted by analyzing a single periodically forced oscillator.

Musicians and the gamma band: a secret affair?

While listening to music, a significant high degree of phase synchrony in the gamma frequency range globally distributed over the brain was found in subjects with musical training (musicians) compared with subjects with no such training (non-musicians). No significant differences were found in other EEG frequency bands. Listening to neutral text did not produce any significant differences in the degree of synchronization between these two groups. For musicians, left-hemispheric dominance was found during listening to music. The right hemisphere was found to be dominant for non-musicians in text listening. The high degree of synchronization in musicians could be due to their high ability to retrieve musical patterns from their acoustic memory, which is a cogent condition for both listening to and anticipating musical sounds.

Synchronization regimes in coupled noisy excitable systems

We study synchronization regimes in a system of two coupled noisy excitable systems which exhibit excitability close to an Andronov bifurcation. The uncoupled system possesses three fixed points: a node, a saddle, and an unstable focus. We demonstrate that with an increase of coupling strength the system undergoes transitions from a desynchronous state to a train synchronization regime to a phase synchronization regime, and then to a complete synchronization regime. Train synchronization is a consequence of the existence of a saddle in the phase space. The mechanism of transitions in coupled noisy excitable systems is different from that in coupled phase-coherent chaotic systems.