Ring intermittency in coupled chaotic oscillators at the boundary of phase synchronization

Wavelet analysis of intermittent dynamics in nocturnal electrocardiography and electroencephalography data

This paper presents the results of a study of the characteristics of phase synchronization between electrocardiography(ECG) and electroencephalography (EEG) signals during night sleep. Polysomnographic recordings of eight generally healthy subjects and eight patients with obstructive sleep apnea syndrome were selected as experimental data. A feature of this study was the introduction of an instantaneous phase for EEG and ECG signals using a continuous wavelet transform at the heart rate frequency using the concept of time scale synchronization, which eliminated the emergence of asynchronous areas of behavior associated with the "leaving" of the fundamental frequency of the cardiovascular system. Instantaneous phase differences were examined for various pairs of EEG and ECG signals during night sleep, and it was shown that in all cases the phase difference exhibited intermittency. Laminar areas of behavior are intervals of phase synchronization, i.e., phase capture. Turbulent intervals are phase jumps of 2π. Statistical studies of the observed intermittent behavior were carried out, namely, distributions of the duration of laminar sections of behavior were estimated. For all pairs of channels, the duration of laminar phases obeyed an exponential law. Based on the analysis of the movement of the phase trajectory on a rotating plane at the moment of detection of the turbulent phase, it was established that in this case the eyelet intermittency was observed. There was no connection between the statistical characteristics of laminar phase distributions for intermittent behavior and the characteristics of night breathing disorders (apnea syndrome). It was found that changes in statistical characteristics in the phase synchronization of EEG and ECG signals were correlated with blood pressure at the time of signal recording in the subjects, which is an interesting effect that requires further research.

Intermittent generalized synchronization and modified system approach: Discrete maps

The present work deals with the intermittent generalized synchronization regime observed near the boundary of generalized synchronization. The intermittent behavior is considered in the context of two observable phenomena, namely, (i) the birth of the asynchronous stages of motion from the complete synchronous state and (ii) the multistability in detection of the synchronous and asynchronous states. The mechanisms governing these phenomena are revealed and described in this paper with the help of the modified system approach for unidirectionally coupled model oscillators with discrete time.

Self-organized bistability on globally coupled higher-order networks

Self-organized bistability (SOB) stands as a critical behavior for the systems delicately adjusting themselves to the brink of bistability, characterized by a first-order transition. Its essence lies in the inherent ability of the system to undergo enduring shifts between the coexisting states, achieved through the self-regulation of a controlling parameter. Recently, SOB has been established in a scale-free network as a recurrent transition to a short-living state of global synchronization. Here, we embark on a theoretical exploration that extends the boundaries of the SOB concept on a higher-order network (implicitly embedded microscopically within a simplicial complex) while considering the limitations imposed by coupling constraints. By applying Ott-Antonsen dimensionality reduction in the thermodynamic limit to the higher-order network, we derive SOB requirements under coupling limits that are in good agreement with numerical simulations on systems of finite size. We use continuous synchronization diagrams and statistical data from spontaneous synchronized events to demonstrate the crucial role SOB plays in initiating and terminating temporary synchronized events. We show that under weak-coupling consumption, these spontaneous occurrences closely resemble the statistical traits of the epileptic brain functioning.

Analysis of the Type V Intermittency Using the Perron-Frobenius Operator

A methodology to study the reinjection process in type V intermittency is introduced. The reinjection probability density function (RPD), and the probability density of the laminar lengths (RPDL) for type V intermittency are calculated. A family of maps with discontinuous and continuous RPD functions is analyzed. Several tests were performed, in which the proposed technique was compared with the classical theory of intermittency, the M function methodology, and numerical data. The analysis exposed that the new technique can accurately capture the numerical data. Therefore, the scheme presented herein is a useful tool to theoretically evaluate the statistical variables for type V intermittency.

Self-organized bistability on scale-free networks

A dynamical system approaching the first-order transition can exhibit a specific type of critical behavior known as self-organized bistability (SOB). It lies in the fact that the system can permanently switch between the coexisting states under the self-tuning of a control parameter. Many of these systems have a network organization that should be taken into account to understand the underlying processes in detail. In the present paper, we theoretically explore an extension of the SOB concept on the scale-free network under coupling constraints. As provided by the numerical simulations and mean-field approximation in the thermodynamic limit, SOB on scale-free networks originates from facilitated criticality reflected on both macro- and mesoscopic network scales. We establish that the appearance of switches is rooted in spatial self-organization and temporal self-similarity of the network's critical dynamics and replicates extreme properties of epileptic seizure recurrences. Our results, thus, indicate that the proposed conceptual model is suitable to deepen the understanding of emergent collective behavior behind neurological diseases.

Intermittency Reinjection in the Logistic Map

Just below a Period-3 window, the logistic map exhibits intermittency. Then, the third iterate of this map has been widely used to explain the chaotic intermittency concept. Much attention has been paid to describing the behavior around the vanished fixed points, the tangent bifurcation, and the formation of the characteristic channel between the map and the diagonal for type-I intermittency. However, the reinjection mechanism has not been deeply analyzed. In this paper, we studied the reinjection processes for the three fixed points around which intermittency is generated. We calculated the reinjection probability density function, the probability density of the laminar lengths, and the characteristic relation. We found that the reinjection mechanisms have broader behavior than the usually used uniform reinjection. Furthermore, the reinjection processes depend on the fixed point.

Calculation of the Statistical Properties in Intermittency Using the Natural Invariant Density

We use the natural invariant density of the map and the Perron–Frobenius operator to analytically evaluate the statistical properties for chaotic intermittency. This study can be understood as an improvement of the previous ones because it does not introduce assumptions about the reinjection probability density function in the laminar interval or the map density at pre-reinjection points. To validate the new theoretical equations, we study a symmetric map and a non-symmetric one. The cusp map has symmetry about x=0, but the Manneville map has no symmetry. We carry out several comparisons between the theoretical equations here presented, the M function methodology, the classical theory of intermittency, and numerical data. The new theoretical equations show more accuracy than those calculated with other techniques.

Intermittent route to generalized synchronization in bidirectionally coupled chaotic oscillators

The type of transition from asynchronous behavior to the generalized synchronization regime in mutually coupled chaotic oscillators has been studied. To separate the epochs of the synchronous and asynchronous motion in time series of mutually coupled chaotic oscillators, a method based on the local Lyapunov exponent calculation has been proposed. The efficiency of the method has been testified using the examples of unidirectionally coupled dynamical systems for which the type of transition is well known. The transition to generalized synchronization regime in mutually coupled systems has been shown to be an on-off intermittency as well as in the case of the unidirectional coupling.

Jump intermittency as a second type of transition to and from generalized synchronization

The transition from asynchronous dynamics to generalized chaotic synchronization and then to completely synchronous dynamics is known to be accompanied by on-off intermittency. We show that there is another (second) type of the transition called jump intermittency which occurs near the boundary of generalized synchronization in chaotic systems with complex two-sheeted attractors. Although this transient behavior also exhibits intermittent dynamics, it differs sufficiently from on-off intermittency supposed hitherto to be the only type of motion corresponding to the transition to generalized synchronization. This type of transition has been revealed and the underling mechanism has been explained in both unidirectionally and mutually coupled chaotic Lorenz and Chen oscillators. To detect the epochs of synchronous and asynchronous motion in mutually coupled oscillators with complex topology of an attractor a technique based on finding time intervals when the phase trajectories are located on equal or different sheets of chaotic attractors of coupled oscillators has been developed. We have also shown that in the unidirectionally coupled systems the proposed technique gives the same results that may obtained with the help of the traditional method using the auxiliary system approach.

Recurrent synchronization of coupled oscillators with spontaneous phase reformation

Self-organizing and spontaneous breaking are seemingly opposite phenomena and hardly captured in a single model. We develop a second order Kuramoto model with phase-induced damping which shows phase locking together with spontaneous synchrony breaking and reformation. In a relatively large regime where the interacting force and the damping ratio are of the same order, the dynamics of the oscillators alternates in an irregular cycle of synchronization, formation-breaking, and reorganization. While the oscillators keep coming back to phase-locked states, their phase distribution repeatedly reforms. Also, the interevent time between bursty deviation from the synchronization states follows a power-law distribution, which implies that the synchronized states are maintained near a tipping point.

Separation of coexisting dynamical regimes in multistate intermittency based on wavelet spectrum energies in an erbium-doped fiber laser

We propose a method for the detection and localization of different types of coexisting oscillatory regimes that alternate with each other leading to multistate intermittency. Our approach is based on consideration of wavelet spectrum energies. The proposed technique is tested in an erbium-doped fiber laser with four coexisting periodic orbits, where external noise induces intermittent switches between the coexisting states. Statistical characteristics of multistate intermittency, such as the mean duration of the phases for every oscillation type, are examined with the help of the developed method. We demonstrate strong advantages of the proposed technique over previously used amplitude methods.

Coexistence of intermittencies in the neuronal network of the epileptic brain

Intermittent behavior occurs widely in nature. At present, several types of intermittencies are known and well-studied. However, consideration of intermittency has usually been limited to the analysis of cases when only one certain type of intermittency takes place. In this paper, we report on the temporal behavior of the complex neuronal network in the epileptic brain, when two types of intermittent behavior coexist and alternate with each other. We prove the presence of this phenomenon in physiological experiments with WAG/Rij rats being the model living system of absence epilepsy. In our paper, the deduced theoretical law for distributions of the lengths of laminar phases prescribing the power law with a degree of -2 agrees well with the experimental neurophysiological data.

Lyapunov exponent corresponding to enslaved phase dynamics: Estimation from time series

A method for the estimation of the Lyapunov exponent corresponding to enslaved phase dynamics from time series has been proposed. It is valid for both nonautonomous systems demonstrating periodic dynamics in the presence of noise and coupled chaotic oscillators and allows us to estimate precisely enough the value of this Lyapunov exponent in the supercritical region of the control parameters. The main results are illustrated with the help of the examples of the noised circle map, the nonautonomous Van der Pole oscillator in the presence of noise, and coupled chaotic Rössler systems.

Intermittency of intermittencies

A phenomenon of intermittency of intermittencies is discovered in the temporal behavior of two coupled complex systems. We observe for the first time the coexistence of two types of intermittent behavior taking place simultaneously near the boundary of the synchronization regime of coupled chaotic oscillators. This phenomenon is found both in the numerical and physiological experiments. The laws for both the distribution and mean length of laminar phases versus the control parameter values are analytically deduced. A very good agreement between the theoretical results and simulation is shown.

Horizontal visibility graphs generated by type-I intermittency

The type-I intermittency route to (or out of) chaos is investigated within the horizontal visibility (HV) graph theory. For that purpose, we address the trajectories generated by unimodal maps close to an inverse tangent bifurcation and construct their associated HV graphs. We show how the alternation of laminar episodes and chaotic bursts imprints a fingerprint in the resulting graph structure. Accordingly, we derive a phenomenological theory that predicts quantitative values for several network parameters. In particular, we predict that the characteristic power-law scaling of the mean length of laminar trend sizes is fully inherited by the variance of the graph degree distribution, in good agreement with the numerics. We also report numerical evidence on how the characteristic power-law scaling of the Lyapunov exponent as a function of the distance to the tangent bifurcation is inherited in the graph by an analogous scaling of block entropy functionals defined on the graph. Furthermore, we are able to recast the full set of HV graphs generated by intermittent dynamics into a renormalization-group framework, where the fixed points of its graph-theoretical renormalization-group flow account for the different types of dynamics. We also establish that the nontrivial fixed point of this flow coincides with the tangency condition and that the corresponding invariant graph exhibits extremal entropic properties.

Generalized synchronization in mutually coupled oscillators and complex networks

We introduce a concept of generalized synchronization, able to encompass the setting of collective synchronized behavior for mutually coupled systems and networking systems featuring complex topologies in their connections. The onset of the synchronous regime is confirmed by the dependence of the system's Lyapunov exponents on the coupling parameter. The presence of a generalized synchronization regime is verified by means of the nearest neighbor method.

On-off intermittency of thalamo-cortical oscillations in the electroencephalogram of rats with genetic predisposition to absence epilepsy

Spike-wave discharges (SWD) are electroencephalographic hallmarks of absence epilepsy. SWD are known to originate from thalamo-cortical neuronal network that normally produces sleep spindle oscillations. Although sleep spindles and SWD are considered as thalamo-cortical oscillations, functional relationship between them is not obvious. The present study describes temporal dynamics of SWD and sleep spindles as determined in 24h EEG recorded in WAG/Rij rat model of absence epilepsy. SWD, sleep spindles (10-15 Hz) and 5-9 Hz oscillations were automatically detected in EEG using wavelet-based algorithm. It was found that non-linear dynamics of SWD fitted well to the law of 'on-off intermittency'. Sleep spindles also demonstrated 'on-off intermittency', in contrast to 5-9 Hz oscillations, whose dynamics could not be classified as having any known type of non-linear behavior. Intermittency in sleep spindles and SWD implies that (1) temporal dynamics of these oscillations are deterministic in nature, and (2) it might be controlled by a system-level mechanism responsible for circadian modulation of neuronal network activity.

Detecting the temporal structure of intermittent phase locking

This study explores a method to characterize the temporal structure of intermittent phase locking in oscillatory systems. When an oscillatory system is in a weakly synchronized regime away from a synchronization threshold, it spends most of the time in parts of its phase space away from the synchronization state. Therefore characteristics of dynamics near this state (such as its stability properties and Lyapunov exponents or distributions of the durations of synchronized episodes) do not describe the system's dynamics for most of the time. We consider an approach to characterize the system dynamics in this case by exploring the relationship between the phases on each cycle of oscillations. If some overall level of phase locking is present, one can quantify when and for how long phase locking is lost, and how the system returns back to the phase-locked state. We consider several examples to illustrate this approach: coupled skewed tent maps, the stability of which can be evaluated analytically; coupled Rössler and Lorenz oscillators, undergoing through different intermittency types on the way to phase synchronization; and a more complex example of coupled neurons. We show that the obtained measures can describe the differences in the dynamics and temporal structure of synchronization and desynchronization events for the systems with a similar overall level of phase locking and similar stability of the synchronized state.

Characterizing the phase synchronization transition of chaotic oscillators

The chaotic phase synchronization transition is studied in connection with the zero Lyapunov exponent. We propose a hypothesis that it is associated with a switching of the maximal finite-time zero Lyapunov exponent, which is introduced in the framework of a large deviation analysis. A noisy sine circle map is investigated to introduce this hypothesis and it is tested in an unidirectionally coupled Rössler system by using the covariant Lyapunov vector associated with the zero Lyapunov exponent.

Ring intermittency near the boundary of the synchronous time scales of chaotic oscillators

In this Brief Report we study both experimentally and numerically the intermittent behavior taking place near the boundary of the synchronous time scales of chaotic oscillators being in the regime of time scale synchronization. We have shown that the observed type of the intermittent behavior should be classified as the ring intermittency.

Scaling and synchronization in a ring of diffusively coupled nonlinear oscillators

Chaos synchronization in a ring of diffusively coupled nonlinear oscillators driven by an external identical oscillator is studied. Based on numerical simulations we show that by introducing additional couplings at (mN(c)+1)-th oscillators in the ring, where m is an integer and N(c) is the maximum number of synchronized oscillators in the ring with a single coupling, the maximum number of oscillators that can be synchronized can be increased considerably beyond the limit restricted by size instability. We also demonstrate that there exists an exponential relation between the number of oscillators that can support stable synchronization in the ring with the external drive and the critical coupling strength ε(c) with a scaling exponent γ. The critical coupling strength is calculated by numerically estimating the synchronization error and is also confirmed from the conditional Lyapunov exponents of the coupled systems. We find that the same scaling relation exists for m couplings between the drive and the ring. Further, we have examined the robustness of the synchronous states against Gaussian white noise and found that the synchronization error exhibits a power-law decay as a function of the noise intensity indicating the existence of both noise-enhanced and noise-induced synchronizations depending on the value of the coupling strength ε. In addition, we have found that ε(c) shows an exponential decay as a function of the number of additional couplings. These results are demonstrated using the paradigmatic models of Rössler and Lorenz oscillators.

Queue-length synchronization in communication networks

We study the synchronization in the context of network traffic on a 2-d communication network with local clustering and geographic separations. The network consists of nodes and randomly distributed hubs where the top five hubs ranked according to their coefficient of betweenness centrality (CBC) are connected by random assortative and gradient mechanisms. For multiple message traffic, messages can trap at the high CBC hubs, and congestion can build up on the network with long queues at the congested hubs. The queue lengths are seen to synchronize in the congested phase. Both complete and phase synchronization are seen, between pairs of hubs. In the decongested phase, the pairs start clearing and synchronization is lost. A cascading master-slave relation is seen between the hubs, with the slower hubs (which are slow to decongest) driving the faster ones. These are usually the hubs of high CBC. Similar results are seen for traffic of constant density. Total synchronization between the hubs of high CBC is also seen in the congested regime. Similar behavior is seen for traffic on a network constructed using the Waxman random topology generator. We also demonstrate the existence of phase synchronization in real internet traffic data.

Zero Lyapunov exponent in the vicinity of the saddle-node bifurcation point in the presence of noise

We consider a behavior of the zero Lyapunov exponent in the vicinity of the bifurcation point that occurs as the result of the interplay between dynamical mechanisms and random dynamics. We analytically deduce the laws for the dependence of this Lyapunov exponent on the control parameter both above and below the bifurcation point. The developed theory is applicable both to the systems with the random force and to the deterministic chaotic oscillators. We find an excellent agreement between the theoretical predictions and the data obtained by means of numerical calculations. We also discuss how the revealed regularities are expected to take place in other relevant physical circumstances.

Length distribution of laminar phases for type-I intermittency in the presence of noise

We consider a type of intermittent behavior that occurs as the result of the interplay between dynamical mechanisms giving rise to type-I intermittency and random dynamics. We analytically deduce the laws for the distribution of the laminar phases, with the law for the mean length of the laminar phases versus the critical parameter deduced earlier [W.-H. Kye and C.-M. Kim, Phys. Rev. E 62, 6304 (2000)] being the corollary fact of the developed theory. We find a very good agreement between the theoretical predictions and the data obtained by means of both the experimental study and numerical calculations. We discuss also how this mechanism is expected to take place in other relevant physical circumstances.

Two types of phase synchronization destruction

In this paper we report that there are two different types of destruction of the phase synchronization (PS) regime of chaotic oscillators depending on the parameter mismatch as well as in the case of the classical synchronization of periodic oscillators. When the parameter mismatch of the interacting chaotic oscillators is small enough, the PS breaking takes place without the destruction of the phase coherence of chaotic attractors of oscillators. Alternatively, due to the large frequency detuning, the PS breaking is accomplished by loss of the phase coherence of the chaotic attractors.