Transition from intermittency to periodicity in lag synchronization in coupled Rössler oscillators

Collective dynamics of coupled Lorenz oscillators near the Hopf boundary: Intermittency and chimera states

We study collective dynamics of networks of mutually coupled identical Lorenz oscillators near a subcritical Hopf bifurcation. Such systems exhibit induced multistable behavior with interesting spatiotemporal dynamics including synchronization, desynchronization, and chimera states. For analysis, we first consider a ring topology with nearest-neighbor coupling and find that the system may exhibit intermittent behavior due to the complex basin structures and dynamical frustration, where temporal dynamics of the oscillators in the ensemble switches between different attractors. Consequently, different oscillators may show a dynamics that is intermittently synchronized (or desynchronized), giving rise to intermittent chimera states. The behavior of the intermittent laminar phases is characterized by the characteristic time spent in the synchronization manifold, which decays as a power law. Such intermittent dynamics is quite general and is also observed in an ensemble of a large number of oscillators arranged in variety of network topologies including nonlocal, scale-free, random, and small-world networks.

Heterogeneity induced splay state of amplitude envelope in globally coupled oscillators

Splay states of the amplitude envelope are stably observed as a heterogenous node is introduced into the globally coupled identical oscillators with repulsive coupling. With the increment of the frequency mismatches between the heterogenous nodes and the rest identical globally coupled oscillators, the formal stable splay state based on the time series becomes unstable, while a splay state based on the new-born amplitude envelopes of time series is stably observed among the rest identical oscillators. The characteristics of the splay state based on the amplitude envelope are numerically and theoretically presented for different parameters of the coupling strength ϵ and the frequency mismatches Δω for small coupling strength and large frequency mismatches. We expect that all these results could reveal the generality of splay states in coupled nonidentical oscillators and help to understand the rich dynamics of amplitude envelopes in multidisciplinary fields.

Sinusoidal and nonsinusoidal patterns in amplitude envelope synchronization

In this work, amplitude envelope synchronization (AES), as a general phenomenon characterized with highly correlated amplitude envelope but uncorrelated phases and frequencies in coupled nonidentical nonlinear systems, is investigated theoretically and numerically. Two different types of AES patterns, including sinusoidal and nonsinusoidal, are widely observable in coupled periodic and/or chaotic oscillators. They both come from modulation of phase mismatch on amplitude but show different patterns due to different behaviors of phase mismatch. With increase of frequency mismatch, the system tends to crossover from nonsinusoidal to sinusoidal AES. With the aid of synchronization manifold and transverse stability analyses of the AES state, the physical mechanism and scale relations for the AES are well revealed. We expect that all these results could uncover the generality of AES in coupled nonlinear oscillators and help to understand the rich dynamics of phase and amplitude coupling in multidisciplinary fields.

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.

Perturbation analysis and comparison of network synchronization methods

In many networked systems, synchronization is important and useful, and how to enhance synchronizability is an interesting problem. Based on the matrix perturbation theory, we analyze five methods of network synchronization enhancement, including the link removal, node removal, dividing hub node, pull control, and pinning control methods, and obtain explicit expressions for eigenvalue changes. By these comparisons, we find that, among all these methods, the pull control method is remarkable, as it can extend the synchronization (coupling strength) region from both the left and right sides, for any controlled node. Extensive simulation results are given to support the accuracy of the perturbation-based analysis.

Anesthetic action on the transmission delay between cortex and thalamus explains the beta-buzz observed under propofol anesthesia

In recent years, more and more surgeries under general anesthesia have been performed with the assistance of electroencephalogram (EEG) monitors. An increase in anesthetic concentration leads to characteristic changes in the power spectra of the EEG. Although tracking the anesthetic-induced changes in EEG rhythms can be employed to estimate the depth of anesthesia, their precise underlying mechanisms are still unknown. A prominent feature in the EEG of some patients is the emergence of a strong power peak in the β-frequency band, which moves to the α-frequency band while increasing the anesthetic concentration. This feature is called the beta-buzz. In the present study, we use a thalamo-cortical neural population feedback model to reproduce observed characteristic features in frontal EEG power obtained experimentally during propofol general anesthesia, such as this beta-buzz. First, we find that the spectral power peak in the α- and δ-frequency ranges depend on the decay rate constant of excitatory and inhibitory synapses, but the anesthetic action on synapses does not explain the beta-buzz. Moreover, considering the action of propofol on the transmission delay between cortex and thalamus, the model reveals that the beta-buzz may result from a prolongation of the transmission delay by increasing propofol concentration. A corresponding relationship between transmission delay and anesthetic blood concentration is derived. Finally, an analytical stability study demonstrates that increasing propofol concentration moves the systems resting state towards its stability threshold.

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.

How the cortico-thalamic feedback affects the EEG power spectrum over frontal and occipital regions during propofol-induced sedation

Increasing concentrations of the anaesthetic agent propofol initially induces sedation before achieving full general anaesthesia. During this state of anaesthesia, the observed specific changes in electroencephalographic (EEG) rhythms comprise increased activity in the δ- (0.5-4 Hz) and α- (8-13 Hz) frequency bands over the frontal region, but increased δ- and decreased α-activity over the occipital region. It is known that the cortex, the thalamus, and the thalamo-cortical feedback loop contribute to some degree to the propofol-induced changes in the EEG power spectrum. However the precise role of each structure to the dynamics of the EEG is unknown. In this paper we apply a thalamo-cortical neuronal population model to reproduce the power spectrum changes in EEG during propofol-induced anaesthesia sedation. The model reproduces the power spectrum features observed experimentally both in frontal and occipital electrodes. Moreover, a detailed analysis of the model indicates the importance of multiple resting states in brain activity. The work suggests that the α-activity originates from the cortico-thalamic relay interaction, whereas the emergence of δ-activity results from the full cortico-reticular-relay-cortical feedback loop with a prominent enforced thalamic reticular-relay interaction. This model suggests an important role for synaptic GABAergic receptors at relay neurons and, more generally, for the thalamus in the generation of both the δ- and the α- EEG patterns that are seen during propofol anaesthesia sedation.

How to induce multiple delays in coupled chaotic oscillators?

Lag synchronization is a basic phenomenon in mismatched coupled systems, delay coupled systems, and time-delayed systems. It is characterized by a lag configuration that identifies a unique time shift between all pairs of similar state variables of the coupled systems. In this report, an attempt is made how to induce multiple lag configurations in coupled systems when different pairs of state variables attain different time shift. A design of coupling is presented to realize this multiple lag synchronization. Numerical illustration is given using examples of the Rössler system and the slow-fast Hindmarsh-Rose neuron model. The multiple lag scenario is physically realized in an electronic circuit of two Sprott systems.

Generalized variable projective synchronization of time delayed systems

We study generalized variable projective synchronization between two unified time delayed systems with constant and modulated time delays. A novel Krasovskii-Lyapunov functional is constructed and a generalized sufficient condition for synchronization is derived analytically using the Lyapunov stability theory and adaptive techniques. The proposed scheme is valid for a system of n-numbers of first order delay differential equations. Finally, a new neural oscillator is considered as a numerical example to show the effectiveness of the proposed scheme.

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.

Generating and enhancing lag synchronization of chaotic systems by white noise

In this paper, we study the crucial impact of white noise on lag synchronous regime in a pair of time-delay unidirectionally coupled systems. Our result demonstrates that merely via white-noise-based coupling lag synchronization could be achieved between the coupled systems (chaotic or not). And it is also demonstrated that a conventional lag synchronous regime can be enhanced by white noise. Sufficient conditions are further proved mathematically for noise-inducing and noise-enhancing lag synchronization, respectively. Additionally, the influence of parameter mismatch on the proposed lag synchronous regime is studied, by which we announce the robustness and validity of the new strategy. Two numerical examples are provided to illustrate the validity and some possible applications of the theoretical result.

Synchronization transitions in coupled time-delay electronic circuits with a threshold nonlinearity

Experimental observations of typical kinds of synchronization transitions are reported in unidirectionally coupled time-delay electronic circuits with a threshold nonlinearity and two time delays, namely feedback delay τ(1) and coupling delay τ(2). We have observed transitions from anticipatory to lag via complete synchronization and their inverse counterparts with excitatory and inhibitory couplings, respectively, as a function of the coupling delay τ(2). The anticipating and lag times depend on the difference between the feedback and the coupling delays. A single stability condition for all the different types of synchronization is found to be valid as the stability condition is independent of both the delays. Further, the existence of different kinds of synchronizations observed experimentally is corroborated by numerical simulations and from the changes in the Lyapunov exponents of the coupled time-delay systems.

Inverse synchronizations in coupled time-delay systems with inhibitory coupling

Transitions between inverse anticipatory, inverse complete, and inverse lag synchronizations are shown to occur as a function of the coupling delay in unidirectionally coupled time-delay systems with inhibitory coupling. We have also shown that the same general asymptotic stability condition obtained using the Krasovskii-Lyapunov functional theory can be valid for the cases where (i) both the coefficients of the Delta(t) (error variable) and Delta(tau)=Delta(t-tau) (error variable with delay) terms in the error equation corresponding to the synchronization manifold are time independent and (ii) the coefficient of the Delta term is time independent, while that of the Delta(tau) term is time dependent. The existence of different kinds of synchronization is corroborated using similarity function, probability of synchronization, and also from changes in the spectrum of Lyapunov exponents of the coupled time-delay systems.

Generalized projective synchronization in time-delayed systems: nonlinear observer approach

In this paper, we consider the projective-anticipating, projective, and projective-lag synchronization in a unified coupled time-delay system via nonlinear observer design. A new sufficient condition for generalized projective synchronization is derived analytically with the help of Krasovskii-Lyapunov theory for constant and variable time-delay systems. The analytical treatment can give stable synchronization (anticipatory and lag) for a large class of time-delayed systems in which the response system's trajectory is forced to have an amplitude proportional to the drive system. The constant of proportionality is determined by the control law, not by the initial conditions. The proposed technique has been applied to synchronize Ikeda and prototype models by numerical simulation.

Transition from phase to generalized synchronization in time-delay systems

The notion of phase synchronization in time-delay systems, exhibiting highly non-phase-coherent attractors, has not been realized yet even though it has been well studied in chaotic dynamical systems without delay. We report the identification of phase synchronization in coupled nonidentical piecewise linear and in coupled Mackey-Glass time-delay systems with highly non-phase-coherent regimes. We show that there is a transition from nonsynchronized behavior to phase and then to generalized synchronization as a function of coupling strength. We have introduced a transformation to capture the phase of the non-phase-coherent attractors, which works equally well for both the time-delay systems. The instantaneous phases of the above coupled systems calculated from the transformed attractors satisfy both the phase and mean frequency locking conditions. These transitions are also characterized in terms of recurrence-based indices, namely generalized autocorrelation function P(t), correlation of probability of recurrence, joint probability of recurrence, and similarity of probability of recurrence. We have quantified the different synchronization regimes in terms of these indices. The existence of phase synchronization is also characterized by typical transitions in the Lyapunov exponents of the coupled time-delay systems.

Delay time modulation induced oscillating synchronization and intermittent anticipatory/lag and complete synchronizations in time-delay nonlinear dynamical systems

Existence of a new type of oscillating synchronization that oscillates between three different types of synchronizations (anticipatory, complete, and lag synchronizations) is identified in unidirectionally coupled nonlinear time-delay systems having two different time-delays, that is feedback delay with a periodic delay time modulation and a constant coupling delay. Intermittent anticipatory, intermittent lag, and complete synchronizations are shown to exist in the same system with identical delay time modulations in both the delays. The transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay with suitable stability condition is discussed. The intermittent anticipatory and lag synchronizations are characterized by the minimum of the similarity functions and the intermittent behavior is characterized by a universal asymptotic -32 power law distribution. It is also shown that the delay time carved out of the trajectories of the time-delay system with periodic delay time modulation cannot be estimated using conventional methods, thereby reducing the possibility of decoding the message by phase space reconstruction.

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

A new type of intermittent behavior is described to occur near the boundary of the phase synchronization regime of coupled chaotic oscillators. This mechanism, called ring intermittency, arises for sufficiently high initial mismatches in the frequencies of the two coupled systems. The laws for both the distribution and the mean length of the laminar phases versus the coupling strength are analytically deduced. Very good agreement between the theoretical results and the numerically calculated data is shown. We discuss how this mechanism is expected to take place in other relevant physical circumstances.

Phase synchronization in time-delay systems

Though the notion of phase synchronization has been well studied in chaotic dynamical systems without delay, it has not been realized yet in chaotic time-delay systems exhibiting non-phase-coherent hyperchaotic attractors. In this paper we report identification of phase synchronization in coupled time-delay systems exhibiting hyperchaotic attractor. We show that there is a transition from nonsynchronized behavior to phase and then to generalized synchronization as a function of coupling strength. These transitions are characterized by recurrence quantification analysis, by phase differences based on a transformation of the attractors, and also by the changes in the Lyapunov exponents. We have found these transitions in coupled piecewise linear and in Mackey-Glass time-delay systems.

Synchronization of chaotic systems with coexisting attractors

Synchronization of coupled oscillators exhibiting the coexistence of chaotic attractors is investigated, both numerically and experimentally. The route from the asynchronous motion to a completely synchronized state is characterized by the sequence of type-I and on-off intermittencies, intermittent phase synchronization, anticipated synchronization, and period-doubling phase synchronization.

Phase synchronization between two essentially different chaotic systems

In this paper, we numerically investigate phase synchronization between two coupled essentially different chaotic oscillators in drive-response configuration. It is shown that phase synchronization can be observed between two coupled systems despite the difference and the large frequency detuning between them. Moreover, the relation between phase synchronization and generalized synchronization is compared with that in coupled parametrically different systems. In the systems studied, it is found that phase synchronization occurs after generalized synchronization in coupled essentially different chaotic systems.

Transition from anticipatory to lag synchronization via complete synchronization in time-delay systems

The existence of anticipatory, complete, and lag synchronization in a single system having two different time delays, that is, feedback delay tau1 and coupling delay tau2, is identified. The transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay tau2 with a suitable stability condition is discussed. In particular, it is shown that the stability condition is independent of the delay times tau1 and tau2. Consequently, for a fixed set of parameters, all the three types of synchronizations can be realized. Further, the emergence of exact anticipatory, complete, or lag synchronization from the desynchronized state via approximate synchronization, when one of the system parameters (b2) is varied, is characterized by a minimum of the similarity function and the transition from on-off intermittency via periodic structure in the laminar phase distribution.

Synchronization of chaotic oscillators due to common delay time modulation

We have found a synchronization behavior between two identical chaotic systems when their delay times are modulated by a common irregular signal. This phenomenon is demonstrated both in two identical chaotic maps whose delay times are driven by a common chaotic or random signal and in two identical chaotic oscillators whose delay times are driven by a signal of another chaotic oscillator. We analyze the phenomenon by using the Lyapunov exponents and discuss it in relation to generalized synchronization.

Detection of mutual phase synchronization in multivariate signals and application to phase ensembles and chaotic data

This work presents a method for the detection of mutual phase synchronization in nonstationary time series. We show how the application of a cluster algorithm that considers spatiotemporal structures of data follows from the general condition of phase-synchronized data. In view of the topology of phasic data, we reformulate the K-means cluster algorithm on a flat torus and apply a segmentation index derived in an earlier work [A. Hutt and H. Riedel, Physica D 177, 203 (2003)]. This index is extended by means of averaging in order to reflect phase synchronization in ensembles of multivariate time series. The method is illustrated using simulated multivariate phase dynamics and arrays of chaotic systems, in which temporal segments of phase-synchronized states are registered. A comparison with results from an existing bivariate synchronization index reveals major advantages of our method.

Complete synchronization and generalized synchronization of one-way coupled time-delay systems

The complete synchronization and generalized synchronization (GS) of one-way coupled time-delay systems are studied. We find that GS can be achieved by a single scalar signal, and its synchronization threshold for different delay times shows the parameter resonance effect, i.e., we can obtain stable synchronization at a smaller coupling if the delay time of the driven system is chosen such that it is in resonance with the driving system. Near chaos synchronization, the desynchronization dynamics displays periodic bursts with the period equal to the delay time of the driven system. These features can be easily applied to the recovery of time-delay systems.

Role of unstable periodic orbits in phase and lag synchronization between coupled chaotic oscillators

An increase of the coupling strength in the system of two coupled Rössler oscillators leads from a nonsynchronized state through phase synchronization to the regime of lag synchronization. The role of unstable periodic orbits in these transitions is investigated. Changes in the structure of attracting sets are discussed. We demonstrate that the onset of phase synchronization is related to phase-lockings on the surfaces of unstable tori, whereas transition from phase to lag synchronization is preceded by a decrease in the number of unstable periodic orbits.

Analysis of generalized synchronization in directionally coupled chaotic phase-coherent oscillators by local minimal fluctuations

The method of local-minimum fluctuation is proposed to analyze generalized synchronization in directionally coupled chaotic phase-coherent oscillators. It is shown that the emergence of generalized synchronization is manifested by the qualitative changes in the statistic of local minimum fluctuations of the receiver oscillator.