Synchronization of chaotic structurally nonequivalent systems

Generalized synchronization mediated by a flat coupling between structurally nonequivalent chaotic systems

Synchronization of chaotic systems is usually investigated for structurally equivalent systems typically coupled through linear diffusive functions. Here, we focus on a particular type of coupling borrowed from a nonlinear control theory and based on the optimal placement of a sensor-a device measuring the chosen variable-and an actuator-a device applying the actuating (control) signal to a variable's derivative-in the response system, leading to the so-called flat control law. We aim to investigate the dynamics produced by a response system that is flat coupled to a drive system and to determine the degree of generalized synchronization between them using statistical and topological arguments. The general use of a flat control law for getting generalized synchronization is discussed.

Periodicity characterization of the nonlinear magnetization dynamics

In this work, we study numerically the periodicity of regular regions embedded in chaotic states for the case of an anisotropic magnetic particle. The particle is in the monodomain regime and subject to an applied magnetic field that depends on time. The dissipative Landau-Lifshitz-Gilbert equation models the particle. To perform the characterization, we compute several two-dimensional phase diagrams in the parameter space for the Lyapunov exponents and the isospikes. We observe multiple transitions among periodic states, revealing complex topological structures in the parameter space typical of dynamic systems. To show the finer details of the regular structures, iterative zooms are performed. In particular, we find islands of synchronization for the magnetization and the driven field and several shrimp structures with different periods.

Synchronizing spatio-temporal chaos with imperfect models: a stochastic surface growth picture

We study the synchronization of two spatially extended dynamical systems where the models have imperfections. We show that the synchronization error across space can be visualized as a rough surface governed by the Kardar-Parisi-Zhang equation with both upper and lower bounding walls corresponding to nonlinearities and model discrepancies, respectively. Two types of model imperfections are considered: parameter mismatch and unresolved fast scales, finding in both cases the same qualitative results. The consistency between different setups and systems indicates that the results are generic for a wide family of spatially extended systems.

Binding under conflict conditions: state-space analysis of multivariate EEG synchronization

Real-world objects are often endowed with features that violate Gestalt principles. In our experiment, we examined the neural correlates of binding under conflict conditions in terms of the binding-by-synchronization hypothesis. We presented an ambiguous stimulus ("diamond illusion") to 12 observers. The display consisted of four oblique gratings drifting within circular apertures. Its interpretation fluctuates between bound ("diamond") and unbound (component gratings) percepts. To model a situation in which Gestalt-driven analysis contradicts the perceptually explicit bound interpretation, we modified the original diamond (OD) stimulus by speeding up one grating. Using OD and modified diamond (MD) stimuli, we managed to dissociate the neural correlates of Gestalt-related (OD vs. MD) and perception-related (bound vs. unbound) factors. Their interaction was expected to reveal the neural networks synchronized specifically in the conflict situation. The synchronization topography of EEG was analyzed with the multivariate S-estimator technique. We found that good Gestalt (OD vs. MD) was associated with a higher posterior synchronization in the beta-gamma band. The effect of perception manifested itself as reciprocal modulations over the posterior and anterior regions (theta/beta-gamma bands). Specifically, higher posterior and lower anterior synchronization supported the bound percept, and the opposite was true for the unbound percept. The interaction showed that binding under challenging perceptual conditions is sustained by enhanced parietal synchronization. We argue that this distributed pattern of synchronization relates to the processes of multistage integration ranging from early grouping operations in the visual areas to maintaining representations in the frontal networks of sensory memory.

The development of generalized synchronization on complex networks

In this paper, we numerically investigate the development of generalized synchronization (GS) on typical complex networks, such as scale-free networks, small-world networks, random networks, and modular networks. By adopting the auxiliary-system approach to networks, we observe that GS generally takes place in oscillator networks with both heterogeneous and homogeneous degree distributions, regardless of whether the coupled chaotic oscillators are identical or nonidentical. We show that several factors, such as the network topology, the local dynamics, and the specific coupling strategies, can affect the development of GS on complex networks.

Synchronization of unidirectionally coupled Mackey-Glass analog circuits with frequency bandwidth limitations

Synchronization of chaotic systems has been studied extensively, and especially, the possible applications to the communication systems motivated many research areas. We demonstrate the effect of the frequency bandwidth limitations in the communication channel on the synchronization of two unidirectionally coupled Mackey-Glass (MG) analog circuits, both numerically and experimentally. MG system is known to generate high dimensional chaotic signals. The chaotic signal generated from the drive MG system is modified by a low pass filter and is then transmitted to the response MG system. Our results show that the inclusion of the dominant frequency component of the original drive signals is crucial to achieve synchronization between the drive and response circuits. The maximum cross correlation and the corresponding time shift reveal that the frequency-dependent coupling introduced by the low pass filtering effect in the communication channel change the quality of synchronization.

Spurious detection of phase synchronization in coupled nonlinear oscillators

Coupled nonlinear systems under certain conditions exhibit phase synchronization, which may change for different frequency bands or with the presence of additive system noise. In both cases, Fourier filtering is traditionally used to preprocess data. We investigate to what extent the phase synchronization of two coupled Rössler oscillators depends on (1) the broadness of their power spectrum, (2) the width of the bandpass filter, and (3) the level of added noise. We find that for identical coupling strengths, oscillators with broader power spectra exhibit weaker synchronization. Further, we find that within a broad bandwidth range, bandpass filtering reduces the effect of noise but can lead to a spurious increase in the degree of phase synchronization with narrowing bandwidth, even when the coupling between the two oscillators remains the same.

Minimization of the energy flow in the synchronization of nonidentical chaotic systems

We argue that maintaining a synchronized regime between different chaotic systems requires a net flow of energy between the guided system and an external energy source. This energy flow can be spontaneously reduced if the systems are flexible enough as to structurally approach each other through an adequate adaptive change in their parameter values. We infer that this reduction of energy can play a role in the synchronization of bursting neurons and other natural oscillators.

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.

Universal critical behavior of the synchronization transition in delayed chaotic systems

We numerically investigate the critical behavior of the synchronization transition of two unidirectionally coupled delayed chaotic systems. We map the problem to a spatially extended system to show that the synchronization transition in delayed systems exhibits universal critical properties. We find that the synchronization transition is absorbing and generically belongs to the universality class of the bounded Kardar-Parisi-Zhang equation, as occurs in the case of extended systems. We also argue that directed percolation critical behavior may emerge for systems with strong nonlinearities.

"Dynamic" connectivity in neural systems: theoretical and empirical considerations

The study of functional interdependences between brain regions is a rapidly growing focus of neuroscience research. This endeavor has been greatly facilitated by the appearance of a number of innovative methodologies for the examination of neurophysiological and neuroimaging data. The aim of this article is to present an overview of dynamical measures of interdependence and contrast these with statistical measures that have been more widely employed. We first review the motivation, conceptual basis, and experimental approach of dynamical measures of interdependence and their application to the study of neural systems. A consideration of boot-strap "surrogate data" techniques, which facilitate hypothesis testing of dynamical measures, is then used to clarify the difference between dynamical and statistical measures of interdependence. An overview of some of the most active research areas such as the study of the "synchronization manifold," dynamical interdependence in neurophysiology data and the putative role of nonlinear desynchronization is then given. We conclude by suggesting that techniques based on dynamical interdependence--or "dynamical connectivity"--show significant potential for extracting meaningful information from functional neuroimaging data.

Interdependency between heart rate variability and sleep EEG: linear/non-linear?

OBJECTIVE

To investigate whether the interdependency between heart rate variability (HRV) and sleep electroencephalogram (EEG) power spectra is linear or non-linear.

METHODS

Heart rate and sleep EEG signals were recorded in 8 healthy young men. Spectral analysis was applied to electrocardiogram and EEG sleep recordings. Synchronization likelihood was computed over the first 3 non-rapid eye movement-rapid eye movement sleep cycles between normalized high frequency of RR intervals (RRI) and all electroencephalographic frequency bands. Comparison to surrogate data of different types was used to attest statistical significance of the coupling between RRI and EEG power bands and its linear or non-linear character.

RESULTS

Synchronization likelihood values were statistically greater than univariate surrogate synchronization for all sleep bands both at the individual and the group levels. With reference to multivariate surrogates, synchronization values were statistically greater at the group level and, in a majority of cases, for individual comparison except for sigma and beta bands.

CONCLUSIONS

While all electroencephalographic power bands are linked to normalized high frequency RRI band, this interdependency is non-linear for delta, theta and alpha bands.

SIGNIFICANCE

Non-linear description is required to capture the full interdependent dynamics of HRV and sleep EEG power bands.

An approach to chaotic synchronization

This paper deals with the chaotic oscillator synchronization. An approach to the synchronization of chaotic oscillators has been proposed. This approach is based on the analysis of different time scales in the time series generated by the coupled chaotic oscillators. It has been shown that complete synchronization, phase synchronization, lag synchronization, and generalized synchronization are the particular cases of the synchronized behavior called "time-scale synchronization." The quantitative measure of chaotic oscillator synchronous behavior has been proposed. This approach has been applied for the coupled Rössler systems and two coupled Chua's circuits.

Irrational phase synchronization

We study the occurrence of physically observable phase locked states between chaotic oscillators and rotors in which the frequencies of the coupled systems are irrationally related. For two chaotic oscillators, the phenomenon occurs as a result of a coupling term which breaks the 2 pi invariance in the phase equations. In the case of rotors, a coupling term in the angular velocities results in very long times during which the coupled systems exhibit alternatively irrational phase synchronization and random phase diffusion. The range of parameters for which the phenomenon occurs contains an open set, and is thus physically observable.

Error function attack of chaos synchronization based encryption schemes

Different chaos synchronization based encryption schemes are reviewed and compared from the practical point of view. As an efficient cryptanalysis tool for chaos encryption, a proposal based on the error function attack is presented systematically and used to evaluate system security. We define a quantitative measure (quality factor) of the effective applicability of a chaos encryption scheme, which takes into account the security, the encryption speed, and the robustness against channel noise. A comparison is made of several encryption schemes and it is found that a scheme based on one-way coupled chaotic map lattices performs outstandingly well, as judged from quality factor.

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.

Synchronization of reconstructed dynamical systems

The problem of constructing synchronizing systems to observed signals is approached from a data driven perspective, in which it is assumed that neither the drive nor the response systems are known explicitly but have to be derived from the observations. The response systems are modeled by utilizing standard methods of nonlinear time series analysis applied to sections of the driving signals. As a result, synchronization is more robust than what might be expected, given that the reconstructed systems are only approximations of the unknown true systems. Successful synchronization also may be accomplished in cases where the driving signals result from nonlinearly transformed chaotic states. The method is readily extended and applied to limited real-time predictions of chaotic signals.

Controlling the complex Lorenz equations by modulation

We demonstrate that a system obeying the complex Lorenz equations in the deep chaotic regime can be controlled to periodic behavior by applying a modulation to the pump parameter. For arbitrary modulation frequency and amplitude there is no obvious simplification of the dynamics. However, we find that there are numerous windows where the chaotic system has been controlled to different periodic behaviors. The widths of these windows in parameter space are narrow, and the positions are related to the ratio of the modulation frequency of the pump to the average pulsation frequency of the output variable. These results are in good agreement with observations previously made in a far-infrared laser system.

Collective phase locked states in a chain of coupled chaotic oscillators

We discuss the emergence of a collective phase locked state in an open chain of N unidirectionally weakly coupled nonidentical chaotic oscillators. Such a regime is characterized by a Lyapunov spectrum where N-1 exponents that were zero in the uncoupled regime assume negative values as the coupling strength increases. The dynamics of such collective state is studied, and a comparison is drawn with the case of phase synchronization of a pair of coupled chaotic oscillators. In particular, it is shown that a full phase synchronized state cannot be constructed without at least partial correlation in the chaotic amplitudes.

n:m phase synchronization with mutual coupling phase signals

We generalize the n:m phase synchronization between two chaotic oscillators by mutual coupling phase signals. To characterize this phenomenon, we use two coupled oscillators to demonstrate their phase synchronization with amplitudes practically noncorrelated. We take the 1:1 phase synchronization as an example to show the properties of mean frequencies, mean phase difference, and Lyapunov exponents at various values of coupling strength. The phase difference increases with 2pi phase slips below the transition. The scaling rules of the slip near and away from the transition are studied. Furthermore, we demonstrate the transition to a variety of n:m phase synchronizations and analyze the corresponding coupling dynamics. (c) 2002 American Institute of Physics.

Reconstructing embedding spaces of coupled dynamical systems from multivariate data

A method for reconstructing dimensions of subspaces for weakly coupled dynamical systems is offered. The tool is able to extrapolate the subspace dimensions from the zero coupling limit, where the division of dimensions as per the algorithm is exact. Implementation of the proposed technique to multivariate data demonstrates its effectiveness in disentangling subspace dimensionalities also in the case of emergent synchronized motions, for both numerical and experimental systems.

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 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.

Synchronization of hyperchaotic harmonics in time-delay systems and its application to secure communication

We present a predictor-feedback method for synchronizing chaotic systems in this paper. By using this method, two structurally equivalent or nonequivalent systems can be synchronized very effectively and quickly. Moreover, the feedback perturbation can be switched on even if trajectories of the two systems are far from each other. Therefore, this method is applicable to real-world experimental systems, especially to some fast experimental systems. The validity of this method is demonstrated by synchronizing hyperchaotic harmonics in a time-delay system. As an application, we introduce how messages can be encoded, transmitted, and decoded using this technique. We suggest taking use of the multistability of time-delay systems to improve the performance of the secure communication.