Information theoretic approach to quantify complete and phase synchronization of chaos

Improving Entropy Estimates of Complex Network Topology for the Characterization of Coupling in Dynamical Systems

A new measure for the characterization of interconnected dynamical systems coupling is proposed. The method is based on the representation of time series as weighted cross-visibility networks. The weights are introduced as the metric distance between connected nodes. The structure of the networks, depending on the coupling strength, is quantified via the entropy of the weighted adjacency matrix. The method has been tested on several coupled model systems with different individual properties. The results show that the proposed measure is able to distinguish the degree of coupling of the studied dynamical systems. The original use of the geodesic distance on Gaussian manifolds as a metric distance, which is able to take into account the noise inherently superimposed on the experimental data, provides significantly better results in the calculation of the entropy, improving the reliability of the coupling estimates. The application to the interaction between the El Niño Southern Oscillation (ENSO) and the Indian Ocean Dipole and to the influence of ENSO on influenza pandemic occurrence illustrates the potential of the method for real-life problems.

Influence of chaotic synchronization on mixing in the phase space of interacting systems

Using the concept of the relative metric entropy, we study the influence of the synchronization phenomenon on mixing rate in the phase space of deterministic and noisy chaotic systems. We show that transition to both complete and phase synchronization of chaos is accompanied by the decrease of the level of mixing induced by internal nonlinear mechanisms of interacting systems as well as by external noise influence. Therefore, the decrease of the mixing rate in the phase space of interacting systems may indicate transition to synchronization. The obtained results are important for time series analysis in various types of real noisy systems (e.g., biological, social, and financial systems).

Synchronization of nonlinear systems under information constraints

A brief survey of control and synchronization under information constraints (limited information capacity of the coupling channel) is given. Limit possibilities of nonlinear observer-based synchronization systems with first-order coders or full-order coders are considered in more detail. The existing and new theoretical results for multidimensional drive-response Lurie systems (linear part plus nonlinearity depending only on measurable outputs) are presented. It is shown that the upper bound of the limit synchronization error (LSE) is proportional to the upper bound of the transmission error. As a consequence, the upper and lower bounds of LSE are proportional to the maximum coupling signal rate and inversely proportional to the information transmission rate (channel capacity). The analysis is extended to networks having a "chain," "star," or "star-chain" topology. Adaptive chaotic synchronization under information constraints is analyzed. The results are illustrated by example: master-slave synchronization of two chaotic Chua systems coupled via a channel with limited capacity.

Chaotic observer-based synchronization under information constraints

Limitations of observer-based synchronization systems under information constraints (limited information capacity of the coupling channel) are evaluated. We give theoretical analysis for multidimensional drive-response systems represented in the Lurie form (linear part plus nonlinearity depending only on measurable outputs). It is shown that the upper bound of the limit synchronization error (LSE) is proportional to the upper bound of the transmission error. As a consequence, the upper and lower bounds of LSE are proportional to the maximum rate of the coupling signal and inversely proportional to the information transmission rate (channel capacity). Optimality of the binary coding for coders with one-step memory is established. The results are applied to synchronization of two chaotic Chua systems coupled via a channel with limited capacity.

Dynamical noise filter and conditional entropy analysis in chaos synchronization

It is shown that, in a chaotic synchronization system whose driving signal is exposed to channel noise, the estimation of the drive system states can be greatly improved by applying the dynamical noise filtering to the response system states. If the noise is bounded in a certain range, the estimation errors, i.e., the difference between the filtered responding states and the driving states, can be made arbitrarily small. This property can be used in designing an alternative digital communication scheme. An analysis based on the conditional entropy justifies the application of dynamical noise filtering in generating quality synchronization.

Quantitative analysis of chaotic synchronization by means of coherence

We use an index of chaotic synchronization based on the averaged coherence function for the quantitative analysis of the process of the complete synchronization loss in unidirectionally coupled oscillators and maps. We demonstrate that this value manifests different stages of the synchronization breaking. It is invariant to time delay and insensitive to small noise and distortions, which can influence the accessible signals at measurements. Peculiarities of the synchronization destruction in maps and oscillators are investigated.

Onset of chaotic symbolic synchronization between population inversions in an array of weakly coupled Bose-Einstein condensates

We investigate the onset of chaotic dynamics of the one-dimensional discrete nonlinear Schrödinger equation with periodic boundary conditions in the presence of a single on-site defect. This model describes a ring of weakly coupled Bose-Einstein condensates with attractive interactions. We focus on the transition to global stochasticity in three different scenarios as the defect is changed. We make use of a suitable Poincaré section and study different families of stationary solutions, where certain bifurcations lead to global stochasticity. The global stochasticity is characterized by chaotic symbolic synchronization between the population inversions of certain pairs of condensates. We have seen that the Poincaré cycles are useful to gain insight in the dynamics of this Hamiltonian system. Indeed, the return maps of the Poincaré cycles have been used successfully to follow the orbit along the stochastic layers of different resonances in the chaotic self-trapping regime. Moreover, the time series of the Poincaré cycles suggests that in the global stochasticity regime the dynamics is, to some extent, Markovian, in spite of the fact that the condensates are phase locked with almost the same phase. This phase locking induces a peculiar local interference of the matter waves of the condensates.

Multiscale analysis of information correlations in an infinite-range, ferromagnetic Ising system

The scale-specific information content of the infinite-range, ferromagnetic Ising model is examined by means of information-theoretic measures of high-order correlations in finite-sized systems. The order-disorder transition region can be identified through the appearance of collective order in the ferromagnetic phase. In addition, it is found that near the transition, the ferromagnetic phase is marked by characteristic information oscillations at scales comparable to the system size. The amplitude of these oscillations increases with the total number of spins, so that large-scale information measures of correlations are nonanalytic in the thermodynamic limit. In contrast, correlations at scales small relative to the system size have a monotonic behavior both above and below the transition point, and a well-defined thermodynamic limit.

Defect-induced spatial coherence in the discrete nonlinear Schrödinger equation

We have considered the discrete nonlinear Schrödinger equation (DNLSE) with periodic boundary conditions in the context of coupled Kerr waveguides. The presence of a defect in the central oscillator equation can induce quasiperiodic or large chaotic amplitude oscillations. As for the quasiperiodic dynamics, an enhancement of the amplitude correlations in certain oscillator pairs can take place. However, when the array dynamics becomes chaotic, these correlations are destroyed, and, for suitable defects, synchronization, in the information sense, of certain signals arises in this Hamiltonian system. A numerical continuation analysis clarifies the onset of this dynamical regime. In this case, phase synchronization follows with a peculiar distribution of the Liapunov exponents. These effects occur for initial conditions in a small neighborhood of a family of stationary solutions. We have also found a regime characterized by persistent localized chaotic amplitudes. We have generalized these results to take into account birefringent effects in waveguides.