Synchronization and coherence in thermodynamic coupled map lattices with intermediate-range coupling

Synchronous slowing down in coupled logistic maps via random network topology

The speed and paths of synchronization play a key role in the function of a system, which has not received enough attention up to now. In this work, we study the synchronization process of coupled logistic maps that reveals the common features of low-dimensional dissipative systems. A slowing down of synchronization process is observed, which is a novel phenomenon. The result shows that there are two typical kinds of transient process before the system reaches complete synchronization, which is demonstrated by both the coupled multiple-period maps and the coupled multiple-band chaotic maps. When the coupling is weak, the evolution of the system is governed mainly by the local dynamic, i.e., the node states are attracted by the stable orbits or chaotic attractors of the single map and evolve toward the synchronized orbit in a less coherent way. When the coupling is strong, the node states evolve in a high coherent way toward the stable orbit on the synchronized manifold, where the collective dynamics dominates the evolution. In a mediate coupling strength, the interplay between the two paths is responsible for the slowing down. The existence of different synchronization paths is also proven by the finite-time Lyapunov exponent and its distribution.

Chaotic communication via temporal transfer entropy

We propose a new perspective on communication using chaos. A binary message is encoded into the temporally causal relations on a coupled maps ring of N chaotic nodes. From the analysis of temporal transfer entropy, the masked information can be recovered from transmitted signals at the receiver. The communication scheme has been demonstrated to be robust against external noise and some traditional attacks.

Paths to globally generalized synchronization in scale-free networks

We apply the auxiliary-system approach to study paths to globally generalized synchronization in scale-free networks of identical chaotic oscillators, including Hénon maps, logistic maps, and Lorentz oscillators. As the coupling strength epsilon between nodes of the network is increased, transitions from partially to globally generalized synchronization and intermittent behaviors near the synchronization thresholds, are found. The generalized synchronization starts from the hubs of the network and then spreads throughout the whole network with the increase of epsilon . Our result is useful for understanding the synchronization process in complex networks.

Influence of noise on the synchronization of the stochastic Kuramoto model

We consider the Kuramoto model of globally coupled phase oscillators subject to Ornstein-Uhlenbeck and non-Gaussian colored noise and investigate the influence of noise on the order parameter of the synchronization process. We use numerical methods to study the dependence of the threshold as well as the maximum degree of synchronization on the correlation time and the strength of the noise, and find that the threshold of synchronization strongly depends on the nature of the noise. It is found to be lower for both the Ornstein-Uhlenbeck and non-Gaussian processes compared to the case of white noise. A finite correlation time also favors the achievement of the full synchronization of the system, in contract to the white noise process, which does not allow that. Finally, we discuss possible applications of the stochastic Kuramoto model to oscillations taking place in biochemical systems.

Transversal dynamics of a non-locally-coupled map lattice

A lattice of coupled chaotic dynamical systems may exhibit a completely synchronized state, which defines a low-dimensional invariant manifold in phase space. However, the high dimensionality of the latter typically yields a complex dynamics with many features like chaos suppression, quasiperiodicity, multistability, and intermittency. Such phenomena are described by considering the transversal dynamics to the synchronization manifold for a coupled logistic map lattice with a long-range coupling prescription.

Empirical mode decomposition and synchrogram approach to cardiorespiratory synchronization

We use the empirical mode decomposition method to decompose experimental respiratory signals into a set of intrinsic mode functions (IMFs), and consider one of these IMFs as a respiratory rhythm. We then use the Hilbert spectral analysis to calculate the instantaneous phase of the IMF. Heartbeat data are finally incorporated to construct the cardiorespiratory synchrogram, which is a visual tool for inspecting synchronization. We perform analysis on 20 data sets collected by the Harvard medical school from ten young (21-34 years old) and ten elderly (68-81 years old) rigorously screened healthy subjects. Our results support the existence of cardiorespiratory synchronization. We also investigate the origin of the cardiorespiratory synchronization by addressing the problem of correlations between regularities of respiratory and cardiac signals. Our analysis shows that regularity of respiratory signals plays a dominant role in the cardiorespiratory synchronization.

Scaling and universality in transition to synchronous chaos with local-global interactions

We study the coupled-map lattice model with both local and global couplings. We find necessary conditions for observing synchronous chaos and investigate the transition to synchronization as a dynamic phase transition. We discover that this transition, if continuous, shows scaling and universal behavior with the dynamic exponent z = 2. We also define and illustrate an interesting quantity similar to persistence at critical point.

Stochastic dynamical model for stock-stock correlations

We propose a model of coupled random walks for stock-stock correlations. The walks in the model are coupled via a mechanism that the displacement (price change) of each walk (stock) is activated by the price gradients over some underlying network. We assume that the network has two underlying structures, describing the correlations among the stocks of the whole market and among those within individual groups, respectively, each with a coupling parameter controlling the degree of correlation. The model provides the interpretation of the features displayed in the distribution of the eigenvalues for the correlation matrix of real market on the level of time sequences. We verify that such modeling indeed gives good fitting for the market data of US stocks.

Analytical results for coupled-map lattices with long-range interactions

We obtain exact analytical results for lattices of maps with couplings that decay with distance as r(-alpha). We analyze the effect of the coupling range on the system dynamics through the Lyapunov spectrum. For lattices whose elements are piecewise linear maps, we get an algebraic expression for the Lyapunov spectrum. When the local dynamics is given by a nonlinear map, the Lyapunov spectrum for a completely synchronized state is analytically obtained. The critical line characterizing the synchronization transition is determined from the expression for the largest transversal Lyapunov exponent. In particular, it is shown that in the thermodynamical limit, such transition is only possible for sufficiently long-range interactions, namely, for alpha<alpha(c)=d, where d is the lattice dimension.

Lyapunov spectrum and synchronization of piecewise linear map lattices with power-law coupling

We study the synchronization properties of a lattice of chaotic piecewise linear maps. The coupling strength decreases with the lattice distance in a power-law fashion. We obtain the Lyapunov spectrum of the coupled map lattice and investigate the relation between spatiotemporal chaos and synchronization of amplitudes and phases, using suitable numerical diagnostics.

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.

Biocomplexity: adaptive behavior in complex stochastic dynamical systems

Existing methods of complexity research are capable of describing certain specifics of bio systems over a given narrow range of parameters but often they cannot account for the initial emergence of complex biological systems, their evolution, state changes and sometimes-abrupt state transitions. Chaos tools have the potential of reaching to the essential driving mechanisms that organize matter into living substances. Our basic thesis is that while established chaos tools are useful in describing complexity in physical systems, they lack the power of grasping the essence of the complexity of life. This thesis illustrates sensory perception of vertebrates and the operation of the vertebrate brain. The study of complexity, at the level of biological systems, cannot be completed by the analytical tools, which have been developed for non-living systems. We propose a new approach to chaos research that has the potential of characterizing biological complexity. Our study is biologically motivated and solidly based in the biodynamics of higher brain function. Our biocomplexity model has the following features, (1) it is high-dimensional, but the dimensionality is not rigid, rather it changes dynamically; (2) it is not autonomous and continuously interacts and communicates with individual environments that are selected by the model from the infinitely complex world; (3) as a result, it is adaptive and modifies its internal organization in response to environmental factors by changing them to meet its own goals; (4) it is a distributed object that evolves both in space and time towards goals that is continually re-shaping in the light of cumulative experience stored in memory; (5) it is driven and stabilized by noise of internal origin through self-organizing dynamics. The resulting theory of stochastic dynamical systems is a mathematical field at the interface of dynamical system theory and stochastic differential equations. This paper outlines several possible avenues to analyze these systems. Of special interest are input-induced and noise-generated, or spontaneous state-transitions and related stability issues.

Synchronous chaos in coupled map lattices with small-world interactions

In certain physical situations, extensive interactions arise naturally in systems. We consider one such situation, namely, small-world couplings. We show that, for a fixed fraction of nonlocal couplings, synchronous chaos is always a stable attractor in the thermodynamic limit. We point out that randomness helps synchronization. We also show that there is a size dependent bifurcation in the collective behavior in such systems.