Limits to the experimental detection of nonlinear synchrony

Introduction to focus issue: Synchronization in large networks and continuous media-data, models, and supermodels

The synchronization of loosely coupled chaotic systems has increasingly found applications to large networks of differential equations and to models of continuous media. These applications are at the core of the present Focus Issue. Synchronization between a system and its model, based on limited observations, gives a new perspective on data assimilation. Synchronization among different models of the same system defines a supermodel that can achieve partial consensus among models that otherwise disagree in several respects. Finally, novel methods of time series analysis permit a better description of synchronization in a system that is only observed partially and for a relatively short time. This Focus Issue discusses synchronization in extended systems or in components thereof, with particular attention to data assimilation, supermodeling, and their applications to various areas, from climate modeling to macroeconomics.

The existence of generalized synchronization of chaotic systems in complex networks

The paper studies the existence of generalized synchronization in complex networks, which consist of chaotic systems. When a part of modified nodes are chaotic, and the others have asymptotically stable equilibriums or orbital asymptotically stable periodic solutions, under certain conditions, the existence of generalized synchronization can be turned to the problem of contractive fixed point in the family of Lipschitz functions. In addition, theoretical proofs are proposed to the exponential attractive property of generalized synchronization manifold. Numerical simulations validate the theory.

Hölder continuity of two types of generalized synchronization manifold

This paper studies the existence of Hölder continuity of generalized synchronization (GS). Based on the modified system approach, GS is classified into three types: equilibrium GS, periodic GS, and C-GS, when the modified system has an asymptotically stable equilibrium, asymptotically stable limit cycles, and chaotic attractors, respectively. The existence of the first two types of Hölder continuous GS inertial manifolds are strictly theoretically proved.

Filters display inverse limit spaces

A rigorous proof that linear filters display the inverse limit spaces of chaotic maps is given.

The structure of synchronization sets for noninvertible systems

Unidirectionally coupled systems (x,y) --> (f(x),g(x,y)) occur naturally, and are used as tractable models of networks with complex interactions. We analyze the structure and bifurcations of attractors in the case the driving system is not invertible, and the response system is dissipative. We discuss both cases in which the driving system is a map, and a strongly dissipative flow. Although this problem was originally motivated by examples of nonlinear synchrony, we show that the ideas presented can be used more generally to study the structure of attractors, and examine interactions between coupled systems.

Robust measure for characterizing generalized synchronization

Generalized synchronization between two coupled systems can be characterized by recently proposed interdependency measures calculated from two simultaneously observed time series from them. However, numerical tests have shown that these measures cannot consistently indicate the direction of the coupling for strongly coupled systems or in situations with a large phase space neighbor size. An interdependency measure is proposed here quantifying how close a conditional neighbor is to a true neighbor in terms of the degree of alignment of their principal axes. Numerical tests are carried out on time series generated from a coupled Hénon map and a Lorenz model driven by a Rossler model. Given that a driving system is more dependent on a response system, the results show that the direction of the coupling is consistently detected by using the proposed measure even in those unfavorable cases for the measures mentioned above.

Detectability of nondifferentiable generalized synchrony

Generalized synchronization of chaos is a type of cooperative behavior in directionally coupled oscillators that is characterized by existence of stable and persistent functional dependence of response trajectories from the chaotic trajectory of driving oscillator. In many practical cases this function is nondifferentiable and has a very complex shape. The generalized synchrony in such cases seems to be undetectable, and only the cases in which a differentiable synchronization function exists are considered to make sense in practice. We show that this viewpoint is not always correct and the nondifferentiable generalized synchrony can be revealed in many practical cases. Conditions for detection of generalized synchrony are derived analytically, and illustrated numerically with a simple example of nondifferentiable generalized synchronization.

The geometry of chaos synchronization

Chaos synchronization in coupled systems is often characterized by a map phi between the states of the components. In noninvertible systems, or in systems without inherent symmetries, the synchronization set--by which we mean graph(phi)--can be extremely complicated. We identify, describe, and give examples of several different complications that can arise, and we link each to inherent properties of the underlying dynamics. In brief, synchronization sets can in general become nondifferentiable, and in the more severe case of noninvertible dynamics, they might even be multivalued. We suggest two different ways to quantify these features, and we discuss possible failures in detecting chaos synchrony using standard continuity-based methods when these features are present.

Detecting generalized synchrony: an improved approach

We examine some of the difficulties involved in detecting generalized synchrony (GS) in systems that exhibit noninvertibility and/or wrinkling. These latter features severely hinder identification of GS by conventional techniques. It is shown that it is possible to greatly improve detection by reducing the pseudofalse neighbors effects. Here we propose the delta(p)-neighbor method to overcome the noninvertibility effect and the delta(p,q) method to detect GS in systems with wrinkled structures.

Topographic organization of nonlinear interdependence in multichannel human EEG

This paper investigates the spatial organization of nonlinear interactions between different brain regions in healthy human subjects. This is achieved by studying the topography of nonlinear interdependence in multichannel EEG data, acquired from 40 healthy human subjects at rest. An algorithm for the detection and quantification of nonlinear interdependence is applied to four pairs of bipolar electrode derivations to detect posterior and anterior interhemispheric and left and right intrahemispheric interdependences. Multivariate surrogate data sets are constructed to control for linear coherence and finite sample size. Nonlinear interdependence is shown to occur in a small but statistically robust number of epochs. The occurrence of nonlinear interdependence in any region is correlated with the concurrent presence of nonlinear interdependence in other regions at high levels of significance. The strength, direction and topography of the interdependences are also correlated. For example, posterior interhemispheric interdependence from right-to-left is strongly correlated with right intrahemispheric interdependence from back-to-front. There is a subtle change in these correlations when subjects open their eyes. These results suggest that nonlinear interdependence in the human brain has a specific topographic organization which reflects simple cognitive changes. It sometimes occurs as an isolated phenomenon between two brain regions, but often involves concurrent interdependences between multiple brain regions.

Generalized synchronization of chaos in noninvertible maps

The properties of functional relation between a noninvertible chaotic drive and a response map in the regime of generalized synchronization of chaos are studied. It is shown that despite a very fuzzy image of the relation between the current states of the maps, the functional relation becomes apparent when a sufficient interval of driving trajectory is taken into account. This paper develops a theoretical framework of such functional relation and illustrates the main theoretical conclusions using numerical simulations.