Conditional exponents, entropies and a measure of dynamical self-organization

In dynamical systems composed of interacting parts, conditional exponents, conditional exponent entropies and cylindrical entropies are shown to be well defined ergodic invariants which characterize the dynamical selforganization and statitical independence of the constituent parts. An example of interacting Bernoulli units is used to illustrate the nature of these invariants.Comment: 6 pages Latex, 1 black and white and 2 color figures, replacement of damaged gif file

"Weak Quantum Chaos" and its resistor network modeling

Weakly chaotic or weakly interacting systems have a wide regime where the common random matrix theory modeling does not apply. As an example we consider cold atoms in a nearly integrable optical billiard with displaceable wall ("piston"). The motion is completely chaotic but with small Lyapunov exponent. The Hamiltonian matrix does not look like one taken from a Gaussian ensemble, but rather it is very sparse and textured. This can be characterized by parameters $s$ and $g$ that reflect the percentage of large elements, and their connectivity, respectively. For $g$ we use a resistor network calculation that has a direct relation to the semi-linear response characteristics of the system, hence leading to a novel prediction regarding the rate of heating of cold atoms in optical billiards with vibrating walls.Comment: 18 pages, 11 figures, improved PRE accepted versio

Synchronization time in a hyperbolic dynamical system with long-range interactions

We show that the threshold of complete synchronization in a lattice of coupled non-smooth chaotic maps is determined by linear stability along the directions transversal to the synchronization subspace. We examine carefully the sychronization time and show that a inadequate observation of the system evolution leads to wrong results. We present both careful numerical experiments and a rigorous mathematical explanation confirming this fact, allowing for a generalization involving hyperbolic coupled map lattices.Comment: 22 pages (preprint format), 4 figures - accepted for publication in Physica A (June 28, 2010

Stability of Synchronized Chaos in Coupled Dynamical Systems

We consider the stability of synchronized chaos in coupled map lattices and in coupled ordinary differential equations. Applying the theory of Hermitian and positive semidefinite matrices we prove two results that give simple bounds on coupling strengths which ensure the stability of synchronized chaos. Previous results in this area involving particular coupling schemes (e.g. global coupling and nearest neighbor diffusive coupling) are included as special cases of the present work.Comment: 9 page

Network synchronization: Spectral versus statistical properties

We consider synchronization of weighted networks, possibly with asymmetrical connections. We show that the synchronizability of the networks cannot be directly inferred from their statistical properties. Small local changes in the network structure can sensitively affect the eigenvalues relevant for synchronization, while the gross statistical network properties remain essentially unchanged. Consequently, commonly used statistical properties, including the degree distribution, degree homogeneity, average degree, average distance, degree correlation, and clustering coefficient, can fail to characterize the synchronizability of networks

Complete Characterization of Stability of Cluster Synchronization in Complex Dynamical Networks

Synchronization is an important and prevalent phenomenon in natural and engineered systems. In many dynamical networks, the coupling is balanced or adjusted in order to admit global synchronization, a condition called Laplacian coupling. Many networks exhibit incomplete synchronization, where two or more clusters of synchronization persist, and computational group theory has recently proved to be valuable in discovering these cluster states based upon the topology of the network. In the important case of Laplacian coupling, additional synchronization patterns can exist that would not be predicted from the group theory analysis alone. The understanding of how and when clusters form, merge, and persist is essential for understanding collective dynamics, synchronization, and failure mechanisms of complex networks such as electric power grids, distributed control networks, and autonomous swarming vehicles. We describe here a method to find and analyze all of the possible cluster synchronization patterns in a Laplacian-coupled network, by applying methods of computational group theory to dynamically-equivalent networks. We present a general technique to evaluate the stability of each of the dynamically valid cluster synchronization patterns. Our results are validated in an electro-optic experiment on a 5 node network that confirms the synchronization patterns predicted by the theory.Comment: 6 figure

Symmetries, Cluster Synchronization, and Isolated Desynchronization in Complex Networks

Synchronization is of central importance in power distribution, telecommunication, neuronal, and biological networks. Many networks are observed to produce patterns of synchronized clusters, but it has been difficult to predict these clusters or understand the conditions under which they form, except for in the simplest of networks. In this article, we shed light on the intimate connection between network symmetry and cluster synchronization. We introduce general techniques that use network symmetries to reveal the patterns of synchronized clusters and determine the conditions under which they persist. The connection between symmetry and cluster synchronization is experimentally explored using an electro-optic network. We experimentally observe and theoretically predict a surprising phenomenon in which some clusters lose synchrony while leaving others synchronized. The results could guide the design of new power grid systems or lead to new understanding of the dynamical behavior of networks ranging from neural to social

Time scale synchronization of chaotic oscillators

This paper presents the result of the investigation of chaotic oscillator synchronization. A new approach for detecting of synchronized behaviour 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. This approach has been applied for the coupled Rossler and Lorenz systems.Comment: 19 pages, 12 figure