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We study synchronization of non-diffusively coupled map networks with arbitrary network topologies, where the connections between different units are, in general, not symmetric and can carry both positive and negative weights. We show that, in contrast to diffusively coupled networks, the synchronous behavior of a non-diffusively coupled network can be dramatically different from the behavior of its constituent units. In particular, we show that chaos can emerge as synchronized behavior although the dynamics of individual units are very simple. Conversely, individually chaotic units can display simple behavior when the network synchronizes. We give a synchronization criterion that depends on the spectrum of the generalized graph Laplacian, as well as the dynamical properties of the individual units and the interaction function. This general result will be applied to coupled systems of tent and logistic maps and to two models of neuronal dynamics. Our approach yields an analytical understanding of how simple model neurons can produce complex collective behavior through the coordination of their actions.
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The linear response of synchronized chaotic units with delayed couplings and feedback to small external perturbations is investigated in the context of communication with chaos synchronization. For iterated chaotic maps, the distribution of distances is calculated numerically and, for some special cases, analytically as well. Depending on model parameters, this distribution has power law tails leading to diverging moments of distances in the region of synchronization. The corresponding linear equations have multiplicative and additive noise due to perturbations and chaos. The response to small harmonic perturbations shows resonances related to coupling and feedback delay times. For perturbation from a binary message the bit error rate is calculated. The bit error rate is not related to the transverse Lyapunov exponents, and it can be reduced when additional noise is added to the transmitted signal. For some special cases, the bit error rate as a function of coupling strength has the structure of a devils staircase, related to an iterated function system. Finally, the security of communication is discussed by comparing uni- and bi-directional couplings.
The linear response of synchronized time-delayed chaotic systems to small external perturbations, i.e., the phenomenon of chaos pass filter, is investigated for iterated maps. The distribution of distances, i.e., the deviations between two synchronized chaotic units due to external perturbations on the transfered signal, is used as a measure of the linear response. It is calculated numerically and, for some special cases, analytically. Depending on the model parameters this distribution has power law tails in the region of synchronization leading to diverging moments of distances. This is a consequence of multiplicative and additive noise in the corresponding linear equations due to chaos and external perturbations. The linear response can also be quantified by the bit error rate of a transmitted binary message which perturbs the synchronized system. The bit error rate is given by an integral over the distribution of distances and is calculated analytically and numerically. It displays a complex nonmonotonic behavior in the region of synchronization. For special cases the distribution of distances has a fractal structure leading to a devils staircase for the bit error rate as a function of coupling strength. The response to small harmonic perturbations shows resonances related to coupling and feedback delay times. A bi-directionally coupled chain of three units can completely filtered out the perturbation. Thus the second moment and the bit error rate become zero.
We study the relationship between synchronization and the rate with which information is exchanged between nodes in a spatio-temporal network that describes the dynamics of classical particles under a substrate Remoissenet-Peyrard potential. We also show how phase and complete synchronization can be detected in this network. The difficulty in detecting phase synchronization in such a network appears due to the highly non-coherent character of the particle dynamics which unables a proper definition of the phase dynamics. The difficulty in detecting complete synchronization appears due to the spatio character of the potential which results in an asymptotic state highly dependent on the initial state.
Networks of chaotic units with static couplings can synchronize to a common chaotic trajectory. The effect of dynamic adaptive couplings on the cooperative behavior of chaotic networks is investigated. The couplings adjust to the activities of its two units by two competing mechanisms: An exponential decrease of the coupling strength is compensated by an increase due to de-synchronized activity. This mechanism prevents the network from reaching a steady state. Numerical simulations of a coupled map lattice show chaotic trajectories of de-synchronized units interrupted by pulses of mutually synchronized clusters. These pulses occur on all scales, sometimes extending to the entire network. Clusters of synchronized units can be triggered by a small group of synchronized units.
Complexity of dynamical networks can arise not only from the complexity of the topological structure but also from the time evolution of the topology. In this paper, we study the synchronous motion of coupled maps in time-varying complex networks both analytically and numerically. The temporal variation is rather general and formalized as being driven by a metric dynamical system. Four network models are discussed in detail in which the interconnections between vertices vary through time randomly. These models are 1) i.i.d. sequences of random graphs with fixed wiring probability, 2) groups of graphs with random switches between the individual graphs, 3) graphs with temporary random failures of nodes, and 4) the meet-for-dinner model where the vertices are randomly grouped. We show that the temporal variation and randomness of the connection topology can enhance synchronizability in many cases; however, there are also instances where they reduce synchronizability. In analytical terms, the Hajnal diameter of the coupling matrix sequence is presented as a measure for the synchronizability of the graph topology. In topological terms, the decisive criterion for synchronization of coupled chaotic maps is that the union of the time-varying graphs contains a spanning tree.
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