No abstract

The paper proposes a new type of small-world networks of cells with chaotic behavior. This network consists of a regular lattice of cells with constant 2K-nearest neighbor couplings and time-dependent on-off couplings between any other pair of cells. In each time interval of duration τ such a coupling is switched on with probability p and the corresponding switching random variables are independent for different links and for different times. At each moment, the coupling structure corresponds to a small-world graph, but the shortcuts change from time interval to time interval, which is a good model for many real-world dynamical networks. It is to be distinguished from networks that have randomly chosen shortcuts, fixed in time.Here, we apply the Connection Graph Stability method, developed in the companion paper ("Connection graph stability method for synchronized coupled chaotic systems", see this issue), to the study of global synchronization in this network with the time-varying coupling structure, in the case where the on-off switching is fast with respect to the characteristic synchronization time of the network. The synchronization thresholds are explicitly linked with the average path length of the coupling graph and with the probability p of shortcut switchings in this blinking model. We prove that for the blinking model, a few random shortcut additions significantly lower the synchronization threshold together with the effective characteristic path length. Short interactions between cells, as in the blinking model, are important in practice. To cite prominent examples, computers networked over the Internet interact by sending packets of information, and neurons in our brain interact by sending short pulses, called spikes. The rare interaction between arbitrary nodes in the network greatly facilitates synchronization without loading the network much. In this respect, we believe that it is more efficient than a structure of fixed random connections.

The paper presents a qualitative analysis of an array of diffusively coupled identical continuous time dynamical systems. The effects of full, partial, antiphase, and in-phase-antiphase chaotic synchronizations are investigated via the linear invariant manifolds of the corresponding differential equations. The existence of various invariant manifolds, a self-similar behavior, and a hierarchy and embedding of the manifolds of the coupled system are discovered. Sufficient conditions for the stability of the invariant manifolds are obtained via the method of Lyapunov functions. Conditions under which full global synchronization cannot be achieved even for the largest coupling constant are defined. The general rigorous results are illustrated through examples of coupled Lorenz-like and Rössler systems.