Invariant cones for strange attractors of Lozi, H\'{e}non and Belykh type maps

We consider strange attractors of two dimensional generalized map with one nonlinearity such that Lozi, H\'{e}non and Belykh maps are particular cases of it. We describe technique of invariant expanding and contracting cones creation for study of hyperbolic properties. Theorems of singular hyperbolic attractors for new modifications of Lozi, H\'{e}non and Belykh-type maps are presented

Dynamics of Stochastically Blinking Systems. Part II: Asymptotic Properties

We study stochastically blinking dynamical systems as in the companion paper (Part I). We analyze the asymptotic properties of the blinking system as time goes to infinity. The trajectories of the averaged and blinking system cannot stick together forever, but the trajectories of the blinking system may converge to an attractor of the averaged system. There are four distinct classes of blinking dynamical systems. Two properties differentiate them: single or multiple attractors of the averaged system and their invariance or noninvariance under the dynamics of the blinking system. In the case of invariance, we prove that the trajectories of the blinking system converge to the attractor(s) of the averaged system with high probability if switching is fast. In the noninvariant single attractor case, the trajectories reach a neighborhood of the attractor rapidly and remain close most of the time with high probability when switching is fast. In the noninvariant multiple attractor case, the trajectory may escape to another attractor with small probability. Using the Lyapunov function method, we derive explicit bounds for these probabilities. Each of the four cases is illustrated by a specific example of a blinking dynamical system. From a probability theory perspective, our results are obtained by directly deriving large deviation bounds. They are more conservative than those derived by using the action functional approach, but they are explicit in the parameters of the blinking system

Synchronization in complex networks with blinking interactions

We propose a new model of small-world networks of cells with a time-varying coupling and study its synchronization properties. In each time interval of length /spl tau/ 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 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. 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

Dynamics of Stochastically Blinking Systems. Part I: Finite Time Properties

We consider dynamical systems whose parameters are switched within a discrete set of values at equal time intervals. Similar to the blinking of the eye, switching is fast and occurs stochastically and independently for different time intervals. There are two time scales present in such systems, namely the time scale of the dynamical system and the time scale of the stochastic process. If the stochastic process is much faster, we expect the blinking system to follow the averaged system where the dynamical law is given by the expectation of the stochastic variables. We prove that, with high probability, the trajectories of the two systems stick together for a certain period of time. We give explicit bounds that relate the probability, the switching frequency, the precision, and the length of the time interval to each other. We discover the apparent presence of a soft upper bound for the time interval, beyond which it is almost impossible to keep the two trajectories together. This comes as a surprise in view of the known perturbation analysis results. From a probability theory perspective, our results are obtained by directly deriving large deviation bounds. They are more conservative than those derived by using the action functional approach, but they are explicit in the parameters of the blinking system

Graph-based criteria for synchronization of diffusively coupled oscillators

We extend the connection graph stability method for complete synchronization, originally developed for symmetrically coupled networks, to the asymmetrical case. First, we study synchronization in asymmetrically connected networks with node balance, the property that the sum of the coupling coefficients of all edges directed to a node equals the sum of the coupling coefficients of all the edges directed outward from the node. We obtain the same criterion as for the network with a symmetrized connection matrix, provided that the condition of node balance is satisfied. Then, we derive a general synchronization criterion for arbitrary asymmetrical networks. The criterion is obtained by symmetrizing the connection graph and associating a weight to the path between any two nodes