Predicting the synchronization time in coupled-map networks

Growth, collapse, and self-organized criticality in complex networks

Network growth is ubiquitous in nature (e.g., biological networks) and technological systems (e.g., modern infrastructures). To understand how certain dynamical behaviors can or cannot persist as the underlying network grows is a problem of increasing importance in complex dynamical systems as well as sustainability science and engineering. We address the question of whether a complex network of nonlinear oscillators can maintain its synchronization stability as it expands. We find that a large scale avalanche over the entire network can be triggered in the sense that the individual nodal dynamics diverges from the synchronous state in a cascading manner within a relatively short time period. In particular, after an initial stage of linear growth, the network typically evolves into a critical state where the addition of a single new node can cause a group of nodes to lose synchronization, leading to synchronization collapse for the entire network. A statistical analysis reveals that the collapse size is approximately algebraically distributed, indicating the emergence of self-organized criticality. We demonstrate the generality of the phenomenon of synchronization collapse using a variety of complex network models, and uncover the underlying dynamical mechanism through an eigenvector analysis.

Convergence time towards periodic orbits in discrete dynamical systems

We investigate the convergence towards periodic orbits in discrete dynamical systems. We examine the probability that a randomly chosen point converges to a particular neighborhood of a periodic orbit in a fixed number of iterations, and we use linearized equations to examine the evolution near that neighborhood. The underlying idea is that points of stable periodic orbit are associated with intervals. We state and prove a theorem that details what regions of phase space are mapped into these intervals (once they are known) and how many iterations are required to get there. We also construct algorithms that allow our theoretical results to be implemented successfully in practice.

Outer synchronization between two complex dynamical networks with discontinuous coupling

In this paper, we study the outer synchronization between two complex networks with discontinuous coupling. Sufficient conditions for complete outer synchronization and generalized outer synchronization are obtained based on the stability theory of differential equations. The theoretical results show that two networks can achieve outer synchronization even if two networks are switched off sometimes and the speed of synchronization is proportional to the on-off rate. Finally, numerical examples are examined to illustrate the effectiveness of the analytical results.

Finite-time stochastic outer synchronization between two complex dynamical networks with different topologies

In this paper, the finite-time stochastic outer synchronization between two different complex dynamical networks with noise perturbation is investigated. By using suitable controllers, sufficient conditions for finite-time stochastic outer synchronization are derived based on the finite-time stability theory of stochastic differential equations. It is noticed that the coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix need not be symmetric. Finally, numerical examples are examined to illustrate the effectiveness of the analytical results. The effect of control parameters on the settling time is also numerically demonstrated.

Synchronization with on-off coupling: Role of time scales in network dynamics

We consider the problem of synchronizing a general complex network by means of the on-off coupling strategy; in this case, the on-off time scale is varied from a very small to a very large value. In particular, we find that when the time scale is comparable to that of node dynamics, synchronization can also be achieved and greatly optimized for the upper bound of the stability region which nearly disappears, and the synchronization speed is accelerated a lot, independent of network topologies. Our study indicates that the time scale for network variation is of crucial importance for network dynamics and synchronization under the comparable time scale which is much more advantageous over other time scales. Both analysis and experiments confirm the conclusions.