Transition to chaos in complex dynamical networks

The transition from a non-chaotic state to a chaotic state is a commonly concerned issue in the study of coupled dynamical networks. In this work, we consider a network consisting of nodes that are in non-chaotic states with parameters in non-chaotic regions before they are coupled together. We show that if these non-chaotic nodes are linked together through a suitable structural topology, positive Lyapunov exponents of the coupled network can be generated by choosing a certain uniform coupling strength, and the threshold for this coupling strength is determined by the complexity of the network topology. Moreover, we show that topological effects of scale-free and random networks, which are two basic types of complex network models, can be visualized based on their topological sensitivity to random failures and intentional attacks. Our simulation results on a 1000-node scale-free network and a 1000-node random network of the Logistic maps have verified that, during the transition from non-chaotic to chaotic states, if the topology is more heterogenous then the coupling strength required to achieve the transition can be decreased.