Transition to chaos in small-world dynamical network

The transition from a non-chaotic state to a chaotic state is an important issue in the study of coupled dynamical networks. In this paper, by using the theoretical analysis and numerical simulation, we study the dynamical behaviors of the NW small-world dynamical network consisting of nodes that are in non-chaotic states before they are coupled together. It is found that, for any given coupling strength and a sufficiently large number of nodes, the small-world dynamical network can be chaotic, even if the nearest-neighbor coupled network cannot be chaotic under the same condition. More interesting, the numerical results show that the measurement 1 of the transition ability from non-chaos to chaos approximately obeys power-law forms as 1/R similar to p(-r1) and 1/R similar to N-r2. Furthermore, based on dissipative system criteria, we obtain the relationship between the network topology parameters and the coupling strength when the network is stable in the sense of Lyapunov (i. s. L.). (c) 2006 Elsevier Ltd. All rights reserved.