Transition to Complex Behavior in Networks of Coupled Dynamical Systems

Abstract

The emergence of complex behavior is studied in a network of coupled dynamical systems whose trajectories converge to a stable equilibrium point. The effects of network topology on the stability of its synchronized behavior is measured in terms of transverse Lyapunov exponents. By choosing a suitable coupling configuration, the transverse Lyapunov exponents are made positive, which may lead to the emergence of complex behavior. Moreover, the effects of the coupling configuration can lead to unbounded trajectories. The relationship between the transverse Lyapunov exponents and the eigenvalues of the connectivity matrix is used to establish upper and lower limit values for the transition to complex behavior. The transition criteria are expressed in terms of coupling strength and the number of nodes in the network. There are two main contributions on this manuscript: (1) the analytical derivation of the relationship between the local Lyapunov exponents and those of the entire network as the number of nodes increases, and (2) to show the existence of an interval of coupling strength values for the transition into complex behavior in networks with homogeneous connectivity, which becomes smaller as the number of nodes in the network growths. Additionally, we also show that in networks with heterogeneous connectivity, the trajectories transit directly from stable equilibrium to unbounded behavior. These results are illustrated with numerical simulations.