Cooperative oscillatory behavior of mutually coupled dynamical systems

In this paper, we make a qualitative study of the dynamics of a network of diffusively coupled identical systems. In particular, we derive conditions on the systems and on the coupling strength between the systems that guarantee the global synchronization of the systems. It is shown that the notion of "minimum phaseness" of the individual systems involved is essential in ensuring synchronous behavior in the network when the coupling exceeds a certain computable threshold. On the other hand, it is shown that oscillatory behavior may arise in a network of identical globally asymptotically stable systems in case the isolated systems are nonminimum phase. In addition, we analyze the synchronization or nonsynchronization of the network in terms of its topology; that is, what happens if either the number of couplings and/or systems increases? The results are illustrated by computer simulations of coupled chaotic systems like the Rossler system and the Lorenz system.