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Emergence of synchronization in complex networks of interacting dynamical systems. (English) Zbl 1130.37347
Summary: We study the emergence of coherence in large complex networks of interacting heterogeneous dynamical systems. We show that for a large class of dynamical systems and network topologies there is a critical coupling strength at which the systems undergo a transition from incoherent to coherent behavior. We find that the critical coupling strength at which this transition takes place is ${k}_{c}={\left(Z\lambda \right)}^{-1}$, where Z depends only on the uncoupled dynamics of the individual systems on each node, while $\lambda$ is the largest eigenvalue of the network adjacency matrix. Thus we achieve a separation of the problem into two parts, one depending solely on the node dynamics, and one depending solely on network topology.
##### MSC:
 37C99 Smooth dynamical systems 94C99 Circuits, networks
##### Keywords:
synchronization; networks