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Synchronization in networks of nonlinear dynamical systems coupled via a directed graph. (English) Zbl 1089.37024
Summary: We study synchronization in an array of coupled identical nonlinear dynamical systems where the coupling topology is expressed as a directed graph and give synchronization criteria related to properties of a generalized Laplacian matrix of the directed graph. In particular, we extend recent results by showing that the array synchronizes for sufficiently large cooperative coupling if the underlying graph contains a spanning directed tree. This is an intuitive yet nontrivial result that can be paraphrased as follows: if there exists a dynamical system which influences directly or indirectly all other systems, then synchronization is possible for strong enough coupling. The converse is also true in general.
MSC:
37C75Stability theory
05C50Graphs and linear algebra
34C15Nonlinear oscillations, coupled oscillators (ODE)
34D20Stability of ODE
93D05Lyapunov and other classical stabilities of control systems
94C15Applications of graph theory to circuits and networks