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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.
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