## 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:

 37C75 Stability theory for smooth dynamical systems 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34D20 Stability of solutions to ordinary differential equations 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 94C15 Applications of graph theory to circuits and networks
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