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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:

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 |