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Synchronization and graph topology. (English) Zbl 1107.34047

The authors consider a network of \(n\) identical oscillators that are linearly coupled \[ \dot x_i = F(x_i) + \sum_{j=1}^{n} \varepsilon_{ij}(t) P x_j, \quad j=1,\dots,n, \] where \(x_i\) is the vector containing the coordinates of the \(i\)th oscillator, the matrix \(P\) determines which variables couple the oscillators, and \((\varepsilon_{ij})\) is an \(n\times n\) symmetric coupling matrix.
The main goal of the paper is to clarify the relation between synchronization and the topology of the considered network. In particular, the authors explicitly link the stability of the synchronization with the average path length of the graph, which corresponds to the network. Finally, the example of coupled Hindmarsh-Rose neuron models is considered.

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
94C15 Applications of graph theory to circuits and networks
34D20 Stability of solutions to ordinary differential equations
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