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Volume bounds for the phase-locking region in the Kuramoto model with asymmetric coupling. (English) Zbl 1467.34040

The author considers the general Kuramoto model given by \[ \frac{d\theta_i}{dt}=\omega_i+\sum_{j=1}^N\gamma_{ij}\sin{(\theta_j-\theta_i)} \] where the weights \(\gamma_{ij}\) may be asymmetric (i.e., \(\gamma_{ij}\neq\gamma_{ji}\)) and some \(\gamma_{ij}\) may be zero, i.e., the network is not fully connected. The interest is in the volume of the set of all natural frequencies (\(\{\omega_i\}_{i=1,\dots N}\)) for which the network is phase locked, i.e., all oscillators have the same frequency but with phases differing from one another in a fixed manner. The volume of the set for which the network has a stable phase locked solution is also of interest.
Generalising previous results for which it was assumed that \(\gamma_{ij}=\gamma_{ji}\) the author derives upper and lower bounds for the volumes of interest. These bounds are determined by the network connectivity, specified by the nonzero \(\gamma_{ij}\). Specifically, they are sums over certain directed subgraphs of the network, using the weights associated with edges in the subgraphs.

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34D06 Synchronization of solutions to ordinary differential equations
05C20 Directed graphs (digraphs), tournaments
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