Topological states in the Kuramoto model. (English) Zbl 1382.05070
Summary: The Kuramoto model is a system of nonlinear differential equations that models networks of coupled oscillators and is often used to study synchronization among the oscillators. In this paper we study steady state solutions of the Kuramoto model by assigning to each steady state a tuple of integers which records how the state twists around the cycles in the network. We then use this new classification of steady states to obtain a “Weyl” type of asymptotic estimate for the number of steady states as the number of oscillators becomes arbitrarily large while preserving the cycle structure. We further show how this asymptotic estimate can be maximized, and as a special case we obtain an asymptotic lower bound for the number of stable steady states of the model.
MSC:
| 05C82 | Small world graphs, complex networks (graph-theoretic aspects) |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34D06 | Synchronization of solutions to ordinary differential equations |
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