Abstract
We study the Kuramoto model with attractive sine coupling. We introduce a complex-valued matrix formulation whose argument coincides with the original Kuramoto dynamics. We derive an exact solution for the complex-valued model, which permits analytical insight into individual realizations of the Kuramoto model. The existence of a complex-valued form of the Kuramoto model provides a key demonstration that, in some cases, reformulations of nonlinear dynamics in higher-order number fields may provide tractable analytical approaches.
Highlights
The dynamics of networks with many nodes and connections poses difficulties for mathematical treatment
In this Letter, we provide a complex-valued matrix formulation of the Kuramoto model (KM) whose argument corresponds to the original KM
We considered the KM on a Watts-Strogatz network (GWS), which is defined as a ring graph, GRG, where each node is first connected to its k neighbors in each direction and each edge is rewired to another node with uniform probability q [12]
Summary
The dynamics of networks with many nodes and connections poses difficulties for mathematical treatment. We note this expression implies θi ∈ C and requires a scaling of the coupling strength (γ = 2κ/π ); as we show, the resulting complex-valued system admits a solution whose argument agrees with the original KM. We can compare the argument of the analytical expression Eq (10) with the result obtained by numerical integration of Eq (1).
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