Synchronization of Kuramoto Oscillators: Inverse Taylor Expansions
This work offers a new theoretical and computational tool for analyzing synchronization in networks of coupled oscillators, which is relevant to physics, engineering, and biology.
The authors propose a power series expansion to compute the synchronized solution of Kuramoto oscillators, providing an efficient approximation scheme and a hierarchy of synchronization tests that generalize existing methods. Numerical experiments demonstrate improved accuracy and computational efficiency over iterative methods.
Synchronization in networks of coupled oscillators is a widely studied topic with extensive scientific and engineering applications. In this paper, we study the frequency synchronization problem for networks of Kuramoto oscillators with arbitrary topology and heterogeneous edge weights. We propose a novel equivalent transcription for the equilibrium synchronization equation. Using this transcription, we develop a power series expansion to compute the synchronized solution of the Kuramoto model as well as a sufficient condition for the strong convergence of this series expansion. Truncating the power series provides (i) an efficient approximation scheme for computing the synchronized solution, and (ii) a simple-to-check, statistically-correct hierarchy of increasingly accurate synchronization tests. This hierarchy of tests provides a theoretical foundation for and generalizes the best-known approximate synchronization test in the literature. Our numerical experiments illustrate the accuracy and the computational efficiency of the truncated series approximation compared to existing iterative methods and existing synchronization tests.