Synchronization of Coupled Oscillators: The Taylor Expansion of the Inverse Kuramoto Map
For researchers studying synchronization in oscillator networks, this provides a computationally efficient method to approximate synchronization thresholds, though it is an incremental improvement over existing numerical approaches.
The paper addresses the lack of sharp closed-form synchronization tests for coupled oscillators by deriving a Taylor expansion of the inverse Kuramoto map, which yields a hierarchy of low-complexity approximate tests for estimating synchronization thresholds and manifold positions.
Synchronization in the networks of coupled oscillators is a widely studied topic in different areas. It is well-known that synchronization occurs if the connectivity of the network dominates heterogeneity of the oscillators. Despite extensive study on this topic, the quest for sharp closed-form synchronization tests is still in vain. In this paper, we present an algorithm for finding the Taylor expansion of the inverse Kuramoto map. We show that this Taylor series can be used to obtain a hierarchy of increasingly accurate approximate tests with low computational complexity. These approximate tests are then used to estimate the threshold of synchronization as well as the position of the synchronization manifold of the network.