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Synchronization in the networks of coupled oscillators is a widely studiedtopic in different areas. It is well-known that synchronization occurs if theconnectivity of the network dominates heterogeneity of the oscillators. Despiteextensive study on this topic, the quest for sharp closed-form synchronizationtests is still in vain. In this paper, we present an algorithm for finding theTaylor expansion of the inverse Kuramoto map. We show that this Taylor seriescan be used to obtain a hierarchy of increasingly accurate approximate testswith low computational complexity. These approximate tests are then used toestimate the threshold of synchronization as well as the position of thesynchronization manifold of the network.

Synchronization of Coupled Oscillators: The Taylor Expansion of the Inverse Kuramoto Map