Synchronization of Kuramoto Oscillators via Cutset Projections

Synchronization in coupled oscillators networks is a remarkable phenomenon of relevance in numerous fields. For Kuramoto oscillators, the loss of synchronization is determined by a tradeoff between coupling strength and oscillator heterogeneity. Despite extensive prior work, the existing sufficient conditions for synchronization are either very conservative or heuristic and approximate. Using a novel cutset projection operator, we propose a new family of sufficient synchronization conditions; these conditions rigorously identify the correct functional form of the tradeoff between coupling strength and oscillator heterogeneity. To overcome the need to solve a nonconvex optimization problem, we then provide two explicit bounding methods, thereby obtaining 1) the best-known sufficient condition for unweighted graphs based on the 2-norm, and 2) the first-known generally applicable sufficient condition based on the \infty-norm. We conclude with a comparative study of our novel \infty-norm condition for specific topologies and IEEE test cases; for most IEEE test cases, our new sufficient condition is one to two orders of magnitude more accurate than previous rigorous tests.