Abstract. The Kuramoto model describes synchronization behavior among coupled oscillators and enjoys successful application in a wide variety of fields. Many of these applications seek phase-coherent solutions, i.e., equilibria of the model. Historically, research has focused on situations where the number of oscillators, n, is extremely large and can be treated as being infinite. More recently, however, applications have arisen in areas such as electrical engineering with more modest values of n. For these, the equilibria can be located by finding the real solutions of a system of polynomial equations utilizing techniques from algebraic geometry. However, typical methods for solving such systems locate all complex solutions even though only the real solutions give equilibria. In this paper, we present an algorithm to locate only the real solutions of the model, thereby shortening computation time by several orders of magnitude in certain situations. This is accomplished by choosing specific equilibria representatives and the consequent algebraic decoupling of the system. The correctness of the algorithm (that it finds only and all the equilibria) is proved rigorously. Additionally, the algorithm can be implemented using interval methods, so that the equilibria can be approximated up to any given precision without significantly more computational effort. We also compare this solution approach to other computational algebraic geometric methods. Furthermore, analyzing this approach allows us to prove, asymptotically, that the maximum number of equilibria grows at the same rate as the number of complex solutions of a corresponding polynomial system. Finally, we conjecture an upper bound on the maximum number of equilibria for any number of oscillators, which generalizes the known cases and is obtained on a range of explicitly provided natural frequencies.

This paper is a follow up to a previous work that presented an algorithm to efficiently find all of the equilibria of the Kuramoto model with nonuniform coupling described by a rank one matrix. The algorithm was shown experimentally to be more efficient than previously used methods, but its performance was not fully characterized. This paper analyzes the effectiveness of the "pruning" method used to skip cases with no solutions. The approach utilized is to construct a weighted graph where every path through the graph corresponds to the algorithm's performance on an input. The maximum weight path then corresponds to the worst case performance of the algorithm. This paper shows that even in the worst case, the pruning method employed is very effective at skipping cases with no solutions.

No abstract