Locating and Counting Equilibria of the Kuramoto Model with Rank-One Coupling

Abstract

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.

Smooth Points on Semi-algebraic Sets

“…The Kuramoto model [40] is a dynamical system used to model synchronization amongst n coupled oscillators. The maximum number of equilibria (i.e., real solutions to steady-state equations) for n ≥ 4 remains an open problem [24]. The following confirms the conjecture in [60] for n = 4 with the rest of the section describing its proof.…”

“…To demonstrate the practical efficiency of our new approach, we present the solution of a conjecture for the first time: counting the equilibria of the Kuramoto model in the n = 4 case [60] (see [40] for the original model and [24] for a detailed historical overview and additional references).…”

Counting Equilibria of the Kuramoto Model Using Birationally Invariant Intersection Index

“…Various algebraic formulations have been used to leverage results from algebraic geometry and numerically find some or all equilibria for certain small graphs [12,32,33,36]. Recently, in the special case of "rank-one coupling", i.e., the matrix [a ij ] has rank 1, a much smaller bound 2 N − 2 was established [14]. Based on the theory of the BKK bound, a search for topology-dependent bounds on the number of solutions to (2) and (3) was initiated in [12,33].…”

Performance Analysis of the Solving Algorithm for the Kuramoto Model with Rank One Coupling

“…This paper is a follow up to a previous work that presented a solving algorithm for the Kuramoto model [2] with rank one coupling [1]. In that paper, the algorithm was experimentally shown to be more efficient than other solving methods, but was not fully characterized.…”