EQUILIBRIA IN KURAMOTO OSCILLATOR NETWORKS: AN ALGEBRAIC APPROACH

The paper proves that if x0 = e^{iθ0} is an eigenvector of a real matrix A with a real eigenvalue λ, then θ0 is an equilibrium of the original Kuramoto model (Proposition 2), using the standard complex-exponential reduction; this matches the model’s Part 1 essentially verbatim . For the complete graph, the paper’s if-and-only-if classification says equilibria are exactly either those with all phases differing by integer multiples of π or those with Σi e^{iθi} = 0 (Proposition 5), which is equivalent to the model’s two-case description via S = Σk e^{iθk} and the two-antipodal-cluster condition; the algebraic manipulations are closely related and reach the same characterization . Minor omissions in the model are only about stating the homogeneous-ω rotating-frame assumption explicitly (set ω = 0), as done in the paper’s setup .