Stability of an [N/2]-dimensional invariant torus in the Kuramoto model at small coupling

Abstract

When the natural frequencies are allocated symmetrically in the Kuramoto model there exists an invariant torus of dimension [N/2]+1 (N is the population size). A global phase shift invariance allows to reduce the model to $N-1$ dimensions using the phase differences, and doing so the invariant torus becomes [N/2]-dimensional. By means of perturbative calculations based on the renormalization group technique, we show that this torus is asymptotically stable at small coupling if N is odd. If N is even the torus can be stable or unstable depending on the natural frequencies, and both possibilities persist in the small coupling limit.

to appear in Physica D

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• The mathematics of asymptotic stability in the Kuramoto model
• Finite-size scaling in globally coupled phase oscillators with a general coupling scheme
• Normal Forms of $C^{\infty }$ Vector Fields based on the Renormalization Group
• Linear stability of the incoherent solution and the transition formula for the Kuramoto-Daido model