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

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 us 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.