Stability of an [N/2][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/2]+1 (NN is the population size). A global phase shift invariance allows us to reduce the model to N−1N−1 dimensions using the phase differences, and doing so the invariant torus becomes [N/2][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 NN is odd. If NN is even the torus can be stable or unstable depending on the natural frequencies, and both possibilities persist in the small coupling limit.