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

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

2009-03-27

arxiv.org/abs/0903.4840v1

Nonlinear Sciences

Adaptation and Self-Organizing Systems

to appear in Physica D

Scientific paper

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.

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