On the Basin of Attractors for the Unidirectionally Coupled Kuramoto Model in a Ring

We present the long-time dynamics of unidirectionally coupled identical Kuramoto oscillators in a ring, when each oscillator is influenced sinusoidally by a single preassigned oscillator. In this situation, for a large system with N ≥ 5, it is well known that the synchronized states and the splay-state are the only stable equilibria by Gershgorin's theorem and linear stability analysis, whereas for low-dimensional systems with $N=2,3$ the synchronized state is the unique stable equilibrium. We present nontrivial proper subsets of synchronized and splay-state basins with positive Lebesgue measure in N-phase space. For the threshold case $N=4$, we show that the splay-state is nonlinearly unstable by explicit construction of perturbations converging toward the synchronized state asymptotically.