On the Łojasiewicz exponent of Kuramoto model

The Łojasiewicz inequality and Łojasiewicz exponent reveal a fundamental relation between a potential function and its gradient. In this paper, we explore the Łojasiewicz exponent of Kuramoto model and prove that the exponent is exactly 1/2 for equilibria located inside a quarter of circle. This implies that the convergence towards such a phase-locked state must be exponentially fast. In contrast, we give an example to see the exponent can be less than 1/2 for other equilibriums. More precisely, we prove that the exponent for the bi-cluster equilibrium, which is located on the boundary of a quarter of circle, is 1/3. This gives an insight for the occurrence of exponential and algebraic convergence of Kuramoto model. We also present a general theorem for exponential convergence of second-order gradient-like system, by which a criterion for the Kuramoto model with inertia is established.