Abstract
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 12 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 12 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 13. 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.
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