On the Łojasiewicz exponent of Kuramoto model

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 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. C 2015 AIP Publishing LLC.The contributions of this paper are summarized as follows. First, we compute the Łojasiewicz exponent for Kuramoto model; more precisely, we will prove a two-side version of Łojasiewicz inequalityfor a large class of θ 0 . Here, the notation "≈" meansfor θ ∈ N (θ 0 ). An example with exponent γ < 1 2 was also given for Kuramoto model. We calculate the Łojasiewicz exponent for the potential f of Kuramoto model consisting of two distinct natural frequencies with threshold coupling. We prove that it is 1 3 for the bi-cluster phased locked equilibriums. In particular, the two-side gradient inequality with exponent 1 3 is true for two-oscillator systems. These results, combining with Theorems A.1 and A.2, illustrate how the Łojasiewicz exponent affects the convergence rate. Second, the applications of Łojasiewicz exponent to the convergence analysis of first-order and second-order Kuramoto model are provided. We will give a general theorem on the exponential convergence of second-order gradient-like systems based on property (1.5). This extends the study of Haraux and Jendoubi 23 in which the convergence was derived but without any information on the convergence rate. Through this general result, we can give a criterion for the exponential convergence of Kuramoto model with inertia.Before we close this section, we recall some definitions for the complete synchronization of Kuramoto model. Definition 1.1. Let θ(t) = (θ 1 (t), . . . , θ N (t)) be an ensemble of phases of Kuramoto oscillators.

Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives

“…are the natural frequencies and m i = 1, then (5.1) is nothing but the Kuramoto model for coupled oscillators This is a classical model that was proposed by Kuramoto as a prototype system exhibiting emergence of synchronization 93,111,115,121,158 . It is intimately related to opinion dynamics and the heterogeneities ν i play the role of a biased tendency of agents to move at their own frequency while being influenced by their neighbors.…”

“…From a mathematical point of view, there have been important contribution in the analysis of phase and frequency synchronization in the system. The reader may want to look in 93,111,115,121,158 and references therein. Here we will sketch some of the most relevant results in the study of synchronization.…”

Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model

“…In [30] it was noticed that the discrete Kuramoto model (1) has a Lyapunov function so that it can be reformulated as a standard gradient system with an analytic potential. Based on this reformulation, in [10,14,20] the Lojasiewicz inequality in finite dimensions was used to prove the convergence towards a phase-locked state for system (1). Two exponents show up for (1), ρ = 1 2 or 1 3 , depending on the configurations of phase-locked states [20].…”

“…Based on this reformulation, in [10,14,20] the Lojasiewicz inequality in finite dimensions was used to prove the convergence towards a phase-locked state for system (1). Two exponents show up for (1), ρ = 1 2 or 1 3 , depending on the configurations of phase-locked states [20]. For other studies on the second-order Kuramoto model with inertia by this inequality we refer to [7,19,21].…”

“…Proof of the Claim (20). If the open set T consists of finite number of open intervals, then as a result of t n ∈ T and t n → +∞, there must be an interval of the form (t * , +∞) such that (t * , +∞) ⊂ T .…”