Title: Quantitative relaxation dynamics from generic initial configurations in the inertial Kuramoto model Show Chinese title

Title: 惯性 Kuramoto 模型中来自通用初始配置的定量松弛动力学

Abstract: We study the relaxation dynamics of the inertial Kuramoto model toward a phase-locked state from a generic initial phase configuration. For this, we propose a sufficient framework in terms of initial data and system parameters for asymptotic phase-locking. It can be roughly stated as set of conditions such as a positive initial order parameter, a coupling strength sufficiently larger than initial frequency diameter and intrinsic frequency diameter, but less than the inverse of inertia. Under the proposed framework, generic initial configuration undergoes three dynamic stages (initial layer, condensation and relaxation stages) before it reaches a phase-locked state asymptotically. The first stage is the initial layer stage in analogy with fluid mechanics, during which the effect of the initial natural frequency distribution is dominant, compared to that of the sinusoidal coupling between oscillators. The second stage is the condensation stage, during which the order parameter increases, and at the end of which a majority cluster is contained in a sufficiently small arc. Finally, the third stage is the persistence and relaxation stage, during which the majority cluster remains stable (persistence) and the total configuration relaxes toward a phase-locked state asymptotically (relaxation). The intricate proof involves with several key tools such as the quasi-monotonicity of the order parameter (for the condensation stage), a nonlinear Gr\"onwall inequality on the diameter of the majority cluster (for the persistence stage), and a variant of the classical {\L}ojasiewicz gradient theorem (for the relaxation stage). Show Chinese abstract

Abstract: 我们研究惯性 Kuramoto 模型从一般的初始相位配置向相位锁定状态的弛豫动力学。 为此，我们提出了一种关于初始数据和系统参数的充分框架，以实现渐近相位锁定。 它可以大致表述为一组条件，例如正的初始序参数、耦合强度足够大于初始频率直径和固有频率直径，但小于惯性的倒数。 在所提出的框架下，通用初始配置在达到渐近相位锁定状态之前会经历三个动态阶段（初始层阶段、凝聚阶段和弛豫阶段）。 第一阶段是类似于流体力学的初始层阶段，在此期间，初始自然频率分布的影响占主导地位，相较于振子之间的正弦耦合影响。 第二阶段是凝聚阶段，在此期间序参数增加，并且在此阶段结束时，一个主要簇被包含在一个足够小的弧中。 最后，第三阶段是持续和弛豫阶段，在此期间主要簇保持稳定（持续），整个配置逐渐弛豫至相位锁定状态（弛豫）。 复杂的证明涉及几个关键工具，例如序参数的准单调性（用于凝聚阶段）、主要簇直径的非线性 Grönwall 不等式（用于持续阶段），以及经典{\L }ojasiewicz 梯度定理的一个变体（用于弛豫阶段）。