标题： Ott-Antonsen 对 D 维 Kuramoto 模型的假设：一种构造性方法 显示英文标题

标题： Ott-Antonsen ansatz for the D-dimensional Kuramoto model: a constructive approach

摘要： Kuramoto的原始模型描述了一组由其相位表示的相互作用振荡器的动力学和同步行为。 该系统也可以被描绘为在二维空间中沿圆周运动的一组粒子，这允许直接推广到在高维球面表面运动的粒子。 2D系统的其中一个关键特征是随着耦合强度的增加，会出现连续的同步相变。 Ott和Antonsen提出了一个振荡器分布的假设，使他们能够用一个微分方程来描述过渡的序参数的动力学。 后来，通过使用2D假设的相同函数形式并调整其参数，为D维模型提出了类似的假设。 在本文中，我们开发了一种构造性方法来寻找假设，类似于在2D中使用的程序。 该方法基于我们之前对3D Kuramoto模型的工作，在该工作中，假设是通过分布函数的球谐分解构建的。 在D维球面上运动的情况下，假设基于超球谐分解。 我们的结果与之前提出的假设不同，并提供了序参数和假设之间更简单、更直接的联系。 显示英文摘要

摘要： Kuramoto's original model describes the dynamics and synchronization behavior of a set of interacting oscillators represented by their phases. The system can also be pictured as a set of particles moving on a circle in two dimensions, which allows a direct generalization to particles moving on the surface of higher dimensional spheres. One of the key features of the 2D system is the presence of a continuous phase transition to synchronization as the coupling intensity increases. Ott and Antonsen proposed an ansatz for the distribution of oscillators that allowed them to describe the dynamics of the transition's order parameter with a single differential equation. A similar ansatz was later proposed for the D-dimensional model by using the same functional form of the 2D ansatz and adjusting its parameters. In this paper we develop a constructive method to find the ansatz, similarly to the procedure used in 2D. The method is based on our previous work for the 3D Kuramoto model where the ansatz was constructed using the spherical harmonics decomposition of the distribution function. In the case of motion in a D-dimensional sphere the ansatz is based on the hyperspherical harmonics decomposition. Our result differs from the previously proposed ansatz and provides a simpler and more direct connection between the order parameter and the ansatz.