非一致多个体耦合系统的动力学性质及应用

本项目利用两类相互联系的非一致多个体耦合系统，Cucker-Smale类模型和Kuramoto类耦合振子模型，来分别探讨非一致个体的速度同步行为和频率同步行为。这里的“非一致”代表群体中个体成员的非均匀特征，如质量或惯性系数、阻尼系数及连接强度。它们为研究一些自然现象，以及工程中的相关问题，如多飞行器编队、电力网络系统等，提供了数学模型。本项目将结合Lyapunov能量方法、梯度不等式、网络和图理论，并运用有关的微分差分方程理论，来研究多个体耦合系统的渐近同步行为。本项目重点关注非一致Cucker-Smale类模型的速度同步行为，及非一致Kuramoto类模型的渐近频率同步性和平衡点吸引域等问题，为工程中的相关问题提供理论依据。

In this project, we will employ two classes of interrelated non-uniform multi-agent systems, Cucker-Smale type model and Kuramoto type model, to investigate the synchronous motion in velocity and frequency respectively. Here, “non-uniform” means that the agents in the systems have heterogeneity in some characterizations, such as mass or inertia coefficients, damping coefficients, and the coupling strength. These systems provide mathematical models for describing some natural phenomena and related problems in engineering, for example, the flying vehicle formation and power grids, etc. In this project, we will combine the Lyapunov energy method, gradient inequality, network and graph theory, and related theory in differential and difference equations, to study the asymptotic collective behavior of the systems. We mainly focus on the velocity synchronization in non-uniform Cucker-Smale type model and the asymptotic frequency synchronization and attraction region of phase-locked equilibria in Kuramoto type model. These studies will help to provide fundamental basis for related problems in the real world.