Collective Behaviors And Asymptotic Properties Of Cucker-smale Model And Related Dynamic Systems

Collective behavior of dynamic systems is ubiquitous,in which self-propelled individuals organize themselves into a particular motion through simple rules.During recent decades,a lot of dynamical models were proposed to describe the collective behaviors,such as Kuramoto model,Winfree model,Vicsek model,Cucker-Smale(C-S)model,etc.These seminal works and models have been extensively studied and widely applied in various areas such as engineering,biology,physics and social science communities.This dissertation focuses on the asymptotical behavior for 1-D singular Cucker-Smale model and the discrete-time Kuramoto model.For the singular C-S model on the real line,we first consider the long range case.We prove the uniqueness of the weak solution,unconditional flocking emergence and the explicit collision behaviors.Then,for short range case,we prove the collision avoidance and construct the uniform-in-time lower bound of the relative distance between particles.Moreover,we provide the sufficient and necessary condition for the emergence of multi-cluster formation.Finally,for critical case,we show the uniform lower bound of the relative distance and unconditional flocking emergence.These results provide a complete classification of the collective behavior for the singular C-S model on the real line.On the other hand,we study the emergent dynamics of the discrete Kuramoto model for generic initial data on the circle.More precisely,we construct the dissipation structure of the discrete gradient flow,which yields the emergence of synchronization of discrete Kuramoto model.