Stability Analysis Of A Class Of Flocking Models

As we all know,there are some group behaviors with common goals in both nature and human society.There is a kind of phenomenon,which can produce group consistent behav-iors only through local interaction among individuals,such as birds,fish,cell organization,etc.,which is called flocking.In the past few decades,scholars from different disciplines,such as biology,physics,traffic mechanics and Social Sciences,have made detailed ob-servation and research on flocking phenomenon,and established many flocking models to describe various types of flocking phenomenon.Recently,people are more concerned about when flocking occurs,how flocking occurs through the interaction between individuals,and how different "participation rules" affect the formation.In this paper,we mainly consider the local stability of the flocking solution family of the following flocking model#12 Where ' is derivatives of time 1,(1-|vi|2)vi is Rayleigh friction,aij=?(|xi-xj|)is a nonnegative,symmetric and differentiable function,which represents the influence function between individuals.Then we mainly discuss the family of flocking solutions as{(x,v)vi=vj=v0,|v0|=1,i,j=1,…,N} and {(x,v)|vi=vj=0,i,j=1,…,N}.In order to achieve our goal,we analyze the stability of the compact invariant manifold formed by the family of equilibrium solutions of the equation#12This paper is divided into three chapters.The first chapter is the introduction,which briefly introduces several classical flocking models and the flocking phenomena described.The second chapter is the preparatory knowledge,including graph theory,tensor product,hyperbolic invariant manifold and so on,so as to make preparation for the later work.The third chapter is the main content of this paper.We will discuss the stability of flocking solution family of the above model in detail then get the corresponding flocking results.