The Mathematical Modeling And Analysis Of Emergent Behavior

In recent years, the emergent behavior of multi-agent systems has attractedmany scholars’ attention. They try to use biology, statistical physics, mathematics,control theory and so on in order to explain the state of the flock occurring inwhich all birds, fish and other biological systems under limited environmentalinformation and simple rules finally move with the same velocities. On one hand,studying the subject can explain the happening of the above phenomenon, andreveal the essence. On the other hand, it has a good extensive application prospect,such as aircraft fleet, formation control of robot and unmanned aerial vehicles.The main work of this paper is as follows:In chapter one, I elaborate the background of the subject, the purpose andsignificance of the research, and the current research at home and abroad. Andthen I focus on the structure of Vicsek model and Cucker-Smale model(C-S model),and simply introduce some subsequent researches about these two models.In chapter two, first of all, I introduce related properties of C-S model, twodefinitions of emergent behavior, and give a sufficient condition under which theemergent behavior will occur. Next I introduce the basic knowledge of Vicsekmodel, which shows that Vicsek model can be transformed into Kuramoto modelwhen1. At last I introduce related knowledge of Kuramoto model.In chapter three, I give the following fact: whether C-S model is connected ornot, the emergent behavior can occur under certain conditions. If the system isconnected, I can use the properties and theorem of C-S model and one lemma ofKuramoto model to prove that the emergent behavior can occur. If the system isdisconnected, I can use the definition of emergent behavior to prove the emergentbehavior can also occur.In chapter four, I work on that in one dimensional space, C-S model has thecharacteristic of sub-swarms when1/2. Under certain conditions, I proof thatthe sub-swarm will occur. It is only guaranteed by the initial velocities andpositions, and I can know the position of the sub-swarm. In particular, I give thecondition that C-S model can be divided into j1sub-swarms. That is to say,when1/2, any sub-swarm phenomenon will appear. Therefore, to somedegree, this reveals the characteristic that the parameter determines whetherthe emergent behavior of C-S model occurs or not.