| Floods of flocking behaviors happen in human life and environment.For instance,hundreds of migrant birds gradually reach to the same velocity in the same direction while they fly from north to south in winter and in human society,people often get the unified conclusion after good communication.Flocking has so huge values that scientists analyze it very much recently.On on hand,it simulates behaviors of animals,the system of robots,UAV team and sensors,etc.On the other hand,it can explain many phenomenons in social science like the evolution of human languages.This article mainly focuses on the interaction of two different ensembles which finally reach flocking under the generalized Cucker-Smale model.We set two homogeneous ensembles as G1=t(x1i,y1i)}i=1 N1,G2={(x2j,v2j)}j=1 N2,Then consider the following system:(?)where xpq and vpq represent the displacement and velocity,·is the derivative to time,K1,K2 and Kd represent the coupling strength of G1,G2 and the interaction of G1 and G2.For simplicity,we set:(H1)ψα:R+→R(α=1,2,d)is Lipschitz continuous and 0<ψα(s)≤ψ∝(0)=1<+∞,ψα(s)∈L1(R+),(ψα(S2)-ψα(s1))(s2-s1)≤0,s1,s2∈R+.(H2)Γα:Rn→Rn(α=1,2,d)is continuous and for any given v∈R+,(1)Γα(v)=-Γα(-v),Γα(v)·v≥0;(2)exists posotive constant Cα,and γ∈[1,3/2),s.t Γα(v)·v≥ Cα‖v‖2γ.We first give some indexes to measure flocking.Then get some inequalities on velocity and displacement and finally use Lyapunov functional method to get the sufficient condition of reaching flocking.The article is divided into three chapters.The first one introduces the background and values of flocking briefly.In next chapter we give sufficient condition for flocking under Cucker-Smale model and the generalized Cucker-Smale model as preliminary knowledge.The last part contains the main result of this paper.Using Lyapunov functional,we get sufficient condition for flocking. |
