Stability Of A Class Of Flocking Models With White Noise

Stochastic differential equation is a cross subject which combines stochastic analysis with differential equation to study the phenomenon of stochastic evolution over time.It originated from British botanist Brown’s study on the irregular movement of pollen in water,Later,after the theoretical work of Einstein,Wiener,Ito and others,it finally became an important subject of modern mathematics,which was widely concerned by mathematicians,physicists,biologists,economists and other scholars in almost all fields.In this paper,we will mainly study the long-term stability of a kind of flocking model with white noise disturbance.In a rough sense,flocking refers to a special phenomenon of group aggregation formed by the evolution of individuals following simple interaction rules over time.For example,we often see wild geese flying in groups in the sky,fishes swimming in a fixed direction in the water,and language formed in the progress of human society.People’s researches on this phenomenon mainly focus on observing the inner biology and mathematical mechanism of this phenomenon,and on this basis,building a simple and practical mathematical model and practical application program,such as UAV,robot formation control and other fields.Our specific research model is as follows:#12(xi,vi)∈ R2d,represents the state of the i th particle at time t,(xi,vi)(0)=(xi0,vi0)represents the initial state of the ith particle.λ>0 denotes the coupling strength,which is a constant greater than 0;N represents the number of particles with unit mass in this system;Ψ denotes Weight of communication,that is,to measure the impact of the jth particle on the ith partical,is a number greater than 0;D is the noise stength,a constant greater than 0;dW(t)denotes white noise;gi(v)＝vi-ve,ve is a constant vector in Rd,xi,vi are vectors in Rd.Let’s first look at the situation where there are only two particles in the system.By analyzing vi(t),the concrete form and its changing trend with time,we finally get the clustering phenomenon of two particles.Then we split the original equation into two parts:macro and micro.By using the formula of Ito and the law of iterated logarithm,we will analyze the case of any N particles in the system and and give the corresponding analysis of long-term dynamic properties.