Analysis Of Large Time Behavior Of Two Kinds Of Stachastic Group Dynimics Models

Emergent collective behaviors are ubiquitous phenomena that had been found in a group of autonomous agents.The terminology flocking represents phenomena in which self-propelled particles using only limited environmental information and simple rules organize into an ordered motion.Whether in natural science or social science,it is much significance for collective behavior.Hence,many scholars focused their efforts on the modeling and research of collective behaviors.We study the collective behaviors of stochastic Motsch-Tadmor model and stochastic and continuum Kuramoto model based on stochastic differential equation theory,quasi-gradient theory and random process and so on.The main results are listed as follows:Firstly,we study the flocking of stochastic Motsch-Tadmor model.The effect on MT model driven by collective behaviors is analyzed and studied.We start to enlarge the critical exponent?that is measuring the strength of the interaction between agents of unconditional flocking,then prove that the system show the unconditional flocking in stochastic MT model driven by the multiplicative white noise for arbitrary exponent?>0,that is,the velocity of every agent is consistent with each other and obtain the conclusion that the relative speed of agents will be exponential convergence.Moreover,we give the convergence of each agent's velocity when the exponent??1/2.Secondly,we respectively study the effort for synchronization based on the discrete Kuramoto model with additive white noise and multiplicative white noise.In the case of additive white noise,we prove that there is no phase synchronization in mean square for any coupling and noise strength in the system.In contrast to additive white noise,we give the sufficient condition to guarantee phase synchronization in Kuramoto model with Multiplicative white noise.Moreover,we also consider the extend Kuramoto model in which the environmental random effect which act on the oscillators continuously in time.For this model,we start to give the definition of critical coupling strength admissible for frequency synchronization with probability p in a given initial area M,then obtain a upper bound estimate of this critical coupling strength when the natural frequency of oscillator follows the given probability distribution.At last,we study the convergence and asymptotic stability using method of?ojasiewicz inequality in infinite dimensional space.We start to give the specific formula of continuum Kuramoto model using strong law of large numbers as N??and prove the convergence and stability of phase-locked solution of system using quasi-gradient flow method based on?ojasiewicz inequality when the initial state and coupling strength satisfy the certain condition.Hence,we present the specific?ojasiewicz inequality about nonlocal integro-differential equation in infinite dimensional space and obtain convergence and stability of quasi-gradient based on this inequality.