Synchronization And Normal Deviation Of Stochastic Differential Equations

The synchronization of coupled system is a common phenomenon in many disciplines such as biology,physics and sociology.It describes the consistency of a certain behavior of the components in the system over time,and explains the dynamic mechanism of chaos in an isolated system and order in a coupled system spontaneously.In mathematics,the research on synchronization has attracted the attention of many scholars.They studied the synchronization of deterministic autonomous and non-autonomous systems based on equilibrium and attractors of coupled system.They also used the concepts of stochastic stationary solution and stochastic attractors to study the influence of noise on the synchronization phenomenon.Moreover,the convergence rate of synchronization is obtained.These studies are closely related to research fields such as stochastic dynamic systems,stochastic analysis and multi-scale methods.On this basis,this dissertation studies the synchronization and convergence rate of a stochastic lattice system driven by additive fractional Brownian motion,as well as the normal deviation of synchronization for stochastic coupled systems.It is organized as follows:One part studies the synchronization of stochastic lattice system driven by additive fractional Brownian motion.We first use the direct coupling method of linear crossover to the original system to get stochastic lattice system.And we then prove stochastic lattice system have stochastic stationary solutions,which are globally asymptotically stable for fixed coupling factor.We finally prove the unique stationary solution of lattice system converges to the unique stationary solution of averaged system as coupling factor tends to infinity.In addition,we also obtain the convergence rate of synchronization for lattice system.This result expands and improves the existing research.As an important tool for obtaining the convergence rate,the uniform boundedness of fractional Ornstein-Uhlenbeck equations with singular parameters is also studied.The research content of this part is detailed in Chapter3.The other part focuses on the normal deviation of the synchronization of stochastic coupled systems.To the best knowledge of the authors,the existing literature about synchronization mainly concentrates on the results of synchronization and the corresponding convergence rate,leaving the normal deviation of synchronization unsolved,which is exactly what we do research into in this part.First,the coupled system will be transformed into a multi-scale system.Second,the martingale method is used to prove normal deviation of general solution of fully coupled multi-scale system.According to the relationship between stationary solution and the solution with fixed initial values,thereby obtaining normal deviation of stationary solution.Finally,using the established relationship between the normal deviation of the multi-scale system and the normal deviation of the synchronization of the coupled system,the normal deviation of the synchronization of the coupled system is obtained.We propose and study the normal deviation of the stationary solution of the multi-scale system.We propose and study the normal deviation of the stationary solution.We have obtained convergence in probability in the case of additive noise,and only weak convergence in the case of multiplicative noise.The details are elaborated in Chapter 4,Chapter 5 in this dissertation.