Application Of Renormalization Group Method To Singular Perturbation Problems Driven By Fractional Brownian Motion

Stochastic noise widely exists in realistic systems in the fields of science and en-gineering.In recent years,stochastic singular perturbation systems,as a description of singular perturbation systems driven by stochastic noise,have received widespread attention.Developing effective methods to study the analytical solutions,approxi-mate solutions or dynamical behavior of such systems has become one of the research hotspots and difficulties in the fields of mathematics and engineering.In this disser-tation,two kinds of singular perturbation problems driven by fractional white noise are studied.The uniformly effective asymptotic solutions of these two are obtained by utilizing the renormalization group method.The results are summarized as follows:(1)The Stratonovich-type singular perturbation problem driven by fractional Brownian motion is studied.To obtain the uniformly valid asymptotic solution of the problem,the renormalization group method is used to construct the asymptotic solution to the order of O(ε~α),and then the uniformly validity is proved.Finally,tak-ing the linear system and slow–fast system driven by additive fractional white noise as examples,the asymptotic solution and exact solution are numerically simulated by MATLAB.(2)The It(?)-type singular perturbation problem driven by multiplicative fractional white noise is investigated.The uniformly effective asymptotic solution of the prob-lem was obtained using the renormalization group method,that is,the O(ε~a)order asymptotic solution was constructed,and the uniformly effective proven was given.Finally,taking a linear system driven by multiplicative fractional white noise and the Lorenz equation as examples,numerical simulations are conducted using MATLAB to investigate the asymptotic and exact solutions.These results extend the application scope of the renormalization group method and reveal the mechanism of action of different types of fractional white noise on singular perturbation systems.