Numerical Stability Of A Class Of Stochastic Hamiltonian Systems

A Hamiltonian system is a dynamical system governed by Hamiltonian equations.In physics,this dynamical system describes the evolution of a physical system such as a plan-etary system or an electron in an electromagnetic field.These systems can be studied in both Hamiltonian mechanics and dynamical systems theory.A Hamiltonian system is a dy-namical system completely described by the scalar function H(p,q,t),the Hamiltonian.The state of the system,is described by the generalized coordinates momentum p and position q where both p and q are vectors with the same dimension n.So,the system is completely described by the 2n dimensional vector.In recent years,with the development of stochastic mechanics theory,the stochastic Hamiltonian systems have attracted more and more schol-ars’attention.The stochastic Hamiltonian systems incorporate white noise on the determin-istic Hamiltonian systems,which has rich physical and geometric properties such as energy conservation and symplectic structure.This paper mainly discusses the numerical stability of a class of stochastic Hamiltonian systems.Firstly,the stability definition includes the stability of equations and the stabil-ity of numerical methods.Starting-ffom the mumerical solution methhod,the experimental equations of stochastic Hamiltonian systems are established.Stochastic Hamiltonian sys-tem with inseparable variables and stability of system equations with single noise,which is transformed into the stability of the stochastic differential equations of the two-dimensional Stratonovich type.In combination with the stability and numerical methods for solving the Ito-type stochastic differential equations before,this paper proposes a numerical simulation using the two-stage stochastic Runge-Kutta method of symplectic structure,and discusses the stability domain and the stability domain of the system equations obtained by numerical methods,and finally obtains the symplectic structure when the coefficient a takes 0 or 1/2.The two-stage stochastic Runge-Kutta method is A-stable in mean-square.