随机平均规范形方法

STOCHASTIC NORMAL FORMS WITH AVERAGING METHODS

计算随机规范形系数是应用随机规范形方法的关键.提出一种应用随机平均计算随机规范形系数的方法.为了说明方法的有效性,对白噪声激励的Duffng系统,经过变换,对于相应的平均方程,比较了精确解、规范形方法解和能量包络方法解的稳态概率密度,结果表明,当非线性项系数较小时,三者完全一致.当非线性项系数较大时,规范形方法所得解与精确解相差不大.

The theory of normal forms for deterministic systems is a method of simplifying a system of differential equations by the use of near-identity change of coordinates. The idea of normal forms goes back to as early as Poincare. Lyapunov, Birkhoff and Arnold were systematically expands in the theory and the methods. The ideas of normal forms were extended to stochastic systems by Knobloch and Wiessenfeld (1983), they applied the idea of normal forms to the Fokker-Planck equation and concluded that the classification of deterministic bifurcation problems carry over to the stochastic case. Coulett (1985) performed the normal transformation directly on the stochastic equations and found addition terms, which can not be eliminated due to 'stochastic resonance'. The applicability of the method of normal forms to nonlinear stochastic systems was extended by Sri Namachchivaya et al. (1990, 1991), they studied the equivalence of stochastic averaging and stochastic normal forms for a specific class of nonlinear systems, and showed that unlike stochastic averaging, stochastic normal forms can be used in the analysis of nilpotent systems. Arnold (1992,1995) do not studied normal forms for random diffeomorphisms and random differential equations, and build e-normal form theory. However, the applicability of the method of normal forms to nonlinear stochastic systems is still problem, the manly difficulty is computation of the coefficients of stochastic normal forms. In general, one has to resolve a lot of stochastic differential equations to obtain the coefficients, however, the equations can not be calculated in most cases. In this paper, we study the normal forms for stochastic systems by use of stochastic averaging methods. The ideal is that we chose the near-identity change of coordinates with stochastic averaging methods, such that the stationary density function of averaging equation can be obtained. Unlike classification stochastic normal form, the methods do not need to resolve stochastic differential equations to obtain the coefficients. We conclude with an example, the stochastic Duffing system, which demonstrates the idea explicitly.