集合平均所构成的动力系统及其特性研究

An approach for analyzing the ensemble mean from a dynamic point of view

摘要:

以Lorenz系统为例,推导出集合平均所定义的完整动力方程(均值方程),将初值的集合平均问题作为一个广义的动力系统问题来进行研究;对于双初值和多初值的均值方程,利用定性理论分析了其吸引中心的位置和个数,并使用数值试验进行了验证,结果表明平均值的吸引子的结构与原解的吸引子位置、数量和结构均有不同.对均值方程的特征矩阵分析表明,定点附近的稳定性与原方程相同,而且特征方程所对应的特征值也与原方程相同.均值方程对应的相流散度为负值且数值上与原系统相同,因此其在相空间中的体积收缩速度和原系统相同,最终趋向一个低纬曲面,均值方程的这个性质使得Lorenz系统的集合平均解趋于一个吸引子.均值方程可以保持原方程的耗散特性、吸引子特性,但稳定点位置和个数发生了变化,非定点处的Jacobian矩阵特征值与原系统也有不同.简而言之,一旦使用了集合平均方法,那么集合数值解并不是原系统的解,仅保持了原系统的部分特征,因而集合平均是否有效需要根据具体问题和其他外部限定条件才能确定.

关键词:Lorenz方程;集合平均;稳定性;Jacobian矩阵

Abstract:

Simultaneous ensemble mean equations for the Lorenz model(LEMEs) are obtained, enabling us to analyze the properties of the ensemble mean from a dynamical point of view.The qualitative analysis for the two-sample and N-sample LEMEs shows that the locations and number of stable points are different from the Lorenz equations(LEs), and the results are validated by numerical experiments.The analysis for the eigenmatrix of stable points of LEMEs indicates that the stability of these stable points is similar to the Les'.The eigenmatrix for non-stable points can be obtained too, but the eigenvalues depend not only on the value of the mean variable but also the other N-1 sample equation's variable, and thus for these points there may be different stabilities compared to the LEs'.The divergence of the LEMEs' flow has a negative value, which is the same as the LEs', and thus the trajectory in phase space approaches zero and the trajectory will be attracted to a low-level dimensional curved surface, i.e., the LEMEs have the attractor property, but the structure of the attractor is not the same as the LEs'.The emsemble mean method only keeps part of the original equations' properties, and thus whether the method is effective depends on the objective problem and other external restrictive conditions.

Key words:Lorenz equation;ensemble mean;stability;Jacobian matrix