混沌系统单变量可预报性研究

Studies of Predictability of Single Variable from Multi-dimensional Chaotic Dynamical System

对于n维的混沌系统,不同变量的可预报性是不同的。为了研究混沌系统中单个变量的可预报性,本文在以前提出的混沌系统整体的非线性局部Lyapunov指数基础上(李建平等,2006),引入了单变量的非线性局部Lyapunov指数及其相关统计量,进一步完善了非线性误差增长理论。通过应用到几个混沌个例,结果表明单变量的非线性局部Lyapunov指数及其相关统计量可以用来定量地研究多维混沌系统中不同变量的可预报性,系统不同变量的可预报性之间不是相互独立的,而是单个变量的可预报期限与系统整体的可预报期限之比都近似保持一个常数,但各个变量的常数值有所不同。

To study the atmospheric predictability from the view of nonlinear error growth dynamics, a new approach, the nonlinear local Lyapunov exponent （NLLE）, is introduced by the authors recently. The NLLE and its derivatives can be used to quantify the predictability of chaotic dynamical systems. For an n-dimensional chaotic dynamical system, the predictability of its different variables is proved to be different. To quantify the predictability of single variable of multi-dimensional chaotic dynamical system, the nonlinear local Lyapunov exponent （NLLE） of single variable is introduced, which is based on the definition of the NLLE of the whole system. Taking the Henon map and Lorenz system as examples, the results indicate that the NLLE of single variable and its derivatives can be used to measure quantitatively the predictability of different variables. In addition, it is not independent among the predictability of different variables. The ratio of the predictability limit of single variable to that of the whole system nearly keeps constant with the change of magnitude of initial error. However, the constant values for different variables are different.