混沌系统可预报期限随初始误差变化规律研究

Study on the regularity of predictability limit of chaotic systems with different initial errors

利用非线性误差增长理论计算了Logistic映射和Lorenz系统可预报期限随初始误差的变化,发现Logistic映射等简单混沌系统的可预报期限与初始误差的对数存在线性关系.在非线性误差增长理论的框架下,理论分析表明,平均误差增长达到一定值时,误差增长进入明显的非线性增长阶段,最终达到饱和;对于一个确定的混沌系统,在控制参数固定的情况下误差增长的饱和值也是固定的,因此可预报期限只依赖于初始误差.在可预报期限与初始误差对数存在的线性函数关系式中,线性系数与最大Lyapunov指数有关,在已知混沌系统的最大Lyapunov指数和某个固定初始误差的可预报期限的条件下,利用可预报期限与初始误差对数的线性函数关系可以外推得到任意固定初始误差的可预报期限.

The predictability limits of the Logistic map and Lorenz system as functions of initial error are calculated by employing the nonlinear error growth dynamics. It is found that there exists a linear relationship between the predictability limit and the logarithm of initial error. It is revealed by the theoretical analysis under the nonlinear error growth dynamics that the growth of average error will enter the nonlinear growth phase after the error reaches a certain critical magnitude and will finally reach saturation. For a given chaotic system, if the control parameters of the system are given, then the saturation of error growth is determined. Therefore, the predictability limit of the system only depends on the initial error. This is different from the linear error growth dynamics, under which the predictability time scale of chaotic system also depends on the upper limit of forecast error. In the linear expression between the predictability limit and the logarithm of initial error, its linear coefficient is relevant to the largest global Lyapunov exponent of chaotic system. If the largest global Lyapunov exponent and the predictability limit corresponding to a fixed initial error are known, the predictability limit corresponding to other initial errors can be extrapolated by the linear function expression between the predictability limit and initial error.