混沌系统平均初始误差增长饱和特性研究

Saturation property of mean growth of initial error for chaps systems

在非线性误差增长理论框架下研究了混沌系统平均初始误差增长饱和特性以及误差饱和值同系统可预报期限的关系.首先探索了Lorenz96系统中平均相对初始误差增长饱和规律,发现平均相对初始误差增长饱和值同初始误差的自然对数存在简单的线性关系:其二者自然对数之和为一常量,且该常量同初始误差无关.实验表明该结论对其他混沌系统也适用.因此对给定混沌系统,在计算出和常数后可以外推得到任意固定初始误差的平均相对误差增长饱和值.为进一步研究误差饱和值同可预报期限的关系,给出了平均绝对误差增长的定义.理论分析表明混沌系统平均绝对误差增长也会达到饱和.其饱和值为常量,与初始误差无关,混沌系统控制参数确定,饱和值就固定.依据上述研究,最后给出一个定量计算可预报期限的模型T_p=1/Λln(Es/δ0)+c,Es为绝对误差增长饱和值.实验研究表明对于复杂的高阶混沌系统,该预报期限模型都能较好地适用.

The saturation property of mean growth of initial error and the relation between saturation value and predictability limit of chaos system are studied in a frame of the nonlinear error growth dynamics. Firstly, the saturation property of mean relative growth of initial error （RGIE） of Lorenz96 system is investigated. It is found that there exists a simple linear relationship between the logarithm of saturation value of mean RGIE and initial error. The sum of logarithms of the two is constant that is independent of the magnitude of the initial error. It is proven by experiment that this conclusion is suitable for other chaotic systems too. With this conclusion, once the constant sum has been determined, the saturation values of mean RGIE at any magnitude of initial error can be calculated easily. Furthermore, to make the study of the relation between error growth saturation and the predictability limits more convenient, just as the definition of the mean RGIE, a definition of the mean absolute growth of initial error （AGIE） is introduced and theoretical analysis reveals that the AGIE has a similar saturation property as RGIE. The saturation value of mean AGIE is constant, which means for a given chaos system, once the control parameters of the system has been determined, the saturation of AGIE is determined. Finally a model for calculating predictability limit quantitatively is given as follows： Tp=1/∧ln（Es/δ0）＋c, where Es is the saturation value of mean AGIE. It is shown that this model can work with complicated and high dimension chaos system very well.