Saturation property of mean growth of initial error for chaos systems

Abstract

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.