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

Abstract

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.