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

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.