Quantifying local instability and predictability of chaotic dynamical systems by means of local metric entropy

A local metric entropy (LME) is introduced and used as a measure of local instability of chaotic dynamical systems. The predictability time scale of a dynamical system during a given period of time can also be estimated with high accuracy by using the LME. It is shown that LME, at any time during the evolution of a dynamical system, can be calculated as the sum of all the positive local Lyapunov exponents (LEs). This conclusion implies that the positive local LEs represent the rates of local information changes along the directions of their respective Lyapunov vectors. LME does not depend upon the amplitudes nor the configurations of initial perturbations; it depends on the positive local LEs which are intrinsic properties of dynamical systems. In addition, the sum of all the local LEs is proven to be equal to the divergence of phase space. Thus for a general chaotic system at any time, the sum of all the local LEs is equal to the sum of all the local growth rates of either instantaneous optimal modes or normal modes. In analyzing local instability, the performance of LME is evaluated by comparing an instability index with LME, the first local LE, locally largest LE, local growth rates of the dominant instantaneous optimal mode and normal mode. When LME is used to estimate the predictability time scales of systems over specified time periods, it is found that the time scales defined by LME are generally closer to the standard predictability times than the Lyapunov times and Kolmogorov-Sinai times for most cases in the two dynamical systems we have tested. Both the instability index and standard predictability time are defined and calculated through a large number of random errors.