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Chaotic Systems Part 3

Tham khảo tài liệu 'chaotic systems part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 3 Relationship between the Predictability Limit and Initial Error in Chaotic Systems Jianping Li and Ruiqiang Ding State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics Institute of Atmospheric Physics Chinese Academy of Sciences Beijing 100029 China 1. Introduction Since the pioneer work of Lorenz on predictability problems 1-2 many studies have examined the relationships between predictability and initial error in chaotic systems 3-7 however these previous studies focused on multi-scale complex systems such as the atmosphere and oceans 4-6 . Because large uncertainties exist regarding the dynamic equations and observational data related to such complex systems there also exists uncertainty in any conclusions drawn regarding the relationship between the predictability of such systems and initial error. In addition multi-scale complex systems such as the atmosphere are thought to have an intrinsic upper limit of predictability due to interactions among different scales 2 4-6 . The predictability time of multi-scale complex systems regardless of the errors in initial conditions cannot exceed their intrinsic limit of predictability. For relatively simple chaotic systems with a single characteristic timescale driven by a small number of variables e.g. the logistic map 7 and the Lorenz63 model 1 their predictability limits continuously depend on the initial errors the smaller the initial error the greater the predictability limit. If the initial perturbation is of size S0 and if the accepted error tolerance A remains small then the largest Lyapunov exponent Z1 gives a rough estimate of the predictability time Tp - ln - - However reliance on the largest Lyapunov exponent commonly proves to be a considerable oversimplification 8 . This generally occurs because the largest Lyapunov exponent A1 which we term the largest global Lyapunov exponent is defined as the long-term average growth rate of a very small initial error. It is