Relationship between the Predictability Limit and Initial Error in Chaotic Systems

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 0 δ and if the accepted error tolerance, Δ , remains small, then the largest Lyapunov exponent 1 Λ gives a rough estimate of the