Abstract
Quantifying the predictability limits of chaotic systems and their forecast models is an important issue with both theoretical and practical significance. This paper introduces three invariant statistical properties of attractors, namely the attractor radius, global attractor radius (GAR), and the global average distance between two attractors, to define the geometric characteristics and average behavior of a chaotic system and its error growth. The GAR is sqrt 2 times the attractor radius. These invariant quantities are applied to quantitatively measure the global and local predictability limits (both have practical and potential predictability limits, which correspond to the attractor radius and GAR, respectively) of both global ensemble average forecasts and one single initial state, respectively. Both the attractor radius and GAR are intrinsic properties of a chaotic system and independent of the forecast model and model errors, and thus provide more accurate, objective metrics to assess the global and local predictability limits of forecast models compared with the traditional error saturation or asymptotic value (AV). Both the Lorenz63 model and operational forecast data are used to demonstrate the theoretical aspects of these geometric characteristics and evaluate the feasibility and effectiveness of their application to predictability analysis.
Highlights
Many nonlinear physical systems in the real world display complex chaotic behavior that makes prediction of their variations challenging
Because a forecast model usually has associated model errors and the model attractor M is different from the real attractor, the saturation level of the global ensemble average root-meansquare error (RMSE) ēM(δ0, t), which corresponds to the traditional asymptotic value (AV) and the global average distance (GAD) between the two attractors and M, is not the same as that of ē(δ0, t)
This paper introduces several geometric properties of attractors, the Local attractor radius (LAR), attractor radius, and global attractor radius (GAR) of an attractor, as well as the Local average distance (LAD) and GAD between two attractors, to characterize features and the average behavior of chaotic systems and evaluate error growth in nonlinear dynamical systems
Summary
Many nonlinear physical systems in the real world (e.g., the atmosphere and the ocean) display complex chaotic behavior that makes prediction of their variations challenging. The error growth in deterministic forecasts is largely determined by model deficiencies for medium- and extended-range forecasts (Orrell et al 2001) This means that the AV of forecast error depends on the numerical model used, and is not an objective threshold to quantify and compare the predictability limits between predictions from different models. This study proposes a unified framework for quantifying global and local predictability limits of both chaotic dynamical systems and their forecast models. 3. Section 4 presents the results of applying the attractor radius and GAR to quantifying global and local predictability limits of an original dynamical system and its forecast model, along with a comparison with the traditional error AV method.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
