Attractor radius for fractional Lorenz systems and their application to the quantification of predictability limits

Abstract

Quantifying the predictability limits of chaotic systems and their forecast models has attracted much interest among scientists. The attractor radius (AR) and the global attractor radius (GAR), as intrinsic properties of a chaotic system, were introduced in the most recent work (Li et al. 2018). It has been shown that both the AR and GAR provide more accurate, objective metrics to access the global and local predictability limits of forecast models compared with the traditional error saturation or the asymptotic value. In this work, we consider the AR and GAR of fractional Lorenz systems, introduced in Grigorenko and Grigorenko [Phys. Rev. Lett. 91, 034101 (2003)] using the Caputo fractional derivatives and their application to the quantification of the predictability limits. A striking finding is that a fractional Lorenz system with smaller Σ, which is a sum of the orders of all involved equal derivatives, has smaller attractor radius and shorter predictability limits. In addition, we present a new numerical algorithm for the fractional Lorenz system, which is the generalized version of the standard fourth-order Runge–Kutta scheme.