Prediction Errors and Local Lyapunov Exponents

TL;DR

This paper investigates how local Lyapunov exponents influence error growth in chaotic systems, revealing non-exponential error scaling and providing a model to predict observed error behaviors relevant to atmospheric predictability.

Contribution

It introduces a simple model that quantitatively links local Lyapunov exponents to error scaling, challenging the assumption that exponential error growth indicates chaos.

Findings

Local Lyapunov exponents vary significantly over attractors.

Error growth often deviates from exponential scaling.

The model accurately predicts error behavior over long times.

Abstract

It is frequently asserted that in a chaotic system two initially close points will separate at an exponential rate governed by the largest global Lyapunov exponent. Local Lyapunov exponents, however, are more directly relevant to predictability. The difference between the local and global Lyapunov exponents, the large variations of local exponents over an attractor, and the saturation of error growth near the size of the attractor---all result in non-exponential scalings in errors at both short and long prediction times, sometimes even obscuring evidence of exponential growth. Failure to observe exponential error scaling cannot rule out deterministic chaos as an explanation. We demonstrate a simple model that quantitatively predicts observed error scaling from the local Lyapunov exponents, for both short and surprisingly long times. We comment on the relevance to atmospheric predictability as studied in the meteorological literature.