[Paper Review] Prediction Errors and Local Lyapunov Exponents

Matthew B. Kennel, Henry D. I. Abarbanel+1 more|ArXiv.org|Mar 9, 1994

Complex Systems and Time Series Analysis3 references3 citations

TL;DR

This paper challenges the conventional assumption that prediction errors in chaotic systems grow exponentially at a rate set by the global Lyapunov exponent. Instead, it proposes a model using finite-time local Lyapunov exponents and a saturation cutoff to accurately predict error scaling at both short and long times, showing that non-exponential error growth is expected due to local exponent variability and attractor size constraints.

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.

Motivation & Objective

• To challenge the assumption that prediction errors grow exponentially at the rate of the global Lyapunov exponent.
• To demonstrate that finite-time local Lyapunov exponents better describe actual error growth in chaotic systems.
• To model prediction error scaling that accounts for both initial non-exponential growth and saturation at attractor size.
• To reconcile discrepancies between observed error growth and exponential scaling in meteorological and dynamical systems.
• To provide a more accurate, physically grounded model for predictability in deterministic chaos.

Proposed method

• Uses finite-time local Lyapunov exponents λ₁(x, L) to compute the expansion factor E(x, L) over L time steps.
• Applies a geometric mean of expansion factors across initial conditions to model average prediction error.
• Introduces a saturation cutoff R to model the physical limit of error growth at the attractor size.
• Employs a thresholded logarithmic average of log E(x, L) to model error evolution, with hard limit ρ = log R.
• Uses the iterated predictor error formula (6) to compute normalized geometric mean error χ(L), normalized by pairwise distances on the attractor.
• Validates the model against observed error scaling in chaotic systems and atmospheric simulations.

Experimental results

Research questions

• RQ1 Why do prediction errors in chaotic systems often fail to show exponential growth despite positive global Lyapunov exponents?
• RQ2 How do local Lyapunov exponents at finite times influence error growth dynamics compared to global exponents?
• RQ3 To what extent does error saturation at the attractor size explain non-exponential error scaling?
• RQ4 Why is the arithmetic mean of errors less reliable than the geometric mean in capturing error dynamics in chaotic systems?
• RQ5 Can a single empirical error growth law accurately describe both initial and saturated error regimes?

Key findings

• Prediction errors do not follow exponential scaling E(t) = E(0)exp(λ₁t) over finite times due to variability in local Lyapunov exponents and saturation effects.
• The model using finite-time local exponents and a saturation cutoff accurately predicts error scaling at both short and long times.
• The average finite-time Lyapunov exponent ¯λ(L) governs early error growth, while local exponent variance and saturation dominate at later times.
• The geometric mean of prediction errors provides a more stable and accurate measure than the arithmetic mean, which is dominated by large errors in the tail.
• The model explains why fitting a single empirical error law (e.g., dE/dt = αE(1 - E/E₀)) fails to capture the true initial growth rate, which depends on local dynamics.
• Empirical validation on low-dimensional chaotic systems and atmospheric simulations confirms the model’s accuracy, showing that non-exponential error growth is expected and not evidence against deterministic chaos.