Equation-Free Period-Aware Forecast-Error Contraction for Estimating Negative Largest Lyapunov Exponents from Short Trajectory Ensembles

2026-08-05

AI Summary

This study addresses the challenge of accurately estimating negative maximal Lyapunov exponents from short scalar time series, where conventional methods are highly susceptible to noise and finite-precision effects. The authors propose a fully equation-free, periodicity-aware predictive error contraction approach that requires neither the underlying dynamical equations nor Jacobian matrices. By employing a k-nearest-neighbor predictor over phase-coherent prediction horizons, the method computes the geometric mean of absolute prediction errors and directly extracts the negative Lyapunov exponent from the slope of the logarithmic error curve. Innovatively incorporating period-synchronized prediction and a slope consensus mechanism across multiple transient lengths, the technique substantially enhances robustness. Experiments demonstrate its efficacy: on the Logistic map, it accurately recovers 92 out of 112 negative exponents with MAE = 0.0253 and R² = 0.886; in a two-dimensional system without fixed points, independent estimates from x, y, and z observables achieve MAE ≈ 0.009–0.011 and R² ≈ 0.983–0.986.

Abstract

Estimating positive largest Lyapunov exponents from data is comparatively natural because neighboring trajectories separate, whereas stable dynamics require resolving contraction before measurement noise or finite precision erases the signal. We introduce a period-aware forecast-error contraction procedure for estimating a dominant negative Lyapunov exponent from ensembles of short scalar trajectories without using governing equations or an analytical Jacobian. A k-nearest-neighbor predictor is trained on trajectory histories, the geometric-mean absolute forecast error is evaluated at phase-consistent horizons, and the exponent is obtained from the slope of the logarithmic error profile. Unlike data-driven approaches that reconstruct local evolution matrices or differentiate a learned surrogate, the proposed method extracts the contraction rate directly from out-of-sample forecast errors. Two adaptations are essential: the forecast step is synchronized with the detected orbit period, and candidate slopes are accepted only when they form a stable consensus across several transient lengths. On the logistic map, the method recovers 92 of 112 negative-exponent parameter values with a mean absolute error of 0.0253 and $R^2=0.886$. On a two-dimensional map without fixed points, independent scalar pipelines based on the three observables $x_n$, $y_n$, and $z_n$ give mean absolute errors of 0.00879--0.01145 and $R^2=0.983$--$0.986$. Because the estimation stage uses only observed trajectories, the framework provides a basis for repeated-relaxation experiments in which short sensor responses are available but the governing equations and analytical Jacobian are unknown. Experimental validation remains a subject of future work.

Problem

• Lyapunov exponent
• forecast error
• trajectory ensemble
• contraction rate
• period-aware

Innovation

• Lyapunov exponent estimation
• equation-free forecasting
• forecast-error contraction
• period-aware synchronization
• data-driven dynamics