🤖 AI Summary
This study addresses the challenge of accurately estimating the positive Lyapunov exponent (LE) from one-dimensional chaotic time series. We propose a data-driven, phase-space-reconstruction-free method that uses the forward prediction error growth rate of machine learning models as a proxy for trajectory divergence, enabling direct modeling of time-series dynamics and estimation of the maximum LE. The approach integrates multiple supervised learning algorithms and operates solely on the raw scalar time series, thereby circumventing the sensitivity of traditional methods to embedding parameters and noise. Evaluated on canonical chaotic maps—including Logistic, Tent, and Chebyshev—the method achieves R²ₚₒₛ values of 0.90–0.999, requiring as few as 200 data points. It significantly improves accuracy and robustness of LE estimation under small-sample conditions. This work establishes an interpretable, deployable paradigm for chaos identification in black-box nonlinear systems.
📝 Abstract
Understanding and quantifying chaos in nonlinear dynamical systems remains a fundamental challenge in science and engineering. The Lyapunov exponent is a key measure of chaotic behavior, but its accurate estimation from experimental data is often hindered by methodological and computational limitations. In this work, we present a novel machine-learning-based approach for estimating the positive Lyapunov exponent (MLE) from one-dimensional time series, using the growth of out-of-sample prediction errors as a proxy for trajectory divergence. Our method demonstrates high scientific relevance, offering a robust, data-driven alternative to traditional analytic techniques. Through comprehensive testing on several canonical chaotic maps - including the logistic, sine, cubic, and Chebyshev maps - we achieved a coefficient of determination R2pos > 0.9 between predicted and theoretical MLE values for time series as short as M = 200 points. The best accuracy was observed for the Chebyshev map (R2pos = 0.999). Notably, the proposed method maintains high computational efficiency and generalizes well across various machine learning algorithms. These results highlight the significance of our approach for practical chaos analysis in both synthetic and experimental settings, opening new possibilities for robust nonlinear dynamics assessment when only time series data are available.