A novel approach for estimating largest Lyapunov exponents in one-dimensional chaotic time series using machine learning.

Abstract

Understanding and quantifying chaos from data remains challenging. We present a data-driven method for estimating the largest Lyapunov exponent (LLE) from one-dimensional chaotic time series using machine learning. A predictor is trained to produce out-of-sample, multi-horizon forecasts; the LLE is then inferred from the exponential growth of the geometrically averaged forecast error across the horizon, which serves as a proxy for trajectory divergence. We validate the approach on four canonical 1D maps-logistic, sine, cubic, and Chebyshev-achieving Rpos2 > 0.99 against reference LLE curves with series as short as M = 450. Among baselines, k-nearest neighbor (KNN) yields the closest fits (KNN-R comparable; random forest larger deviations). By design the estimator targets positive exponents: in periodic/stable regimes, it returns values indistinguishable from zero. Noise robustness is assessed by adding zero-mean white measurement noise and summarizing performance vs the average signal-to-noise ratio (SNR) over parameter sweeps: accuracy saturates for SNRm ≳ 30 dB and collapses below ≈27 dB, a conservative sensor-level benchmark. The method is simple, computationally efficient, and model-agnostic, requiring only stationarity and the presence of a dominant positive exponent. It offers a practical route to LLE estimation in experimental settings where only scalar time-series measurements are available, with extensions to higher-dimensional and irregularly sampled data left for future work.