# A Novel Approach for Estimating Largest Lyapunov Exponents in One-Dimensional Chaotic Time Series Using Machine Learning

arXiv: https://arxiv.org/abs/2507.04868 · HTML: https://mlsanity.com/paper/2507.04868/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 44 references that are in the map.
- First posted on arXiv 7 Jul 2025.

## Builds on (most cited first, 3 of 3)

- [Attractor reconstruction with reservoir computers: The effect of the reservoir's conditional Lyapunov exponents on faithful attractor reconstruction](https://mlsanity.com/paper/2401.00885/) (2023)
- [A Method for Medical Data Analysis Using the LogNNet for Clinical Decision Support Systems and Edge Computing in Healthcare](https://mlsanity.com/paper/2108.02428/) (2021)
- [Neural Network Entropy (NNetEn): Entropy-Based EEG Signal and Chaotic Time Series Classification, Python Package for NNetEn Calculation](https://mlsanity.com/paper/2303.17995/) (2023)

## Built on by (most cited first, 6 of 6)

- [Optimizing Parallel Schemes with Lyapunov Exponents and kNN-LLE Estimation](https://mlsanity.com/paper/2601.13604/) (2026)
- [FEG-Pro: Forecast-Error Growth Profiling for Finite-Horizon Instability Analysis of Nonlinear Time Series](https://mlsanity.com/paper/2605.17282/) (2026)
- [Interpretable AI-Assisted Early Reliability Prediction for a Two-Parameter Parallel Root-Finding Scheme](https://mlsanity.com/paper/2603.16980/) (2026)
- [Unified Geometry-Guided ML-FTLE for Tracking Transient Chaos from Scalar Time Series](https://mlsanity.com/paper/2606.07385/) (2026)
- [Equation-Free Period-Aware Forecast-Error Contraction for Estimating Negative Largest Lyapunov Exponents from Short Trajectory Ensembles](https://mlsanity.com/paper/2608.05522/) (2026)
- [Beyond RMSE and Granger Gain: Multi-Horizon Forecast-Error Entropy for Nonlinear Predictive Coupling](https://mlsanity.com/paper/2610.18390/) (2026)

## Abstract (arXiv)

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 (GMAE) 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 R2pos > 0.99 against reference LLE curves with series as short as M = 450. Among baselines, KNN yields the closest fits (KNN-R comparable; RF 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 versus the average 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.