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.
Largest Lyapunov exponents (LLEs) quantify exponential sensitivity, but data-driven estimates are often obtained from finite-precision trajectories. We show that increasing the length of a single reduced-precision chaotic record can eventually degrade a forecast-error LLE estimate. Using the logistic map at r=4, an ESP32 single-precision trajectory is reproduced bit-for-bit by NumPy float32. Across 10,000 random float32 initial conditions, every trajectory reaches an exact recurrence before iteration 7612. For one long float32 record, the estimated LLE changes from 0.6853 at N=15,000 to 0.1827 at N=20,000 and approximately zero at N=30,000 as exact train-test histories saturate. At N=100,000, the long float32 record gives 0.0016, whereas independently restarted length-100 trajectories give 0.6917; matched float64 controls remain near ln(2)=0.6931. The collapse is reproduced for 28 representative initial conditions, and its onset is strongly correlated with the recurrence scale set by transient length and digital period (Pearson r=0.982). Thus, finite-state recurrence can turn additional samples into duplicate futures rather than new dynamical information, while independent restarts substantially delay this saturation.
We study the reconstruction of an unknown dynamical system from a single noisy scalar time series. The goal is to recover the underlying dynamics for forecasting. We introduce a method that uses differential embedding coordinates to identify a rational closure of the embedding dynamics directly from data. The closure is identified through a weak-form regression pipeline, which avoids unstable pointwise differentiation of noisy data. When applied to noise-free Lorenz and R\"ossler systems, the method recovers closures that support long forecasts across a broad ensemble of realizations ($18.1$ and $7.1$ Lyapunov times respectively). Under $15$--$30\%$ additive Gaussian noise, performance becomes system-dependent. For the Lorenz system, forecast horizons remain short even in the best cases, whereas the R\"ossler system generally performs better in absolute terms, though not once normalized by the Lyapunov time. Our proposed method recovers directly interpretable closure coefficients which we compared against the known analytic closures of the Lorenz and R\"ossler systems.
Fractional calculus is a powerful framework for modeling nonlocal behavior in complex systems. However, the identification of fractional dynamics from measured time series remains challenging, as most existing approaches require knowledge of the underlying governing equations. In this work, we propose a data-driven diagnostic pipeline that detects fractional signatures directly from scalar observations using a multi-horizon k-nearest neighbors (kNN) forecast-error growth framework. The central idea is that fractional systems exhibit power-law or Mittag-Leffler error growth, in contrast to the exponential divergence characteristic of chaotic integer-order systems. By comparing the empirical error-growth curve against exponential and Mittag-Leffler models, and by examining the local slope of the logarithmic curve, we construct a preliminary fractionality indicator. The method is evaluated on a fractional chaotic system and in a controlled stable fractional relaxation setting, including a kNN-based contraction test. On a fractional chaotic system the Mittag-Leffler model achieved a 58% reduction in RMSE over the exponential model, with $\Delta>0$ in 100% of bootstrap replicates. In the stable relaxation setting, Mittag-Leffler decay strongly outperformed the exponential alternative; in the kNN contraction test, the free-order Mittag-Leffler model reduced the RMSE from $4.810\times 10^{-3}$ to $5.14\times10^{-4}$. The fitted Mittag-Leffler order should be interpreted as an effective shape parameter of the error-growth curve rather than as a direct estimate of the true system order, the recovery of which remains a more difficult inverse problem. Our results demonstrate that multi-horizon forecast-error geometry can serve not only for forecasting and chaos detection, but also for dynamical characterization in fractional systems.
We study the approximation of dynamical systems by semi-autonomous neural ordinary differential equations (SA-NODEs) over long time horizons. For a single network trained on the whole horizon, the available error bound deteriorates double exponentially in the horizon length. We develop two training strategies that avoid this barrier, each built on a reset of the state. The model predictive strategy partitions the horizon adaptively and restarts every window from observed data: when training meets a prescribed tolerance on every window, the composite model meets it uniformly in time, with a parameter budget linear in the horizon for targets with a bounded, uniformly regular reachable tube. The Floquet strategy addresses autonomous targets with a stable limit cycle and uses no data at deployment: a certified contraction of the learned return map confines the error to linear growth in the number of elapsed periods. For the time-periodic architecture we deploy, the scalar certificate degenerates; we prove instead a uniform-in-time orbital guarantee whose hypotheses are measured on the trained model, and an obstruction showing that, for an exactly periodic learned field, small one-period error and a contracting stroboscopic map cannot hold at once. Numerical experiments on four benchmarks confirm the predicted error laws and measure the hypotheses of every guarantee.
ABSTRACT Reliable 24–48 hr long-term wind-speed forecasts are critical for the safe operation of energy systems. Wind variability is shaped by chaotic dynamics and predictability limits, motivating chaos-aware modeling of Weather Research and Forecasting (WRF) outputs. This paper proposes a double-stage modeling method based on the chaos-noise decoupling, which effectively improves the chaotic modeling performance and prediction accuracy of the WRF for long-term wind speed prediction. In the first step, a dynamic reinforcement learning model guided by the Gramian angular field is used for the WRF ensemble by tracking the time-varying chaotic process of wind speed. In the second step, a closed-loop correction algorithm guided by predictability iteratively extracts the remaining predictable components until the residuals become approximately decorrelated. Experimental results show that (1) Compared to the WRF model, the proposed method yields Lyapunov and Hurst exponents closer to those of the observed data and produces residuals with a median Hurst exponent approaching 0.5, indicating that the residuals are approximately decorrelated. (2) Through comparisons with ablation and existing models, the proposed framework shows improved forecasting accuracy and robustness with over 95% confidence, with long-term prediction horizons of 24 to 48 hours. HIGHLIGHTS A chaos-noise decoupling framework based on predictability is designed to improve the chaotic modeling and prediction performance of WRF. Lyapunov and Hurst exponents are employed to validate dynamic fidelity performance by effective decoupling of chaotic and noisy components. The method is validated using wind-speed data from 10 German stations and an additional public GEFS–ISD-Lite dataset for 24–48 h forecasting.
Zhu Duan, Hui Liu, N. Nikitas· International Journal of Gre...· 0 citations
This paper proposes an uncertainty-quantified surrogate framework for ordinary differential equations (ODEs) based on ensembles of extreme learning machines (ELMs). The method constructs M=100 independently randomized ELM surrogates for a given ODE trajectory, uses the ensemble mean as the predictor, and defines a pointwise ensemble spread as a preliminary uncertainty measure. To obtain statistically valid prediction intervals, a split-conformal calibration procedure is applied to the ensemble spread, yielding finite-sample marginal coverage under exchangeability while preserving computational efficiency, since each ELM is trained via a single ridge-regression solve. Theoretical results establish exact satisfaction of the prescribed initial condition, stability of the ensemble mean and variance with respect to perturbations in the training data, and almost-sure convergence of the empirical ensemble variance to the corresponding random-feature prediction variance. The hidden-layer sampling hypothesis is made explicit: the experiments use bounded hyperbolic-tangent features with independent uniform draws as the default and independent normal draws in sensitivity tests, both of which satisfy the finite-moment assumptions required by the convergence theorem. Comparisons with capacity-matched Bayesian random-feature neural surrogates and Monte Carlo dropout clarify differences in uncertainty representation. Numerical experiments on exponential, logistic, and damped oscillator dynamics demonstrate accurate reconstruction and calibrated uncertainty quantification in sparse and noisy regimes. Additional ablation studies quantify the effect of the denominator safeguard, calibration-sample size, ensemble size, training time, noise level, and hidden-parameter distribution.
S. A. Sheikh, L. Wani· Mathematics· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
