Equation-Free Screening of Mittag-Leffler-Compatible Dynamics from Scalar Time Series via kNN Multi-Horizon Profiles

Abstract

Fractional models provide a natural description of systems with memory, but a noninteger derivative should not be introduced solely because a time series is curved or slowly relaxing. We develop an equation-free preliminary screening framework that asks whether a scalar time series produces a multi-horizon k-nearest-neighbor (kNN) profile more compatible with Mittag-Leffler-type behavior than with selected conventional alternatives. In an ideal matched Caputo-relaxation benchmark, the complete generation-kNN-profile-model-comparison pipeline reproduces the expected Mittag-Leffler geometry and recovers the generating order to within approximately $10^{-3}$; this is interpreted as controlled calibration rather than as general fractional-order identification. Under 3% trajectory-specific observational noise, the held-out Mittag-Leffler preference is most consistent when the generating dynamics are well separated from the integer-order limit and becomes progressively less decisive as $\alpha\rightarrow1$. The fitted order $\alpha_{\mathrm{fit}}$, however, shows substantially larger realization-to-realization variability. Thus, relative model compatibility is more robust than single-realization order estimation in the present noisy benchmark. Noise-free nonfractional controls show a separate limitation of specificity: a stretched exponential can generate a strongly Mittag-Leffler-compatible profile, whereas inclusion of the generating rational/Hill family recovers that family and its parameters to numerical precision in the matched setting. A positive Mittag-Leffler-versus-exponential screen therefore does not uniquely establish fractional origin. A fractional chaotic system is treated only as an exploratory extension: the Mittag-Leffler growth family gives lower finite-window RMSE than exponential and logistic/saturating alternatives over the detected pre-transition interval.