Differ Hug: ODE Research Platform

Curve Annotations:

• Each curve is annotated with its idx at the final point.
• Table shows robust curvature statistics (median, p10, p90) and path length.
• FTLE diagnostics include R² of ln(dist) fit (ftle_r2). Anomaly score combines FTLE, path length, max curvature and R² reliability.

Column Descriptions:

• idx — Index of the trajectory, starting from 0
• ftle — Finite-Time Lyapunov Exponent, computed as the slope of linear fit of ln(d(t)) vs t on a central window (25%–75% of t_eval) after clipping d(t) to a minimum (1e-12)
• ftle_r2 — Coefficient of determination (R²) of the linear fit for FTLE, indicating reliability of the ftle estimate
• amp — Amplitude (max−min of radial distance √(x²+y²))
• final_dist — Final distance between the original trajectory and its companion trajectory with tiny perturbation
• hurst — Hurst exponent using the rescaled range (R/S) method calculated for x(t) and y(t) and averaged
• curv_rad_mn — Mean curvature radius computed from 1/κ(t) where κ(t) is the curvature
• curv_rad_med — Median curvature radius computed from 1/κ(t) where κ(t) is the curvature
• curv_rad_std — Standard deviation of curvature radius
• curv_p10 — 10th percentile of curvature radius
• curv_p90 — 90th percentile of curvature radius
• curv_ct_fin — Number of finite radius samples
• initial_x — X coordinate of the initial point
• initial_y — Y coordinate of the initial point
• path_len — Total arclength computed as sum of Euclidean distances between consecutive points along the trajectory
• max_kappa — Maximum finite curvature value, useful to detect sharp bends
• frac_high_curv — Fraction of time points with κ(t) above a threshold (default κ > 0.1, i.e., radius < 10), measuring density of sharp bends along the trajectory
• curv_rad_lcl_z — Local z-score of curve median curvature radius relative to nearest neighbors in initial condition space
• anomaly_score — Aggregated score combining robust z-scores (IQR-based) of ftle, path_len, max_kappa, and ftle_r2 (ftle + path_len + max_kappa − ftle_r2)

$$\begin{cases} \displaystyle \frac{dx}{dt} = \frac{K\,x^{1/\alpha}}{b^{1/\alpha} + x^{1/\alpha}} - \gamma_1\,x,\\[10pt] \displaystyle \frac{dy}{dt} = \frac{K\,y^{1/\alpha}}{b^{1/\alpha} + y^{1/\alpha}} - \gamma_2\,y. \end{cases}$$