Fractal dimension

Generate a two-dimensional orbit or paste x,y pairs separated by spaces, commas or semicolons. Use decimal points, with optional scientific notation. Sample order matters when excluding neighbours. Up to8000 samples are counted in a cancellable worker. Progress measures eligible pairs actually processed; resizing only redraws the saved result.

The coordinates are translated and uniformly scaled by L, the larger bounding-box span; x and y share one scale. Radii are r/L. Box grids double from the first to the last chosen size b; ε=L/b. The top/right boundary belongs to the last cell. D₀ is the fitted slope of ln N(b) against ln b.

C(r) is the fraction of eligible unordered pairs with Euclidean distance strictly less than r. W excludes index separations≤W (a Theiler window). Its denominator is the number of remaining pairs, not all pairs. Duplicate samples retain their weights and order. Zero counts have no logarithm; zero and fully saturated fractions are omitted from the correlation fit. D₂ is the slope of ln C(r) against ln(r/L), with at least three usable scales. Orange lines show the fitted interval. R² describes this fit, not its uncertainty or correctness.

These are finite-scale estimates, not proven dimensions or proof of chaos. A finite point set eventually has dimension zero as scale vanishes. Sample size, chosen interval, temporal correlations, noise, periodicity and floating-point collapse can change the estimates. Try different intervals and W and inspect the curves. No automatic reliable scaling-range detection, delay embedding, confidence interval or periodic-distance metric. Hénon/Tinkerbell parameters can yield divergent trajectories; a partial orbit is rejected. Cat/baker orbits may become periodic or collapse in finite precision.

PNG exports the active plots with axes and estimates. Valid map settings can be saved locally; pasted coordinates stay only in memory and are not stored in the preference cookie. Reload waits for Compute.

Grassberger–Procaccia · Scholarpedia