Grassberger–Procaccia Correlation Dimension

The correlation dimension D₂ characterizes the fractal structure of strange attractors. For a point cloud from a dynamical system, C(r) = (pairs with distance < r) / (total pairs) scales as C(r) ~ r^D₂. The GP algorithm estimates D₂ from the slope of log C(r) vs log r. This reveals the attractor's fractal dimension — Lorenz ~2.06, Rössler ~2.01, Hénon ~1.26.

Attractor

Points N: 800

Embedding dim m: 3

Estimated D₂
—

Slope from log-log
—

Expected D₂: —

GP method: for each pair of points, count how many are within distance r. Plot log C(r) vs log r. Slope = D₂.