Category: Nonlinear Dyn.

Summary: Mapping how closely logistic-map orbits follow Benford's law across the bifurcation diagram.

Benford's law predicts a specific uneven distribution of first digits in many naturally generated data sets. This experiment asks how the first-digit statistics of the logistic map change across control parameter values, especially inside periodic windows embedded in chaos.

The script scans the bifurcation diagram, computes first-digit frequencies from orbit data, and measures the divergence from Benford's law as a function of the map parameter r. It also compares those deviations with the Lyapunov exponent to test whether stronger chaos coincides with better Benford compliance.

That combination links nonlinear dynamics with a digit-statistics benchmark that is usually studied in applied settings rather than as a fine-grained dynamical observable. The resulting map highlights where periodic structure leaves a visible fingerprint in the digits.

Method: High-resolution logistic-map simulations across r in the bifurcation diagram, with first-digit counts and divergence-from-Benford analysis.

What is measured: First-digit distribution, KL divergence from Benford's law, dependence on r, periodic-window structure, and relation to the Lyapunov exponent.