Feigenbaum Constants

Feigenbaum Constants

One of the most surprising discoveries in chaos theory is that the period-doubling route to chaos is universal — the same quantitative ratios appear in virtually every system that undergoes this transition, from logistic maps to dripping faucets to electronic circuits.

As the parameter r of the logistic map increases, the stable period doubles: 1 → 2 → 4 → 8 → … Each period-doubling occurs at a specific value r_n. The Feigenbaum constant δ is defined by:

$\delta = \lim_{n \to \infty} \frac{r_n - r_{n-1}}{r_{n+1} - r_n} \approx 4.6692$

This constant appears universally in one-dimensional maps with a quadratic maximum. It was discovered by Mitchell Feigenbaum in 1975 and was one of the first indicators that chaos has deep mathematical structure.

Implement the following functions:

• find_period(r, x0, n_transient, n_test, tol) — find the period of the logistic map orbit at the given r value, after discarding n_transient transient iterations
• find_bifurcation(p_target, r_lo, r_hi, x0, n_transient, n_test, tol, n_bisect) — find the bifurcation point where the period first reaches p_target, using bisection
• feigenbaum_ratio(r1, r2, r3) — compute δ ≈ (r2 - r1) / (r3 - r2) from three consecutive bifurcation points