First period-doubling bifurcation of logistic map at r=3 (|f'(x*)| = 1)

Closed-form bifurcation analysis. For the logistic map f(x) = rx(1−x), non-zero fixed point x* = 1 − 1/r is stable whenever |f'(x*)| < 1. Computing the derivative: f'(x) = r − 2rx, so f'(x*) = r − 2r(1−1/r) = r − 2r + 2 = 2 − r. …