# Logistic map bifurcation
import numpy as np
import matplotlib.pyplot as plt

def logistic_bifurcation(r_min=2.5, r_max=4.0, steps=1000):
    r_vals = np.linspace(r_min, r_max, steps)
    x = 0.5
    for r in r_vals:
        for _ in range(100):  # transient
            x = r * x * (1 - x)
        for _ in range(50):   # attractor
            x = r * x * (1 - x)
            yield r, x