Coding a bifurcation diagram

I was wondering what was wrong with my code as I keep getting a weird but similar result to the actual bifurcation diagram. I am using the iterative equation xn+1 = xn * r(1-xn).

Here is my code:

import numpy as np 
import matplotlib.pyplot as plt
from functools import lru_cache

@lru_cache(maxsize = 1000)
def bifunc():
    R_val = []
    X_val = []
    R = np.linspace(0.5,4,1000)
    for r in R:
        x = 0.5
        for iterations in range(1001):
            x = x*r*(1-x)
            R_val.append(r)
            X_val.append(x)

    plt.plot(R_val, X_val, ls = '', marker = ',')
    plt.show()

bifunc()

Here is the image that keeps coming up:

Any help would be appreciated. Thank you.

As the first few iterations don't follow the pattern, they can be left out via an if-test. Best also remove the lru_cache as it has no real function here and can interfere with testing.

import numpy as np 
import matplotlib.pyplot as plt

def bifunc():
    R_val = []
    X_val = []
    R = np.linspace(0.5, 4, 1000)
    for r in R:
        x = 0.5
        for iterations in range(1001):
            x = x*r*(1-x)
            if iterations > 100:
                R_val.append(r)
                X_val.append(x)
    plt.plot(R_val, X_val, ls='', marker=',')
    plt.show()

bifunc()

This results in a plot similar to many found on the web. Usually the plots don't start before R=1 to avoid displaying the flat part between 0.5 and 1.