Take your time to understand this python program...

It generates one of the most interesting graphs in all of math. At certain points it is predictable, at others, it is quite "random". It is actually very hard to create a function that outputs random numbers on a computer, since computer programs are essentially deterministic. 

This program uses the python module matplotlib a.k.a. "math plot library" to draw the graph. It's the python graphing calculator. You may need to download this module.

from matplotlib import pyplot as plt
import numpy as np

plt.rcParams['axes.facecolor'] = '#22bb88'
plt.rcParams['figure.figsize'] =  40,10

# x and y values to be plotted
a_values = []
outputs = []

def iterator(a, x):
    global a_values, outputs
    old = x
    output_set = set()
    # wait to stablize
    for _ in range(1000):
        new = a*old - a*old**2
        old = new
    # either cycle is found or goes into chaos
    while new not in output_set\
            and len(output_set) < 100:
        output_set.add(new)
        new = a*old - a*old**2
        old = new
    outputs.extend(list(output_set))
    a_values.extend([a]*len(output_set))


# generate points for range of a values
STEP = 0.001
INITIAL_SEED = 0.5  # arbitrary starting pt 0<x<1
for a in np.arange(0, 4, STEP):
    iterator(a, INITIAL_SEED)

title = 'test'
plt.scatter(a_values, outputs, color = 'white', s = 0.2)
plt.savefig(title + '.png', format='png', dpi = 300, bbox_inches = 'tight')
plt.show()