Goal: introduction to matplotlib and numpy with a classic example

We are going to study the suite produced by the iteration of the function f, defined on the real numbers and parameterized by k:

f(x) = k * x (1-x)

Where

• k is a number between 0 and 4
• x is a float between 0 and 1

The suites we'll study will start at 0.5 and then apply f, like this:

index term
  0    0.5
  1    f(0.5)
  2    f(f(0.5))
  3    f(f(f(0.5)))

More specifically, we'll study what happens to the suites when the k value changes.

Create a virtual environment containing the matplotlib and numpy packages

Specification: write a command line program that, given the k value as argument, plots the 1000 first values of the suite. Use 0.5 for the initial value.

Here are a few hints to help you get started:

import numpy

def suite(initial, k, iterations):
    """
    :param initial: the initial value (x0)
    :param k: the `k` parameter
    :param iterations: how many iterations to compute
    :return: a numpy array contains exactly `iterations` values
    """
    x = initial
    res = []
    for i in range(0, iterations):
        res.append(x)
        x = x * k * (1 - x)
    return numpy.array(res)

import sys
def main_single():
    k = float(sys.argv[1])
    # If run with `python main.py 3.2`, now `k` equals 3.2

ts = numpy.linspace(start, end, iterations)

from matplotlib import pyplot

xs = ...
ys = ...
# Plot the graph using the xs array as X axis,
# and the ys array as Y axis
pyplot.plot(xs, ys)
# Show the graph: must be called once, at the end
# of the script
pyplot.show()

Run the main script with the following values for k: 2, 2, 2.8, 3, 3.5, 3.8, 3.86, 4

In the step above, you may have found that the value 3.86 had a "strange" behavior.

In the same graph, draw to lines. One for the series starting at 0.5 with k=3.86 and an other one starting at 0.44449 with k still set to k=3.86.

The idea is to show in a single graph the behavior of the series where k changes.

This time you will need a scatter plot, where k is on the X axis, and the Y axis contains all the different values of the series, starting at 1000 iterations.

For instance, if after 1000 iterations when k = 3.53 the possible values are 0.36, 0.47, 0.80 and 0.88, then the scatter plot should contain:

The suite is actually called "The Logistic Map". See the corresponding Wikipedia page for more info.