Bifurcation and Chaos

Let’s evaluate that formula on a regular grid and create a plot to see a pattern.

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%matplotlib widget
import matplotlib.pyplot as plt
import seaborn as sns
sns.set()
plt.rcParams['figure.dpi'] = 300

# prepare list of numbers to analyze
x_min, x_max = -2.1, .5
max_iterations = 50
resolution = 300

x = [x_min + (x_max - x_min) / (resolution - 1) * index for index in range(resolution)]

# updated calculate function to track all intermediate results
def calculate(x, max_iterations):
    x0 = x
    
    results= []
    for index in range(max_iterations):
        x = x**2 + x0
        if x > 10:
            break

        results.append(x)

    return results

# Run the calculations

y = [calculate(_x, max_iterations) for _x in x]

fig, axes = plt.subplots()
for _x, _y in zip(x, y):
    __x = [_x for _ in _y]

    axes.plot(__x, _y, "b.", markersize=1)
    
axes.set_ylim(-2.2, 2.2)
plt.tight_layout()

Quite impressive. A great mixture between order and chaos.

Benoit Mandelbrot went a step further and analyzed this formula for the complex plane instead of the real numbers alone.

So let’s add this axis as well to our problem.