Butterfly effect

The butterfly effect is a concept in chaos theory that suggests that small changes in initial conditions can have significant long-term effects on a complex system. Here’s a simple Python program that simulates the butterfly effect using the logistic map equation:

pythonCopy codeimport matplotlib.pyplot as plt

def logistic_map(x, r):
    return r * x * (1 - x)

def simulate_butterfly_effect(r, x0, num_iterations):
    results = []
    x = x0
    for i in range(num_iterations):
        x = logistic_map(x, r)
        results.append(x)
    return results

# Define the parameters
r = 3.8
x0 = 0.5
num_iterations = 1000

# Simulate the butterfly effect
results1 = simulate_butterfly_effect(r, x0, num_iterations)
results2 = simulate_butterfly_effect(r, x0 + 0.0001, num_iterations)

# Plot the results
plt.plot(results1, label='x0 = 0.5')
plt.plot(results2, label='x0 = 0.5001')
plt.legend()
plt.show()

In this program, we define the logistic map equation and a function to simulate the butterfly effect. We then define the parameters of the simulation, including the value of the parameter r, the initial condition x0, and the number of iterations to simulate.

We then simulate the butterfly effect by running the simulate_butterfly_effect function twice, once with x0 equal to 0.5 and once with x0 equal to 0.5001. We plot the results of both simulations on the same graph to visualize the effect of the small change in initial conditions.

When we run this program, we can see the butterfly effect in action. Even though the initial conditions are very similar, the two simulations quickly diverge and produce very different results over time. This is a simple example of how small changes in initial conditions can have a significant impact on a complex system over time.