Illustration of the logistic map

Illustration of the logistic map, $x_{n+1}=f\left(x_n\right)=r\cdot x_n\cdot (1-x_n)$

At $r_1=3$, a stable period $2^1=2$ orbit is born. At $r_2=3.449$ a stable period $2^2=4$ orbit is born. As $r$ continues to increase, the period doublings continue until $r_{\infty }\approx 3.56995$ after which chaotic dynamics begin to occur, interspersed with periodic windows.

The Feigeinbaum constant $\delta =4.6692\dots$ is the ratio of subsequent differences between the values of $r_n$ at which the period doubles, as $n$ approaches infinity.

import numpy as np
import matplotlib.pyplot as plt
import pylab

def f(x, R):
    return R * x * (1 - x)

def run_simulation(R, x_0, num_steps):
    x_list = np.zeros(num_steps)
    x_list[0] = x_0   
    for t in range(num_steps-1):
        x_list[t+1] = f(x_list[t], R)       
    return x_list

def plot_two(x_list, y_list):
    plt.plot(x_list)
    plt.plot(y_list)

x_list = run_simulation(R=4, x_0=0.7, num_steps=50)
y_list = run_simulation(R=4, x_0=0.70001, num_steps=50)

plot_two(x_list, y_list)