Logistic Map

A Logistic Map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behavior can arise from very simple non-linear dynamical equations. It is defined by the recurrence relation:

$$x_{n+1}=r{x}_n(1-x_n)$$

and is used to model population growth in an environment with limited resources. Here, (x_n) represents the population at generation (n), scaled between 0 and 1, and (r) is a positive parameter representing the growth rate.

In the Dynamical Systems context, the logistic map exhibits a range of behaviors depending on the value of the parameter $r$ and the initial condition $x_0$:

$$f:x\mapsto rx(1-x)$$

It can be seen as a tool for studying bifurcations, chaos, and the transition from order to chaos in dynamical systems. As $r$ increases, the system undergoes a series of bifurcations leading to chaotic behavior.

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Plotting Bifurcations

Something interesting happens when we plot the long-term behavior of the logistic map as a function of the parameter (r). This is known as a bifurcation diagram.

Python

import numpy as np import matplotlib.pyplot as plt # Params r_min, r_max = 2.5, 4.0 n_r = 10000 n_transient = 1000 n_last = 100 r = np.linspace(r_min, r_max, n_r) x = 0.5 * np.ones(n_r) for i in range(n_transient): x = r * x * (1 - x) r_list = [] x_list = [] for i in range(n_last): x = r * x * (1 - x) r_list.append(r) x_list.append(x) r_plot = np.array(r_list).flatten() x_plot = np.array(x_list).flatten() plt.figure(figsize=(12, 8)) plt.scatter(r_plot, x_plot, s=0.1, c='black', alpha=0.1) plt.title('Bifurcation Diagram of the Logistic Map') plt.xlabel('Growth Rate ($r$)') plt.ylabel('Population ($x$)') plt.xlim(r_min, r_max) plt.ylim(0, 1) plt.grid(False) plt.show()

Output

This interesting behavior illustrates how simple deterministic systems can exhibit complex and unpredictable dynamics, a hallmark of chaos theory.