Note
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Animated: the logistic map’s cobweb as r grows#
A cobweb diagram iterates \(x_{k+1} = rx_k(1 - x_k)\) graphically: go up to the parabola, across to the diagonal, repeat. Sweeping \(r\) from 2.8 to 4 shows the period-doubling route to chaos as the red web settles onto a fixed point, then a square (a period-2 cycle), then nested squares (period 4, 8, …), and from \(r_\infty \approx 3.5699\) a web that never closes. The bifurcation diagram on the right marks where each frame sits.
import matplotlib.pyplot as plt
import numpy as np
from mathematicskit.ode_dynamics.visualizers import animate_logistic_cobweb
# In Jupyter or JupyterLite, show animations as an HTML/JavaScript player
# (no ffmpeg needed).
plt.rcParams["animation.html"] = "jshtml"
From a fixed point to chaos#
The sweep slows down through the period-doubling cascade, where the action is.
r_values = np.concatenate([np.linspace(2.8, 3.4, 12), np.linspace(3.4, 3.6, 24), np.linspace(3.6, 4.0, 14)[1:]])
anim = animate_logistic_cobweb(r_values, x0=0.2, n_transient=60, n_keep=60)
anim
plt.show()
Total running time of the script: (0 minutes 5.889 seconds)
