Henon Map

The Henon map is a simple 2D quadratic map,

\[x_{n+1} = 1 - a x_n^2 + y_n, \qquad y_{n+1} = b x_n\]

whose classic parameters (\(a=1.4\), \(b=0.3\)) produce a strange attractor with the map’s signature fractal, self-similar structure. Unlike the (area-preserving) standard and baker’s maps, the Henon map is dissipative – its Jacobian determinant is the constant \(-b\), with \(|b|<1\) – so nearby points contract in area on average even while they stretch apart along the attractor’s unstable direction. This example iterates a single long trajectory and scatter-plots it.

import matplotlib.pyplot as plt
import numpy as np

from physicskit.chaos.systems.maps import HenonMap
from physicskit.chaos.visualizers import theme
from physicskit.chaos.visualizers.bifurcation import map_bifurcation_sampler, plot_bifurcation_diagram

system = HenonMap(a=1.4, b=0.3)

The finished attractor

Discard a short initial transient so the plotted points already lie on the attractor, then plot the full, much longer orbit at once to reveal the map’s signature fractal, self-similar structure – especially visible when zooming into the folds.

traj = system.trajectory(np.array([0.0, 0.0]), n_iter=50000)
traj = traj[100:]

fig, ax = plt.subplots(figsize=(7, 5))
ax.scatter(traj[:, 0], traj[:, 1], s=0.3, color=theme.PRIMARY, alpha=0.6)
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_title("Henon map attractor")

plt.show()

The route there: a bifurcation diagram over a

The single attractor above is only the end state at one fixed a (b held at its classic 0.3). Sweeping a and, for each value, plotting the surviving long-term x values – exactly the same plot_bifurcation_diagram() machinery used for the Logistic Map – shows how that attractor is reached: a period-doubling cascade out of simple fixed points and cycles, opening into the fractal chaotic band at the classic a = 1.4 used above (dashed line), interrupted by visible periodic windows.

a_values = np.linspace(0.8, 1.42, 1200)
sampler = map_bifurcation_sampler(lambda a: HenonMap(a=a, b=0.3), state0=np.array([0.0, 0.0]), n_transient=300, n_keep=150)

fig2, ax2 = plot_bifurcation_diagram(a_values, sampler)
ax2.axvline(1.4, color=theme.MUTED, ls="--", lw=1.0, label="a = 1.4 (shown above)")
ax2.set_xlabel("a")
ax2.set_ylabel("x (long-term values)")
ax2.set_title("Henon map bifurcation diagram (b = 0.3)")
ax2.legend(fontsize=8)

plt.show()