Rossler Attractor

The Rossler system is three coupled nonlinear autonomous ODEs, deliberately designed to be simpler than Lorenz while still chaotic:

\[\begin{split}\dot{x} &= -y - z \\ \dot{y} &= x + a y \\ \dot{z} &= b + z (x - c)\end{split}\]

It produces a simpler-looking, ribbon-like strange attractor than Lorenz, built from a single stretch-and-fold mechanism: trajectories spiral outward in the \((x, y)\) plane (driven by the linear \(-y-z\) and \(x+ay\) terms) until \(x\) exceeds \(c\), at which point the \(z (x - c)\) term kicks the trajectory up and folds it back down onto the spiral. This example integrates a trajectory with physicskit.chaos.systems.continuous.Rossler.trajectory() and plots it in 3D, using the classic chaotic parameters \(a=0.2\), \(b=0.2\), \(c=5.7\).

import matplotlib.pyplot as plt

from physicskit.chaos.systems.continuous import Rossler
from physicskit.chaos.visualizers.section import plot_poincare_map

system = Rossler(a=0.2, b=0.2, c=5.7)

Integrate

t, states = system.trajectory(n_steps=40000, dt=0.02)
states = states[1000:]

Plot

fig = plt.figure(figsize=(7, 6))
ax = fig.add_subplot(projection="3d")
ax.plot(states[:, 0], states[:, 1], states[:, 2], lw=0.4, color="teal")
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_zlabel("z")
ax.set_title("Rossler attractor")

plt.show()

Poincare section: the stretch-and-fold mechanism, laid bare

Sampling the flow only at the instants it crosses the plane \(y = 0\) moving outward (\(\dot{y} > 0\)) – the natural section for a spiral that winds around the \(z\)-axis – turns the ribbon-like 3D attractor into a near-one-dimensional curve of successive crossings’ \((x, z)\). That the points collapse almost onto a single folded curve (rather than filling a 2D patch) is the direct visual signature of the single stretch-and-fold mechanism described above: each pass around the spiral maps an interval of \(x\) values into a slightly different, folded-over interval, iteration after iteration.

fig2, ax2 = plot_poincare_map(
    system,
    system.initial_state(),
    coord=1,
    value=0.0,
    direction=1.0,
    plot_coords=(0, 2),
    t_max=400.0,
    dt=0.005,
    s=4,
)
ax2.set_xlabel("x")
ax2.set_ylabel("z")
ax2.set_title("Rossler Poincare section at y = 0 (ascending)")

plt.show()