Poincaré Plots

Generate and plot Poincaré sections by intersecting 3D trajectories with 2D cutting planes.

Poincaré sections simplify the analysis of high-dimensional continuous systems by intersecting the 3D continuous trajectory with a 2D transversal cutting plane. This converts continuous flow into a discrete map, clarifying the fractal geometry of chaotic attractors.

Cutting Planes

A Poincaré section requires defining a plane equation: $A x + B y + C z + D = 0$

Common planes include simple coordinate cuts (e.g., $x = 0$, $y = 0$, or $z = 0$) with a specified crossing direction (positive-going crossings, negative-going crossings, or both).

Generating Poincaré Sections

Use the analysis package to extract crossing points from a trajectory array:

from hidden_attractors.analysis import compute_poincare_crossings

# Compute crossings on the z = 0 plane in the positive direction
crossings = compute_poincare_crossings(
    trajectory,
    plane_normal=(0.0, 0.0, 1.0),
    plane_offset=0.0,
    direction="positive"
)

Plotting Poincaré Sections

The plotting module renders the discrete points clearly:

from hidden_attractors.plotting import plot_poincare_section

plot_poincare_section(
    crossings,
    output_path="outputs/analysis/poincare_z0.png",
    x_axis="x",
    y_axis="y",
    title="Poincare Section at z = 0",
    color="#8b5cf6",
    s=4.0
)

Reading the Map

• Single Point: Represents a periodic orbit of period 1.
• N Points: Represents a sub-harmonic periodic orbit of period N.
• Closed Loop: Indicates quasi-periodic behavior (torus).
• Fractal Set: Confirms chaotic behavior (strange attractor).