Poincaré Return Map

System

Lorenz r: 28

Rössler c: 5.7

Points N: 3000

Section plane

Poincaré 1899: Record xₙ₊₁ = f(xₙ) each time the orbit crosses a surface Σ. Chaos → cloud of points. Period-n → exactly n points. Periodic orbit → fixed point.

3D Trajectory (projected)

Return Map: xₙ₊₁ vs xₙ

Poincaré Section (crossing points on Σ)