OGY Chaos Control — Stabilizing Unstable Periodic Orbits

The Ott-Grebogi-Yorke (1990) method applies tiny perturbations to a system parameter to keep a chaotic trajectory near an unstable periodic orbit (UPO) embedded in the attractor.

Control on/off

Max perturbation ε 0.050

Hénon a 1.40

Hénon b 0.30

Target UPO period

Step: 0
Mode: Chaos
Distance to UPO: —
Perturbation applied: 0

Hénon map:
xₙ₊₁ = 1 − a·xₙ² + yₙ
yₙ₊₁ = b·xₙ

OGY method (1990): near a UPO, stable/unstable manifolds intersect. When trajectory is close, apply δa to push it onto the stable manifold.

δaₙ = −(eᵤᵀΔxₙ)/(eᵤᵀ·∂f/∂a·eᵤ) × λᵤ

Applications: cardiac arrhythmias (Garfinkel 1992), lasers, chemical reactors, mechanical systems — chaos control is practical and experimentally verified.

Hénon attractor. Blue=chaotic, green=controlled near UPO. Yellow circle = UPO target.

Return map xₙ vs xₙ₊₁. Controlled orbit shown in green; diagonal = fixed point.