OGY Chaos Control: Stabilizing Unstable Periodic Orbits

OGY Method (1990): Chaotic attractors contain infinitely many unstable periodic orbits (UPOs) as a dense set. The key insight: rather than suppressing chaos, exploit it. Small time-dependent perturbations to a system parameter p → p+δp can stabilize a chosen UPO.

Algorithm: (1) Identify the UPO of desired period T (here period-1,2,4 fixed points of f^T(x)=x). (2) At each iterate, compute distance to the stable manifold of the UPO. (3) Apply perturbation δr = −(x_n − x*) · λ_u/(∂f/∂r) only when |x_n − x*| < ε. (4) Perturbation → 0 as orbit locks on.

Logistic map: x_{n+1} = r·x_n·(1−x_n). For r=3.9, fully chaotic. Fixed point x*=1−1/r. Period-2 points satisfy f(f(x))=x. Control works because the unstable manifold has only one direction in 1D maps.

Left: Time series (blue=free chaos, green=controlled). Right: Cobweb diagram showing convergence to UPO.