An integrable area-preserving map — each circle rotates by a frequency ω(r). Rational frequencies break into island chains; irrational ones persist as KAM tori.
Controls
ε = 0: All circles are KAM tori
Twist map: (r, θ) → (r, θ + ω(r)) where ω is the rotation function.
Irrational ω: Dense orbit, quasi-periodic — fills the circle uniformly (Weyl equidistribution).
Rational ω = p/q: Periodic orbit with period q. Under perturbation: Poincaré-Birkhoff theorem guarantees 2q fixed points appear (alternating stable/unstable).
KAM theorem: For sufficiently irrational ω (Diophantine condition: |ω - p/q| > c/q²), the invariant circle persists under small perturbations.