Chirikov Standard Map
Resonance overlap criterion: KAM tori, island chains, and chaos onset. Each dot is an orbit trajectory in (θ, p) phase space.
Standard map: θ_{n+1} = θ_n + p_{n+1} (mod 2π), p_{n+1} = p_n + K·sin(θ_n).
Chirikov criterion: chaos when K ≳ 0.97 (resonances overlap). Below: KAM tori survive.
At K≈0.97 the last KAM torus (golden ratio) breaks. Above K~4: nearly fully chaotic.