The Chirikov standard map is the canonical model of Hamiltonian chaos:
p_{n+1} = p_n + K sin(θ_n)
θ_{n+1} = θ_n + p_{n+1} (mod 2π)
It arises from a particle on a ring kicked periodically. For K=0: integrable (circles). As K increases, KAM tori break down progressively.
At the Kolmogorov-Arnold-Moser (KAM) threshold K_c ≈ 0.9716 (Greene 1979), the last invariant torus (golden-ratio winding number) breaks, and global chaos begins.
Below K_c: mix of KAM tori (smooth curves) and chaotic seas. Above K_c: single connected chaotic region — the Chirikov diffusion regime with ⟨(Δp)²⟩ ~ K²t/2.
Click canvas to add a trajectory at that phase-space point.