KAM Torus Generator

Interactive phase space explorer based on the Chirikov standard map. Click the canvas to place orbits. Adjust the perturbation strength to watch tori dissolve into chaos.

θ p
KAM tori (quasi-periodic)
Chaotic orbits
Click anywhere on the canvas to place an orbit
0.80
8000

The Chirikov Standard Map

pn+1 = pn + K · sin(θn)   mod 2π      θn+1 = θn + pn+1   mod 2π

The standard map is the simplest Hamiltonian system that exhibits the full transition from order to chaos. At K = 0, every orbit lies on an invariant torus — a closed curve in phase space. As K increases, resonant tori break into island chains (Poincaré-Birkhoff theorem), while irrational tori persist — these are the KAM tori, guaranteed by the Kolmogorov-Arnold-Moser theorem.

At K ≈ 0.9716 (the Greene critical value), the last KAM torus — with winding number equal to the golden ratio — is destroyed. Above this threshold, chaotic orbits can diffuse across the entire phase space.

This transition from integrable order to deterministic chaos is fundamental to celestial mechanics, plasma confinement, particle accelerators, and the stability of quantum computing gates.