KAM Theory: The Standard Map

The Chirikov standard map $p'=p+K\sin\theta$, $\theta'=\theta+p'$ is the canonical model of the transition from order to chaos. It is area-preserving (Jacobian determinant exactly 1), so this Poincare section is a faithful snapshot of a Hamiltonian flow. At $K=0$ the action $p$ is conserved and the section is horizontal invariant tori. The KAM theorem says sufficiently irrational tori survive small $K$; as $K$ grows they break, rational ones first, in island chains, until the very last one, the golden-mean torus, is destroyed at Greene's $K_c\approx0.9716$. Above $K_c$ no curve spans the cylinder and $p$ diffuses freely.