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.

Figure 1. Poincare section of the Chirikov standard map: invariant KAM tori below K_c, the golden torus, island chains and the chaotic sea above. Method: exact area-preserving twist map iterated from a seed grid; the golden-mean orbit highlighted.

stochasticity K0.50

orbits22

iterations420

WHAT TO TRY

Let it sweep from K=0: watch horizontal invariant tori first wrinkle, then break into island chains around rational frequencies, while the white golden torus holds on longest.
Stop near K_c = 0.9716 with the slider: the golden torus is the last unbroken curve spanning the section. Just above it, that barrier is gone and p diffuses across the whole strip.
Read the bottom plot: the golden-torus p-spread stays flat while the torus confines the orbit, then jumps right at K_c. That is Greene's criterion, the quantitative onset of global chaos.