Standard Map: 1st Poincaré Integral Invariant
The first Poincaré integral invariant
\[I_{1} = \oint_{\gamma} p \, d\theta\]
of the Chirikov standard map, for four values of the stochasticity parameter K. K ≈ 0.971635 is the critical value at which the last invariant torus breaks up.
The map is symplectic, so I₁ is conserved exactly; what the figures show is how well the loop integral can still be evaluated numerically as the flow folds the loop. In the regular regime it is conserved to machine precision, and in the chaotic regime the quadrature loses resolution exponentially.
K = 0.6
K = 0.971635
K = 1.2
K = 2.0
