The figure displays a Poincaré section of the Hénon-Heiles
Hamiltonian,

H=(1/2)(x_{1}^{2}+x_{2}^{2}+
y_{1}^{2}+y_{2}^{2})+
(1/3)x_{1}^{3}-x_{1}x_{2}^{2},
using x_{2}=0 as surface of section, and (x_{1},y_{1})
as coordinates. In the figure the x_{1} positive axis points
downwards and the y_{1} one to the right. As initial
points we have taken 400 points on y_{1}=0 equally spaced between the
two elliptic points (on that line) which correspond to elliptic periodic orbits.
Note that also the boundary of the section is a periodic orbit. For each one
of the initial points 5,000 iterates of the Poincaré map are computed
and
plotted. In all 2,000,000 Poincaré iterates have been computed. In all these
iterates the maximal error in the conservation of the energy is 1.31E-16.
Estimated total number of arithmetic operations: 90.65E9.
Total (manual cronometer) time: 45 seconds.