From dynamics to statistical mechanics in the Hénon-Heiles model: Dynamics

The equations of motion of the H\'enon-Heiles model have been numerically integrated for 100 different starting conditions on the critical energy surface E=1/6. The truncation error in the data was monitored using the separation in phase space of two numerical histories with the same initial conditions, based on different time steps, h=2^-13 and h'=2^-14. From the data for which the truncation error is less than 1% it is found that the 100 histories fall into three categories, (a) regular (or quasiperiodic), (b) irregular (or chaotic), and (c) regular-irregular. The three-dimensional phase space portraits prove to be the most useful tool in distinguishing between regular and irregular behavior. In category (c) the orbit switches reversibly from (to) regular to (from) irregular behavior. The data suggest that all 100 orbits, followed for long enough times, will show regular-irregular behavior and moreover the phase points in these orbits will spend, on average, equal amounts of time in the quasiperiodic and chaotic regimes.