Performance of two methods for solving separable Hamiltonian systems

We make qualitative comparisons of fixed step symplectic and variable step nonsymplectic integrations of the separable Hénon–Heiles Hamiltonian system. Emphasis is given to interesting numerical phenomena. Particularly, we observe the relationship of the error in the computed Hamiltonian to the presence and absence of chaos, when computing with a symplectic (fixed step) method, qualitative phenomena in the Hamiltonian error for a variable step method, and the sensitivity of the chaotic behavior and of the computation of features in Poincaré sections to very small changes in initial conditions, step sizes and error tolerances.