The numerical integration of Hamiltonian systems with different time scales and exponentially diverging trajectories is a challanging task. In particular, standard forward error analysis cannot clarify the question if the obtained numerical results make any sense for long-term simulations. The recent progress in the backward error analysis of symplectic methods for Hamiltonian problems might change this unsatisfying situation. In my talk I will apply backward error analysis, normal form theory, and the concept of shadowing to show that for hyperbolic Hamiltonian systems and systems with adiabatic invariants that are due to separated time scales, the numerical results can be interpreted in a meaningful way.

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