Structure preserving numerical methods for time-dependent differential systems have received a lot of attention recently. The challenge is to find for a given dynamical system discretization methods which yield discrete dynamical systems possessing ``important properties'' of their continuous couterpart to a good accuracy even without closely reproducing all exact system's features (especially without pointwise accuracy).

Occasionally, it is possible to analyze the discrete dynamical system by transforming it and considering the passage of the transformed discrete system to a nearby continuous limit. If the limit ``ghost differential system'' behaves well then the approximate solution is qualitatively good. But if not then by following these ``wrong'' features the discrete dynamical system may become unstable, or worse, produce deceptively looking, qualitatively wrong solutions.

We will discuss several examples of Hamiltonian systems, including certain PDEs in one space variable and highly oscillatory ODEs.