Due to the presence of parasitic roots in linear multistep methods, the numerical solution of differential (and differential-algebraic) equations gives rise to non-physical oscillations. For strictly stable methods these oscillations are rapidly damped, so that the numerical solution behaves like that of a one-step method. The presented results have been obtained in collaboration with Christian Lubich and Paola Console. For symmetric methods these oscillations, although with small amplitude in the beginning, can grow exponentially with time and soon dominate the error in the numerical approximation. Certain symmetric multistep methods for second order differential equations, when applied to (constrained) Hamiltonian systems, have the feature that these oscillations remain bounded and small (below the discretization error of the smooth solution) over very long time intervals. Numerical experiments are presented and a proof of the long-time behaviour is outlined. The technique of proof is backward error analysis combined with modulated Fourier expansions. The presented results have been obtained in collaboration with Christian Lubich and Paola Console.