On the qualitative behaviour of symplectic integrators Part I: Perturbed linear systems

We consider a dissipative perturbation of non–resonant harmonic oscillators. Under the perturbation the system admits a weakly attractive invariant torus. We apply a Runge-Kutta method to the system. If the integration method is symplectic then it also admits an attractive invariant torus, the step-size being independent of the perturbation parameter. For non–symplectic methods the discrete system only admits an attractive invariant torus if the step-size is so small such that the discretisation error is smaller than the perturbation.