Dissipatively Perturbed Hamiltonian and Reversible Systems

Abstract

Symplectic integrators also show a favourable long-time behaviour when they are applied to non-Hamiltonian perturbations of Hamiltonian systems. The same is true for symmetric methods applied to non-reversible perturbations of reversible systems. In this chapter we study the behaviour of numerical integrators when they are applied to dissipative perturbations of integrable systems, where only one invariant torus persists under the perturbation and becomes weakly attractive. The simplest example of such a system is Van der Pol’s equation with small parameter, which has a single limit cycle in contrast to the infinitely many periodic orbits of the unperturbed harmonic oscillator.