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Numerical Integration of Nearly Integrable Systems

Perturbed Linear Systems: Consider a dissipative perturbation of a system of non-resonant harmonic oscillators. We assume that the perturbed system admits a weakly attractive invariant torus. Assume that this system is integrated numerically using a Runge-Kutta scheme with constant step-size. If the integration method is symplectic the discrete system admits an attractive invariant torus as well, with no restriction on the step size by the size of the perturbation. For nonsymplectic schemes however, the discrete system admits an attractive invariant torus only if the discretisation error is small compared to the perturbation, for details see D. Stoffer: ''On the qualitative behaviour of symplectic integrators. Part I: Perturbed linear systems'', Numer. Math. 77 (1997), no. 4, 535--547.
Perturbed Integrable Systems: Consider a dissipative perturbation of an arbitrary integrable Hamiltonian system, the smallness of the perturbation being described with the help of a perturbation parameter. Again the perturbed system is assumed to admit a (weakly) attractive torus. Assume that the system is integrated numerically using a symplectic integrator. Then the discrete system also admits an attractive invariant torus for sufficiently small step size. But how small? We show that it may be chosen remarkably large. Essentially the step size can be taken as large as the logarithm of the perturbation parameter. This means that the discrete system provides a faithful description of the qualitative behaviour of the underlying differential equation even for large step size. For details see D. Stoffer: "On the qualitative behaviour of symplectic integrators. II. Integrable systems, J. Math. Anal. Appl. 217 (1998), no. 2, 501--520.

Contacts:

PD Dr. D. Stoffer, Prof. U. Kirchgraber

Electronic Contacts:

stoffer@math.ethz.ch

Last Update:

04/07/97

Responsible:

Professor or Project Leader:           : Prof. Dr. Urs Kirchgraber
Institute or Independent Professorship : Independent Professorship of Mathematics
Department                             : Department of Mathematics

Comments to the ETH Research Report administration : Sat Jun 19 16:30:35 1999