Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems

TL;DR

This paper investigates the persistence of invariant tori in integrable Hamiltonian systems under symplectic integrators, extending previous results to Rüssmann's weaker non-degeneracy condition and providing measure estimates for these tori.

Contribution

It generalizes existing theorems on invariant tori persistence to Rüssmann's non-degeneracy condition, offering new insights into symplectic integrator stability.

Findings

Invariant tori persist under symplectic integrators with Rüssmann's condition.

The measure of phase space occupied by invariant tori is estimated.

The one-step map on an invariant torus is conjugate to a family of linear rotations.

Abstract

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying R\"{u}ssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus, the one-step map of the scheme is conjugate to a one parameter family of linear rotations with a step size dependent frequency vector in terms of iteration. These results are a generalization of Shang's theorems (1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov. In comparison, R\"{u}ssmann's condition is the weakest non-degeneracy condition for the persistence of invariant tori in Hamiltonian systems. These results provide new insight into the nonlinear stability of symplectic integrators.