Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems
For researchers in numerical analysis and Hamiltonian dynamics, this work extends the theoretical foundation of symplectic integrators to the weakest non-degeneracy condition, but it is an incremental generalization of existing theorems.
This paper proves the persistence of invariant tori for integrable Hamiltonian systems under Rüssmann's non-degeneracy condition when symplectic integrators are applied, generalizing previous results that assumed Kolmogorov non-degeneracy. It also provides an estimate of the measure of the set occupied by these tori.
In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rüssmann'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, Rüssmann'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.