The elliptical invariant tori of nearly integrable Hamiltonian system through symplectic algorithms

In this paper we apply symplectic algorithms to nearly integrable Hamiltonian system, and prove it can maintain lots of elliptic lower dimensional invariant tori. We are committed to consider the elliptic lower dimensional invariant tori for symplectic mapping with a small twist under the R\"{u}ssmann's non-degenerate condition, and focus on its measure estimation. And then apply it to the nearly integrable Hamiltonian system to obtain lots of elliptic lower dimensional invariant tori.