Exponential Stability Estimate of Symplectic Integrators for Integrable Hamiltonian Systems

TL;DR

This paper establishes exponential stability estimates for symplectic integrators applied to integrable Hamiltonian systems, enhancing understanding of their long-term behavior and stability in nonlinear dynamics.

Contribution

It proves a Nekhoroshev-type theorem for nearly integrable symplectic maps and derives bounds on perturbations and action variable variations, advancing stability analysis of symplectic algorithms.

Findings

Exponential bounds for perturbations and action variations

Enhanced stability estimates for symplectic integrators

Deeper understanding of nonlinear dynamical behavior

Abstract

We prove a Nekhoroshev-type theorem for nearly integrable symplectic map. As an application of the theorem, we obtain the exponential stability symplectic algorithms. Meanwhile, we can get the bounds for the perturbation, the variation of the action variables, and the exponential time respectively. These results provide a new insight into the nonlinear stability analysis of symplectic algorithms. Combined with our previous results on the numerical KAM theorem for symplectic algorithms (2018), we give a more complete characterization on the complex nonlinear dynamical behavior of symplectic algorithms.