Exponential Stability Estimate of Symplectic Integrators for Integrable Hamiltonian Systems
For researchers in numerical analysis and Hamiltonian dynamics, this provides a theoretical foundation for the long-term stability of symplectic integrators in integrable systems.
The authors prove a Nekhoroshev-type theorem for nearly integrable symplectic maps and apply it to show exponential stability of symplectic integrators, providing bounds on perturbation, action variation, and exponential time. This extends previous work on numerical KAM theory for symplectic algorithms.
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.