Some stability properties of Hamiltonian Poisson integrators

TL;DR

This paper explores the stability properties of Hamiltonian Poisson integrators, emphasizing the role of a modified Hamiltonian in both integrable and non-integrable systems, supported by numerical analysis of a 5D Lotka-Volterra model.

Contribution

It demonstrates the significance of a modified Hamiltonian for the stability of Poisson integrators in complex systems, including non-integrable cases.

Findings

Modified Hamiltonian plays a crucial role in stability analysis.

Numerical investigations on a 5D Lotka-Volterra system reveal stability characteristics.

Illustrates importance in both integrable and non-integrable Hamiltonian systems.

Abstract

Hamiltonian Poisson integrators are Poisson integrators that admit a modified Hamiltonian. In this article, we illustrate the importance of the existence of a modified Hamiltonian for Poisson integrators in the context of integrable and non-integrable systems. Examples of Hamiltonian systems are provided by Lotka-Volterra dynamics; in order to investigate stability properties of Hamiltonian Poisson integrators on non-integrable systems, we exhibit a non-integrable $5$-dimensional Lotka-Volterra system and pursue numerical investigations of it.