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Some stability properties of Hamiltonian Poisson integrators

Oscar Cosserat

TL;DR

The article develops and applies Hamiltonian Poisson integrators to Poisson-geometric and integrable settings, showing that a modified Hamiltonian exists and that backward error analysis yields exponentially long-time stability around Liouville tori. It generalizes KAM-type long-time estimates to Poisson systems and analyzes stability near non-resonant elliptic singularities, illustrating quasi-periodic behavior through a Lotka-Volterra testbed. Using a 5D integrable LV system and a constructed non-integrable LV-like system, the work demonstrates both theoretical stability and practical limitations, including transitions to chaotic dynamics in non-integrable regimes. The results have implications for structure-preserving numerics in Poisson geometry and potentially for control theory, where respecting foliations and invariants improves long-time qualitative behavior.

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.

Some stability properties of Hamiltonian Poisson integrators

TL;DR

The article develops and applies Hamiltonian Poisson integrators to Poisson-geometric and integrable settings, showing that a modified Hamiltonian exists and that backward error analysis yields exponentially long-time stability around Liouville tori. It generalizes KAM-type long-time estimates to Poisson systems and analyzes stability near non-resonant elliptic singularities, illustrating quasi-periodic behavior through a Lotka-Volterra testbed. Using a 5D integrable LV system and a constructed non-integrable LV-like system, the work demonstrates both theoretical stability and practical limitations, including transitions to chaotic dynamics in non-integrable regimes. The results have implications for structure-preserving numerics in Poisson geometry and potentially for control theory, where respecting foliations and invariants improves long-time qualitative behavior.

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 -dimensional Lotka-Volterra system and pursue numerical investigations of it.

Paper Structure

This paper contains 18 sections, 15 theorems, 51 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Hamiltonian Poisson integrators on integrable systems
  3. Hamiltonian Poisson integrators
  4. Completely integrable systems in Poisson geometry
  5. Exponentially long times estimates for the discrete trajectory
  6. The Lotka-Volterra model
  7. The evolution equation and related Hamiltonian structures
  8. Hamiltonian Poisson integrators for Lotka-Volterra systems
  9. Examples of integrable systems and numerical simulations
  10. Around non-resonant elliptic singularities
  11. Backward error analysis for the stability of Hamiltonian Poisson integrators around elliptic orbits
  12. A toy example: the Euler symplectic scheme for the harmonic oscillator
  13. A non-integrable Hamiltonian system
  14. Numerical investigations of a non-integrable system
  15. Near the family of periodic orbits $\Pi^1$
  16. ...and 3 more sections

Key Result

Proposition 2.1

A Hamiltonian Poisson integrator stays on a symplectic leaf along iterations. It preserves consequently any Casimir.

Figures (10)

  • Figure 1: The $10^2$ first iterations of the Hamiltonian Poisson integrator \ref{['eq:Pois_Ham_1']} applied to the system \ref{['eq:LV_5d_integrable']} with time-step $\Delta t = 1$ and initial point $x_0 = (2,2,2,2,2)$
  • Figure 2: The behavior of a Hamiltonian Poisson integrator of order $k$ applied to a Hamiltonian system admitting an elliptic non-resonant singularity on a symplectic leaf of dimension $2$
  • Figure 3: The first $100$ iterations of the numerical method \ref{['eq:euler_sym']} with a time-step $\epsilon$
  • Figure 4: Iterates of the coordinates $(x_1, x_2, x_3)$
  • Figure 5: Iterates of the coordinates $(x_4, x_5)$
  • ...and 5 more figures

Theorems & Definitions (37)

  • Remark 2.1: Backward error analysis
  • Remark 2.2
  • Definition 2.1: Hamiltonian Poisson integrator
  • Proposition 2.1
  • Definition 2.2: Liouville Integrable system
  • Theorem 2.1: Action-angle variables
  • Theorem 2.2: Long run estimates
  • proof
  • Remark 2.3
  • Remark 3.1: Applications of Lotka-Volterra systems
  • ...and 27 more