# On the dynamics of a Hamilton-Poisson system

## TL;DR

This paper explores the dynamics of a three-dimensional Hamilton-Poisson system, introducing a new system with multiple realizations, analyzing stability, periodic orbits, and heteroclinic connections through both theoretical and numerical methods.

## Contribution

It constructs a novel Hamilton-Poisson system with infinitely many realizations and investigates its dynamic properties, including stability and heteroclinic orbits.

## Key findings

- Identification of equilibrium point stability
- Existence of periodic orbits
- Discovery of heteroclinic orbit pairs through numerical integration

## Abstract

The dynamics of a three-dimensional Hamilton-Poisson system is closely related to its constants of motion, the energy or Hamiltonian function $H$ and a Casimir $C$ of the corresponding Lie algebra. The orbits of the system are included in the intersection of the level sets $H=constant$ and $C=constant$. Furthermore, for some three-dimensional Hamilton-Poisson systems, connections between the associated energy-Casimir mapping $(H,C)$ and some of their dynamic properties were reported. In order to detect new connections, we construct a Hamilton-Poisson system using two smooth functions as its constants of motion. The new system has infinitely many Hamilton-Poisson realizations. We study the stability of the equilibrium points and the existence of periodic orbits. Using numerical integration we point out four pairs of heteroclinic orbits.