On Integrals, Hamiltonian and Metriplectic Formulations of 3D Polynomial Systems

TL;DR

This paper explores the integrability and Hamiltonian structures of various 3D polynomial systems, including the Darboux method, and examines their Hamiltonian, Nambu-Poisson, and metriplectic properties.

Contribution

It applies Darboux integrability to identify first integrals and Hamiltonian formulations for several complex 3D polynomial systems, highlighting their geometric structures.

Findings

Identified first integrals for multiple 3D polynomial systems.

Established Hamiltonian, Nambu-Poisson, and metriplectic frameworks for these systems.

Enhanced understanding of the geometric and integrable properties of these models.

Abstract

We apply the Darboux integrability method to determine first integrals and Hamiltonian formulations of three dimensional polynomial systems; namely the reduced three-wave interaction problem, the Rabinovich system, the Hindmarsh-Rose model, and the oregonator model. Additionally, we investigate their Hamiltonian, Nambu-Poisson and metriplectic characters.