Hamiltonian structure and Darboux theorem for families of generalized Lotka-Volterra systems

TL;DR

This paper establishes a Poisson structure for generalized Lotka-Volterra systems, revealing their algebraic hierarchy and deriving key features like symplectic foliation and Darboux form through matrix methods.

Contribution

It translates the algebraic hierarchy of generalized Lotka-Volterra systems into a Poisson framework, providing explicit structures and canonical forms.

Findings

Poisson structure for generalized Lotka-Volterra systems established

Symplectic foliation and Darboux form derived from matrix manipulations

Algebraic hierarchy translated into Poisson geometric structures

Abstract

This work is devoted to the establishment of a Poisson structure for a format of equations known as Generalized Lotka-Volterra systems. These equations, which include the classical Lotka-Volterra systems as a particular case, have been deeply studied in the literature. They have been shown to constitute a whole hierarchy of systems, the characterization of which is made in the context of simple algebra. Our main result is to show that this algebraic structure is completely translatable into the Poisson domain. Important Poisson structures features, such as the symplectic foliation and the Darboux' canonical representation, arise as result of rather simple matrix manipulations.