Hypergraphs and Lotka-Volterra systems with linear Darboux polynomials

TL;DR

This paper explores the relationship between hypergraphs and Lotka-Volterra systems with linear Darboux polynomials, introducing new superintegrable systems and analyzing their equivalence classes for small dimensions.

Contribution

It establishes a correspondence between hypergraphs and Lotka-Volterra systems with linear Darboux polynomials and presents a new superintegrable 5-component system.

Findings

Introduces a 13-parameter 5-component superintegrable Lotka-Volterra system.

Studies equivalence classes of hypergraphs for systems with up to 5 components.

Conjectures non-equivalence of tree-systems associated with nonisomorphic trees for n<9.

Abstract

We associate parametric classes of $n$-component Lotka-Volterra systems which admit $k$ additional linear Darboux polynomials, with admissible loopless hypergraphs of order $n$ and size $k$. We study the equivalence relation on admissible hypergraphs induced by linear transformations of the associated LV-systems, for $n\leq 5$. We present a new 13-parameter 5-component superintegrable Lotka-Volterra system, i.e. one that is not equivalent to a so-called tree-system. We conjecture that tree-systems associated with nonisomorphic trees are not equivalent, which we verified for $n<9$.