Liouvillian integration of the Lotka-Volterra system
type: Item
instance of: scholarly article
title: Liouvillian integration of the Lotka-Volterra system
zbMATH Open document ID: 1084.34001
DOI: 10.1007/BF02969345
author: Jean Moulin Ollagnier
published in: Qualitative Theory of Dynamical Systems
publication date: 2005-04-23T00:00:00Z
review text: The system of differential equations under consideration is \[ x' = x(Cy+z),\quad y' = y(Az+x),\quad z' = z(Bx+y), \] where \(A\), \(B\), and \(C\) are nonzero complex constants. This system is of broad interest because it typically serves as a normal form for ``factored quadratic systems,'' quadratic homogeneous systems such that \(\alpha\) factors out of \(\alpha'\) for each \(\alpha \in \{x, y, z \}\). A first integral for the system is a function that is constant on trajectories. The chief result of this paper is a characterization of those triples \((A, B, C)\) for which the system of ordinary differential equations has a Liouvillian first integral. Specific first integrals, or Darboux polynomials, are provided. The author also addresses the problem of Liouvillian integration of factored quadratic systems that cannot be placed in the form above.
reviewed by: Douglas S. Shafer
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 2161121
zbMATH Keywords: first integral
zbMATH Keywords: Darboux polynomial
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02969345
OpenAlex ID: W2011939732
description: scientific article; zbMATH DE number 2161121