Integrability conditions for Lotka-Volterra planar complex quartic systems having homogeneous nonlinearities
type: Item
instance of: scholarly article
title: Integrability conditions for Lotka-Volterra planar complex quartic systems having homogeneous nonlinearities
zbMATH Open document ID: 1280.34034
author: Brigita Ferčec
author: Valery G. Romanovski
author: Jaume Giné
author: Yirong Liu
published in: Acta Applicandae Mathematicae
publication date: 2013-04-11T00:00:00Z
review text: The (complex) center-focus problem is investigated for the system \[ \begin{aligned} \dot x &= x(1- a_{30} x^3- a_{21} x^2y- a_{12} xy^2- a_{03} y^3),\\ \dot y &=-y(1- b_{30} x^3- b_{21} x^2y- b_{12} xy^2- b_{03} y^3).\end{aligned} \] Necessary conditions for a center, that is, conditions for the existence of a first integral of the form \(\Psi=xy+h\cdot o\cdot t\) are obtained. For all of them but one it is proved that they are also the sufficient conditions for the existence of such an integral. In the unsolved case, the system can be written as \[ \begin{aligned} \dot x &= x\Biggl(1- {3\over 16} x^3+ {1\over 2} x^2 y- xy^2+ {8\over 9} y^3\Biggr),\\ \dot y &-y\Biggl(1+ {9\over 16} x^3- x^2y+{5\over 3} xy^2+ {16\over 9} y^3\Biggr)\end{aligned} \] and by the substitution \(x\to ax\), \(y\to by\), where \(a= 2({2\over 3})^{1/3}\), \(b= ({3\over 2})^{2/3}\), the latter system is transformed to \[ \begin{aligned} \dot x &= x(1-(x- y)(x^2- xy+ 2y^2)),\\ \dot y &= y(-1-8x^3+ (5x+ 4y)(x^2- y^2)).\end{aligned} \]
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 6153123
zbMATH Keywords: integrability
zbMATH Keywords: linearizability
zbMATH Keywords: polynomial vector field
zbMATH Keywords: polynomial differential system
reviewed by: A. P. Sadovskiĭ
describes a project that uses: SINGULAR
describes a project that uses: primdec
describes a project that uses: FGb
MaRDI profile type: Publication
full work available at URL: s10440-012-9772-5
OpenAlex ID: W2042337985
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DOI: 10.1007/S10440-012-9772-5
description: scientific article; zbMATH DE number 6153123