Integrability conditions for Lotka–Volterra planar complex quintic systems

Abstract

In this paper we obtain necessary and sufficient integrability conditions at the origin for the Lotka–Volterra complex quintic systems which are linear systems perturbed by fifth degree homogeneous polynomials, i.e., we consider systems of the form View the MathML sourceẋ=x(1−a40x4−a31x3y−a22x2y2−a13xy3−a04y4), View the MathML sourceẏ=−y(1−b40x4−b31x3y−b22x2y2−b13xy3−b04y4). The necessity of these conditions is derived from the first nine focus-saddle quantities and their sufficiency is proved by finding an inverse integrating factor or a first integral.

The first author is partially supported by a MCYT/FEDER grant number MTM2008-00694 and by a CIRIT grant number 2005SGR 00550. The second author acknowledges the support of the Slovenian Research Agency, of the Nova Kreditna Banka Maribor, of TELEKOM Slovenije, and of the Transnational Access Programme at RISC-Linz of the European Commission Framework 6 Programme for Integrated Infrastructures Initiatives under the project SCIEnce (Contract No. 026133).