Integrability conditions for Lotka–Volterra planar complex quintic systems

Giné, Jaume; Romanovski, Valery G.

doi:https://doi.org/10.1016/j.nonrwa.2009.06.002

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Issue date

2010

Author

Giné, Jaume

Romanovski, Valery G.

Suggested citation

Giné, Jaume; Romanovski, Valery G.; . (2010) . Integrability conditions for Lotka–Volterra planar complex quintic systems. Nonlinear Analysis: Real World Applications, 2010, vol. 11, núm. 3, p. 2100-2105. https://doi.org/10.1016/j.nonrwa.2009.06.002.

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.

URI

http://hdl.handle.net/10459.1/58671

DOI

https://doi.org/10.1016/j.nonrwa.2009.06.002

Is part of

Nonlinear Analysis: Real World Applications, 2010, vol. 11, núm. 3, p. 2100-2105