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Integrability conditions for Lotka-Volterra planar complex quintic systems. (English) Zbl 1194.34003

Summary: We obtain necessary and sufficient integrability conditions at the origin for Lotka-Volterra complex quintic systems which are linear systems perturbed by fifth degree homogeneous polynomials, i.e., we consider systems of the form \(\dot x = x(1-a_{40}x^4-a_{31}x^3y-a_{22}x^2y^2 -a_{13}xy^3-a_{04}y^4),\dot y = -y (1-b_{40}x^4 -b_{31}x^3y-b_{22}x^2y^2-b_{13}xy^3-b_{04}y^4)\). 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.

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

Software:

SINGULAR; primdec
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References:

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