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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.

34A05Methods of solution of ODE
34C05Location of integral curves, singular points, limit cycles (ODE)
primdec; SINGULAR
Full Text: DOI
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