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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 x ˙=x(1-a 40 x 4 -a 31 x 3 y-a 22 x 2 y 2 -a 13 xy 3 -a 04 y 4 ),y ˙=-y(1-b 40 x 4 -b 31 x 3 y-b 22 x 2 y 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:
34A05Methods of solution of ODE
34C05Location of integral curves, singular points, limit cycles (ODE)
Software:
SINGULAR
References:
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