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The \({1:-q}\) resonant center problem for certain cubic Lotka-Volterra systems. (English) Zbl 1365.34006

Summary: Necessary conditions and distinct sufficient conditions are derived for the system \[ \begin{aligned} \dot x = x(1-a_{20}x^2-a_{11}xy-a_{02}y^2), \\ \dot y=y (-q+b_{20}x^2+b_{11}xy+b_{02}y^2) \end{aligned} \] to admit a first integral of the form \(\varPhi(x, y)=x^qy+\cdots\) in a neighborhood of the origin, in which case the origin is termed a \(1 : -q\) resonant center. Necessary and sufficient conditions are obtained for odd \(q\), \(q \leqslant 9\); necessary conditions, most of which are also sufficient, are obtained for even \(q\), \(q \leqslant 8\). Key ideas in the proofs are computation of focus quantities for the complexified systems and decomposition of the variety of the ideal generated by an initial string of them to obtain necessary conditions, and the theory of Darboux first integrals to show sufficiency.

MSC:

34A05 Explicit solutions, first integrals of ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations

Software:

FGb; SINGULAR; primdec
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References:

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