Abstract
We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in ℝ2 having an invariant ellipse. More precisely, a quadratic system having an invariant ellipse can be written into the form \(\dot x = x^2 + y^2 - 1 + y\left( {ax + by + c} \right)\), \(\dot y = - x\left( {ax + by + c} \right)\), and the ellipse becomes x 2 + y 2 = 1. We prove that
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(i)
this quadratic system is analytic integrable if and only if a = 0
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(ii)
if x 2+y 2 = 1 is a periodic orbit, then this quadratic system is Liouvillian integrable if and only if x 2 + y 2 = 1 is not a limit cycle; and
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(iii)
if x2 +y 2 = 1 is not a periodic orbit, then this quadratic system is Liouvilian integrable if and only if a = 0.
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The first author is partially supported by the MINECO/FEDER (Grant No. MTM2008-03437), AGAUR (Grant No. 2009SGR-410), ICREA Academia and FP7-PEOPLE-2012-IRSES 316338 and 318999; the second author is supported by Portuguese National Funds through FCT — Fundação para a Ciência e a Tecnologia within the project PTDC/MAT/117106/2010 and by CAMGSD
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Llibre, J., Valls, C. Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse. Acta. Math. Sin.-English Ser. 30, 453–466 (2014). https://doi.org/10.1007/s10114-014-2484-1
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DOI: https://doi.org/10.1007/s10114-014-2484-1