TY - JOUR

T1 - Liouvillian and analytic integrability of the quadratic vector fields having an invariant ellipse

AU - Llibre, Jaume

AU - Valls, Claudia

PY - 2014/3/1

Y1 - 2014/3/1

N2 - 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 ẋ = x2 + y2 - 1 + y (ax + by + c}, ẏ = - x(ax + by + c}, and the ellipse becomes x 2 + y 2 = 1. We prove that (i) this quadratic system is analytic integrable if and only if a = 0 (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 (iii) if x2 +y 2 = 1 is not a periodic orbit, then this quadratic system is Liouvilian integrable if and only if a = 0. © 2014 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.

AB - 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 ẋ = x2 + y2 - 1 + y (ax + by + c}, ẏ = - x(ax + by + c}, and the ellipse becomes x 2 + y 2 = 1. We prove that (i) this quadratic system is analytic integrable if and only if a = 0 (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 (iii) if x2 +y 2 = 1 is not a periodic orbit, then this quadratic system is Liouvilian integrable if and only if a = 0. © 2014 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.

KW - Liouvillian integrability

KW - invariant ellipse

KW - quadratic planar polynomial vector fields

UR - https://ddd.uab.cat/record/150717

U2 - https://doi.org/10.1007/s10114-014-2484-1

DO - https://doi.org/10.1007/s10114-014-2484-1

M3 - Article

VL - 30

SP - 453

EP - 466

JO - Acta Mathematica Sinica, English Series

JF - Acta Mathematica Sinica, English Series

SN - 1439-8516

IS - 3

ER -