TY - JOUR

T1 - Limit cycles for a generalization of polynomial Liénard differential systems

AU - Llibre, Jaume

AU - Valls, Clàudia

PY - 2013/1/1

Y1 - 2013/1/1

N2 - © 2012 Elsevier Ltd We study the number of limit cycles of the polynomial differential systems of the form x˙=y-f1(x)y,y˙=-x-g2(x)-f2(x)y,where f1(x) = ɛf11(x) + ɛ2f12(x) + ɛ3f13(x), g2(x) = ɛg21(x) + ɛ2g22(x) + ɛ3g23(x) and f2(x) = ɛf21(x) + ɛ2f22(x) + ɛ3f23(x) where f1i, f2i and g2i have degree l, n and m respectively for each i = 1, 2, 3, and ɛ is a small parameter. Note that when f1(x) = 0 we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that this differential system can have bifurcating from the periodic orbits of the linear center x˙=y,y˙=-x using the averaging theory of third order.

AB - © 2012 Elsevier Ltd We study the number of limit cycles of the polynomial differential systems of the form x˙=y-f1(x)y,y˙=-x-g2(x)-f2(x)y,where f1(x) = ɛf11(x) + ɛ2f12(x) + ɛ3f13(x), g2(x) = ɛg21(x) + ɛ2g22(x) + ɛ3g23(x) and f2(x) = ɛf21(x) + ɛ2f22(x) + ɛ3f23(x) where f1i, f2i and g2i have degree l, n and m respectively for each i = 1, 2, 3, and ɛ is a small parameter. Note that when f1(x) = 0 we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that this differential system can have bifurcating from the periodic orbits of the linear center x˙=y,y˙=-x using the averaging theory of third order.

UR - https://ddd.uab.cat/record/150621

U2 - https://doi.org/10.1016/j.chaos.2012.11.010

DO - https://doi.org/10.1016/j.chaos.2012.11.010

M3 - Article

VL - 46

SP - 65

EP - 74

JO - Chaos, Solitons and Fractals

JF - Chaos, Solitons and Fractals

SN - 0960-0779

ER -