Limit cycles for a generalization of polynomial Liénard differential systems

© 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.