On the limit cycles of polynomial vector field

In this paper we study the limit cycles which can bifurcate from the periodic orbits of the center located at the origin of the quadratic polynomial di®erential system x = -y(1+x), y= x(1+x), and of the cubic polynomial di®erential system x = -y(1-x2-y2), y= x(1-x 2-y2), when we perturb them in the class of all polynomial vector fields with quadratic and cubic homogenous nonlinearities, respectively. For doing this study we use the averaging theory. Copyright © 2011 Watam Press.