Limit Cycles For Quadratic And Cubic Planar Differential Equations Under Polynomial Perturbations Of Small Degree

In this paper we consider planar systems of differential equations of the form { (x) over dot = -y + delta p(x,y) + epsilon P-n(x,y) , (y) over dot = x + delta q(x,y) + epsilon Q(n)(x,y), where delta, epsilon are small parameters, (p, q) are quadratic or cubic homogeneous polynomials such that the unperturbed system (epsilon = 0) has an isochronous center at the origin and P-n, Q(n) are arbitrary perturbations. Estimates for the maximum number of limit cycles are provided and these estimatives are sharp for n <= 6 (when p, q are quadratic). When p, q are cubic polynomials and P-n ,Q(n) are linear, the problem is addressed from a numerical viewpoint and we also study the existence of limit cycles.37633533386Octave development communit