Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree

Abstract

In this paper we consider planar systems of differential equations of the form ẋ = −y + δp(x, y) + εPn(x, y), y = x + δq(x, y) + εQn(x, y), where δ, ε are small parameters, (p, q) are quadratic or cubic homogeneous polynomials such that the unperturbed system (ε = 0) has an isochronous center at the origin and Pn, Qn 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 Pn, Qn are linear, the problem is addressed from a numerical viewpoint and we also study the existence of limit cycles.