On the limit cycles for continuous and discontinuous cubic differential systems (Q1677693)

Summary: We study the number of limit cycles for the quadratic polynomial differential systems \(\dot{x} = - y + x^2\), \(\dot{y} = x + x y\) having an isochronous center with continuous and discontinuous cubic polynomial perturbations. Using the averaging theory of first order, we obtain that 3 limit cycles bifurcate from the periodic orbits of the isochronous center with continuous perturbations and at least 7 limit cycles bifurcate from the periodic orbits of the isochronous center with discontinuous perturbations. Moreover, this work shows that the discontinuous systems have at least 4 more limit cycles surrounding the origin than the continuous ones.