On the number of limit cycles coming from a uniform isochronous center with continuous and discontinuous quartic perturbations (Q6557956)

The purpose of this work is the number of limit cycles bifurcating from the periodic orbits of a cubic uniform isochronous center with continuous and discontinuous quartic polynomial perturbations in real autonomous systems defined as follows \N\[\N\frac{dx}{dt} = -y + x^2y + \varepsilon p(x,y), \quad \frac{dy}{dt} = x + xy^2 + \varepsilon q(x,y),\N\]\Nwhere the real parameter \(\varepsilon\) is sufficiently small. In the case of discontinuous perturbations the system consists of two parts separated by the axes \(y=0\). Applying the averaging theory of first order for continuous and discontinuous differential systems and comparing the obtained results, the authors conclude that the discontinuous systems can have at least \(6\) more limit cycles than the continuous ones.