Limit Cycles of the Discontinuous Piecewise Differential Systems Separated by a Nonregular Line and Formed by a Linear Center and a Quadratic One

During the last decades, the study of discontinuous piecewise differential systems has become an interesting subject of research due to the important applications of this kind of systems to model natural phenomena. In the qualitative theory of differential equations, one of the interesting problems is the detection of the number of limit cycles and their configurations which remains open to date, except for very particular families of differential equations. Here, we are inspired to study the maximum number of limit cycles of the discontinuous piecewise differential systems separated by a nonregular line and formed by a linear center and one of the four classes of quadratic centers. The main tool used to prove our main results is based on the first integrals of such systems. All the computations of this paper are verified using the algebraic manipulator Mathematica.