Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations

In this paper, by using Picard-Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): x˙=y−12x2+16y2, y˙=−x−16xy, and (r20): x˙=y+4x2, y˙=−x+16xy, and the periodic orbits of the quadratic isochronous centers (S1):x˙=−y+x2−y2, y˙=x+2xy, and (S2):x˙=−y+x2, y˙=x+xy. The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system (S1) and (S2) are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line y=0. It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively 4n−3(n≥4) and 4n+3(n≥3) counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers (S1) and (S2) are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively.