This paper deals with the problem of limit cycle bifurcations for two kinds of quadratic reversible differential systems, when they are perturbed inside all discontinuous polynomials of degree n. The switching lines are \(x=1\) and \(y=0\). Firstly, we derive the algebraic structure of the first order Melnikov function M(h) by computing its generating functions, which is more complicated than the Melnikov function corresponding to the perturbations with one switching line. Then, we obtain the detailed expression of M(h) by solving the Picard–Fuchs equations that the generating functions satisfy. Finally, we derive the upper bounds of the number of limit cycles by using the derivation-division algorithm for \(n\ge 2\) and the lower bounds of the number of limit cycles by linear independence for \(n=2\), counting the multiplicity.

本文研究了两种二次可逆微分系统在所有n 次不连续多项式中被摄动时的极限环分岔问题。切换线是\(x=1\)和\(y=0\)。首先，我们通过计算其生成函数推导出一阶Melnikov函数M ( h )的代数结构，这比Melnikov函数对应一个开关线的扰动更复杂。然后，我们得到了M ( h) 通过求解生成函数满足的 Picard-Fuchs 方程。最后，我们通过对\(n\ge 2\)使用推导-除法算法来推导出极限环数的上界，对\(n=2\) 使用线性独立性来推导极限环数的下界，计算多重性。