Limit cycles for classes of piecewise smooth differential equations separated by the unit circle

The paper proves that systems in XS1_0 ∪ XS1_1 have at most one crossing limit cycle meeting S1 in two points, and systems in XS1_2 ∪ XS1_3 ∪ XS1_4 have at most two, via “closing equations” and Bézout counts for intersections of conics; see their Main Results (Theorem A/B) and the class-by-class theorems (XS1_0: Theorem 2; XS1_1: Theorem 6; XS1_2: Theorem 10; XS1_3: Theorem 16; XS1_4: Theorem 20) . The model’s solution reaches the same bounds using an angular parametrization of S1, the quadratic first integrals for trace-zero linear fields, and a “reflection law” for constant fields, then solves the two-point constraints directly. The two arguments are aligned in conclusions but use different methods.