Bi-center conditions and bifurcation of limit cycles in a class of Z2-equivariant cubic switching systems with two nilpotent points

The paper proves six necessary-and-sufficient bi-center conditions (I–VI) and, under any one of them, constructs perturbations that yield 9 small crossing cycles at each nilpotent center plus 1 more per center via a symmetric pseudo-Hopf, totaling 20; this is explicitly stated and supported by detailed generalized Lyapunov-constant computations and a nonvanishing Jacobian determinant for nine independent parameters, as well as an explicit pseudo-Hopf analysis (e.g., Theorems 2.2 and 2.3, determinant calculation, and equations (59)–(61) in the uploaded PDF . The candidate solution outlines the same end result and mechanism (9 focus-type cycles + 1 pseudo-Hopf at each symmetric point) but argues at a higher level using a quasi-homogeneous nilpotent Poincaré map and a triangular unfolding-in-ε scheme rather than the paper’s Bogdanov–Takens-based construction and explicit symbolic computations. Hence both are consistent in claims and logic, with different proof styles.