Bifurcation analysis of multiple limit cycles created in boundary equilibrium bifurcations in hybrid systems

The paper constructs a smooth Poincaré map via blow-up and proves that, at a codimension-two BEB, a simple unit eigenvalue (+1) yields a saddle-node curve of single-impact limit cycles and a simple eigenvalue (−1) with c^2+f≠0 yields a period-doubling curve, for µ>0 small. These are Theorems 2 and 3, respectively, and rely on the smoothness of P, center-manifold reduction, and standard normal forms for maps . The candidate solution proves the same bifurcation statements by Lyapunov–Schmidt reduction (for +1) and a second-iterate computation (for −1), obtaining the same nondegeneracy and transversality conditions and the same correspondence between fixed points/2-cycles of P and single-impact cycles of the hybrid system. Minor discrepancies are not substantive: the paper states P is C^{k−1} (if F,R are C^k), whereas the candidate’s outline informally says C^k; and the paper uses center manifolds while the candidate uses Lyapunov–Schmidt. Overall, both arguments are correct and reach the same conclusions.