Limit cycles bifurcating from periodic integral manifold in non-smooth differential systems

The paper’s Theorem 1 establishes persistence and local uniqueness of a crossing periodic orbit near a periodic integral manifold for small ε, under (H1), the nondegeneracy det(Π2·β(u))≠0, and a nondegenerate zero of the Melnikov-type map M(u) defined via M(u)=Π1(Id−β(Π2β)^{-1}Π2)α(u). The paper constructs impact-time maps t_±, defines the Σ-to-Σ displacement Δ, computes first variations α and β using variation-of-constants and event-time differentiation (Propositions 7–8), and reduces via Lyapunov–Schmidt to obtain M; existence and uniqueness then follow by the implicit function theorem (see Theorem 1 and formulas (2)–(3) for M, α, β; hypotheses (H1); and the displacement/derivative computations). The candidate solution reproduces the same program: event-time maps, displacement Δ, first variations α, β, the elimination projector E(u)=Id−β(Π2β)^{-1}Π2, and the reduced equation F(u,ε)=Π1E(u)Δ(u,ε)=εM(u)+o(ε), followed by IFT and reconstruction of the periodic orbit, including the crossing property and local uniqueness. Minor differences are expository (the paper phrases the reduction via a Lyapunov–Schmidt lemma, the candidate via an explicit projector), but the logical structure and key ingredients coincide. Thus both are correct with substantially the same proof, and the model’s argument aligns tightly with the paper’s derivation and hypotheses (Theorem 1, (H1), and Propositions 7–8) .