LIMIT CYCLES BIFURCATING FROM A PERIODIC ANNULUS IN DISCONTINUOUS PLANAR PIECEWISE LINEAR HAMILTONIAN DIFFERENTIAL SYSTEM WITH THREE ZONES

The paper proves that at least three limit cycles bifurcate from a periodic annulus in the stated three-zone piecewise linear near-Hamiltonian system by (i) deriving a first-order Melnikov function as a sum of four line integrals along the unperturbed closed orbit (Theorem 2; equation (5)) and (ii) reducing to a normal form that simplifies the coefficients (Proposition 3; equation (7)), yielding an explicit combination (equation (9)) of computable functions whose linear independence is established (via Wronskians) to force ≥3 simple zeros and hence ≥3 limit cycles (Theorem 1 and the concluding corollaries) . By contrast, the candidate solution asserts a Green’s-theorem-based decomposition of the Melnikov function into area terms plus boundary jump terms and then claims linear independence of {AL, AC, ΔyL, ΔyR} without proof; this unproven independence is the key step needed to enforce alternating signs at prescribed points and hence three zeros. Moreover, after the paper’s normal-form reduction, certain jump contributions vanish (S_L=S_R=0), undercutting the candidate’s chosen parameter specialization and leaving only three area-type basis functions, which cannot support the proposed four-point sign-assignment argument. Consequently, the model’s proof is incomplete, while the paper’s argument is complete and correct.