Local and Global Bifurcation for Periodic Solutions of Hamiltonian Systems via Comparison Theory for the Spectral Flow

The uploaded paper states and proves Theorem 2.2 using a variational setup on H^{1/2}(S^1, R^{2n}), a comparison principle for spectral flow, explicit constant-coefficient computations via crossing forms, and a parity-versus-spectral-flow link to obtain global bifurcation in the planar case. The definitions, assumptions, and proof steps are clearly laid out and consistent, including the variational functional ψ, the Hessians L_λ, the paths M and N with B_λ and C_λ, the comparison inequality sf(M) ≤ sf(L) ≤ sf(N), the explicit counts sf(M)=2nΔ(β−,α+) and sf(N)=2nΔ(α−,β+), and the global bifurcation criterion via odd spectral flow and parity (Theorems 3.2, 3.4; Prop. 3.3) . By contrast, the candidate solution has multiple sign and order errors: it uses a minus sign in the potential term of the action (yielding the wrong Euler–Lagrange sign), assigns the wrong sign to the spectral flow of the constant path corresponding to C^+(s), and then applies the comparison inequality in the wrong direction in case (i) for β+<α−, making the conclusion invalid as stated. It also asserts an “exact parity formula” mod 2 for sf(L) without justification, whereas the paper correctly derives oddness via two-sided bounds and parity. Therefore the paper’s argument is correct and complete for its claims, while the model’s solution contains critical mistakes.