Exponential Dichotomy and Rotation Number for Linear Hamiltonian Systems

Let x′ = H(t)x be a non-autonomous Hamiltonian linear differential system. We show that if the rotation number α(λ) for the associated family of systems x′ = (H(t) + λJγ(t))x (where J = (0In−In0) and γ(t) = γ*(t) ≥ 0 for all t) is constant on an interval containing λ = 0, then x′ = H(t)x has an exponential dichotomy.