Morse theory for asymptotically linear Hamiltonian systems

Let H∈C2(R2N×R/TZ), the T-periodic Hamiltonian system dz/dt = J▽H(z(t),t) (equation 0.1) is considered. A linear T-periodic Hamiltonian system is said to be T-non-resonant if it has no T-periodic solutions, except from the equilibrium point 0. The system (0.1) is said to be T-non-resonant at infinity if the linear system dw/dt = JA∞(t)w(t) (equation 0.2) is T-non-resonant. A T-periodic solution z of Eq. (0.1) is said to be T-non-resonant if the linearized system near z is dw/dt = JD2H(z(t),t)w(t) is T-non-resonant.