Morse Theory for Forced Oscillations of Hamiltonian Systems on n ×

Considering a large class of periodically time dependent Hamiltonian systems on the cotangent bundle T*( n) of the n-dimensional torus n, we prove the existence of at least 2n forced oscillations in every homotopy class of loops, provided these periodic solutions are non-degenerate. Moreover, given α n such that all the periodic solutions having rotation vector α are non-degenerate, there exist at least 2n of them. We assume the Hamiltonian system to be asymptotically linear in the fibres, and the norm of the Hessian is required to have at most polynomial growth.