A previous exact formal and practical solution for the time-dependent Schrödinger equation for a two-level non-degenerate system in an arbitrarily strong sinusoidal field is transformed for an N-level system to a Floquet form. Using the Floquet form of the solution computationally convenient expressions are obtained for the phase and the phase-long time-averaged induced transition probabilities. The method exploits the time periodicity of the Schrödinger equation and the solution over the initial period of the hamiltonian provides all the information needed to obtain both the unaveraged and the averaged transition probabilities for the system. The method is illustrated by specific examples which are used to discuss some of the temporal features of the multi-photon spectra of two-level systems.