Stable dynamics in forced systems with sufficiently high/low driving frequency

We consider parametrically forced Hamiltonian systems with one-and-a-half degrees of freedom and study the stability of the dynamics when the frequency of the forcing is relatively high or low. We show that, provided the frequency is sufficiently high, Kolmogorov-Arnold-Moser (KAM) theorem may be applied even when the forcing amplitude is far away from the perturbation regime. A similar result is obtained for sufficiently low frequency, but in that case we need the amplitude of the forcing to be not too large; however, we are still able to consider amplitudes which are outside of the perturbation regime. In addition, we find numerically that the dynamics may be stable even when the forcing amplitude is very large, well beyond the range of validity of the analytical results, provided the frequency of the forcing is taken correspondingly low.