Existence of quasi-periodic solutions of the real pendulum equation

The pendulum equation x¨=-αx-δẋ-(1+f0cosω1t)sinx+f1sinω2t is considered in this paper, where f0,f1 and δ are small real parameters, the ratio of ω1 and ω2 is irrational, and frequencies ω1 and ω2 satisfy the Diophantine condition. The unperturbed system (f0=f1=δ=0) has several fixed points for different parameter α. We use KAM theory to prove that the perturbed system possesses quasi-periodic solutions in neighborhoods of those fixed points.