Existence of quasi-periodic solutions of the real pendulum equation

The pendulum equation x = -delta x -delta x (1 + f(o) cos omega(1)t) sinx +f(1), sin omega(2)t is considered in this paper, where f(o),f(1) and delta are small real parameters, the ratio of oh and oh is irrational, and frequencies omega(1), and omega(2) satisfy the Diophantine condition. The unperturbed system (f(o) = f(1) = delta = 0) has several fixed points for different parameter alpha. We use KAM theory to prove that the perturbed system possesses quasi-periodic solutions in neighborhoods of those fixed points. (c) 2014 Elsevier Ltd. All rights reserved.