Dynamical behaviors for generalized pendulum type equations withp-Laplacian

We consider a pendulum type equation withp-Laplacian (phi(p)(x '))' +G '(x)(t, x) = p(t), where phi(p)(u) = vertical bar u vertical bar(p-2)u, p >1,G(t, x) andp(t) are 1-periodic about every variable. The solutions of this equation present two interesting behaviors. On the one hand, by applying Moser's twist theorem, we find infinitely many invariant tori whenever integral(1)(0)p(t)dt = 0, which yields the boundedness of all solutions and the existence of quasi-periodic solutions starting at t = 0 on the invariant tori. On the other hand, if p(t) = 0 and G '(x)(t, x) has some specific forms, we find a full symbolic dynamical system made by solutions which oscillate between any two different trivial solutions of the equation. Such chaotic solutions stay close to the trivial solutions in some fixed intervals, according to any prescribed coin-tossing sequence.