Quasi-periodic solutions for an asymmetric oscillation

In this paper we study the dynamical behaviour of the differential equation x''+ax+−bx−=f(t), where , , a and b are two different positive constants, and f (t) is a smooth quasi-periodic function. For this purpose, firstly, we have to establish some variants of the invariant curve theorem of planar smooth quasi-periodic mappings, which was proved recently by the authors (see Huang et al preprint). Then we will discuss the boundedness of all solutions and the existence of quasi-periodic solutions for the above asymmetric oscillation.