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In this paper, the existence of Aubry-Mather sets and quasi-periodic solutions of the oscillator \(x''+\alpha{x^{+}}^{3}-\beta{x^{-}}^{3}+f(x)q(x')+\psi(x)=p(t) \) are established, where f, q, and ψ belong to the class \(C^{1}(R) \), p is a continuous 2π-periodic function. Under some assumptions on the parities of f, ψ, and p, we prove that there are infinitely many generalized quasi-periodic solutions by a result of Shuinee Chow and Mingliang Pei from the Aubry-Mather theorem of reversible mappings.

Quasi-periodic solutions for second order differential equation with superlinear asymmetric nonlinearities and nonlinear damping term