Response solutions for a class of degenerate oscillators under quasi-periodic perturbations with a small parameter

In this paper we consider a quasi-periodically forced degenerate oscillator equation \(\ddot{x}+x^{3}=\epsilon g(\epsilon ; \omega t, x,\dot{x})\) $\ddot{x}+x^3=ϵg(ϵ;\omega t,x,\dot{x})$ , where \(\epsilon \) $ϵ$ is a small parameter and \(\omega \) $\omega$ is a diophantine frequency. We prove that there exists a sufficiently small \(\epsilon _0>0\) ${ϵ}_0>0$ and a non-empty subset \(E\subset (0, \epsilon _0)\) $E\subset (0,{ϵ}_0)$ such that for \(\epsilon \in E\) $ϵ\in E$ , the equation has a response solution, moreover, E has a positive Lebesgue measure with 0 as a Lebesgue dense point. The proof is based on the existence of a piecewise differentiable path in the set of all the real roots for an approximating 3-degree real polynomial with a parameter.