Since Littlewood works in the 1960's, the boundedness of solutions of Duffing-type equations $\ddot{ x}+g(x)=p(t)$ has been extensively investigated. More recently, some researches have focused on the family of non-smooth forced oscillators $ \ddot{ x}+\text{ sgn}(x)=p(t)$, mainly because it represents a simple limit scenario of Duffing-type equations for when $g$ is bounded. Here, we provide a simple proof for the boundedness of solutions of the non-smooth forced oscillator in the case that the forcing term $p(t)$ is a $T$-periodic Lebesgue integrable function with vanishing average. We reach this result by constructing a sequence of invariant tori whose union of their interiors covers all the $(t,x,\dot x)$-space, $(t,x,\dot{ x})\in \mathbb{ S}^1\times\mathbb{ R}^2$.
Categories: math.DS