Invariant tori and boundedness of solutions of non-smooth oscillators with Lebesgue-integrable forcing term.

Since Littlewood works in the 1960s, the boundedness of solutions of Duffing-type equations x ¨ + g (x) = p (t) has been extensively investigated. More recently, some researches have focused on the family of non-smooth forced oscillators x ¨ + sign (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 , x ˙) -space, (t , x , x ˙) ∈ S 1 × R 2 . [ABSTRACT FROM AUTHOR]