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

TL;DR

This paper proves the boundedness of solutions for a class of non-smooth oscillators with Lebesgue integrable forcing, using invariant tori to cover the phase space, extending classical results to non-smooth cases.

Contribution

It provides a simple proof of boundedness for non-smooth forced oscillators with Lebesgue integrable forcing, constructing invariant tori to cover the entire phase space.

Findings

Solutions are bounded for the non-smooth oscillator with Lebesgue integrable forcing.

Invariant tori can be constructed to cover the entire phase space.

The method extends classical boundedness results to non-smooth systems.

Abstract

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, $(t,x,\dot{x})\in \mathbb{S}^1\times\mathbb{R}^2$.