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arXiv · 2305.13457

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

Douglas D. Novaes·Luan V. M. F. Silva
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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, $(t,x,\dot{x})\in \mathbb{S}^1\times\mathbb{R}^2$.

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Douglas D. Novaes, Luan V. M. F. Silva. 2023-05-22. Invariant tori and boundedness of solutions of non-smooth oscillators with Lebesgue integrable forcing term. https://doi.org/10.1007/s00033-023-02152-0

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