Abstract:

In this paper we study the boundedness and unboundedness of the solutions of the smooth and discontinuous (SD) oscillator

\begin{equation*}

x''+f(x)x'+x-\frac{x}{\sqrt{x^{2}+\alpha^{2}}}=p(t).

\end{equation*}

Since $f(x)\neq 0$, the system is non-Hamiltonian, so we have to introduce some reversibility assumptions to apply a suitable twist theorem, for reversible maps with small twist. Moreover, when the nonnegative parameter $\alpha$ decreases to 0, the system becomes discontinuous. In this case, we need to introduce some suitable transformations to overcome the lack of regularity. We will prove that for any nonnegative parameter $\alpha$, when $p(t)$ is an odd periodic function satisfying $\left|\int^{2\pi}_{0}p(t)\sin t\,\dif\,t\right|<4$, all the solutions are bounded; when $p(t)$ satisfies $\left|\int^{2\pi}_{0}p(t)\sin t\,\dif\,t\right|>4$, the SD oscillator has unbounded solutions, and when $p(t)$ satisfies $\left|\int^{2\pi}_{0}p(t)\sin t\,\dif\,t\right|\geqslant 4+|F|_{\infty}$, all the solutions are unbounded.

Key words: SD oscillator, Reversible system, Bounded solution, Unbounded solution