Bounded and unbounded solutions of a discontinuous oscillator at resonance

First, we have studied a specific discontinuous differential equation for a smooth and discontinuous (SD) oscillator ẍ+x(1−1∕a2+x2)=p(t), where p(t) is a given 2π-periodic forcing function and a is a real parameter. When the forcing p(t)∈C7(R∕2πZ) and ∫02πp(t)e−itdt<4, all solutions of the oscillator are shown to be bounded, i.e., lim|t|→+∞x(t)2+ẋ(t)2<+∞.Moreover, the oscillator has at least one harmonic solution. When the forcing p(t)∈C0(R∕2πZ) and∫02πp(t)e−itdt≥4, all solutions of the oscillator are unbounded, i.e., lim|t|→+∞x(t)2+ẋ(t)2=+∞.The two conditions are sharp because they complement each other. Moreover, a bifurcation that connects the bounded solutions with unbounded ones occurs, and 4 is a bifurcation value. Inspired by this special discontinuous oscillator, we have found the conditions for the existence of bounded and unbounded solutions for a more general discontinuous oscillator. Finally, we present interesting physical models to illustrate our results.