Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8947668 | International Journal of Non-Linear Mechanics | 2018 | 15 Pages |
Abstract
First, we have studied a specific discontinuous differential equation for a smooth and discontinuous (SD) oscillator xÌ+x(1â1âa2+x2)=p(t), where p(t) is a given 2Ï-periodic forcing function and a is a real parameter. When the forcing p(t)âC7(Râ2ÏZ) and â«02Ïp(t)eâitdt<4, all solutions of the oscillator are shown to be bounded, i.e., lim|t|â+âx(t)2+xÌ(t)2<+â.Moreover, the oscillator has at least one harmonic solution. When the forcing p(t)âC0(Râ2ÏZ) andâ«02Ïp(t)eâitdtâ¥4, all solutions of the oscillator are unbounded, i.e., lim|t|â+âx(t)2+xÌ(t)2=+â.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.
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Mechanical Engineering
Authors
Hebai Chen, Jinqiao Duan,