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Oscillations in a secondorder discontinuous system with delay
1.  School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel 
2.  Department of Electrical Engineering & Systems, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel 
3.  Division of Postgraduate and Investigation, Chihuahua Institute of Technology, Chihuahua, Chi, C.P. 31160, Mexico 
$\alpha x''(t)=x'(t)+F(x(t),t)$sign$x(th),\quad\alpha=$const$>0,\ $ $h=$const$>0,$
which is a model for a scalar system with a discontinuous negative delayed feedback, and study the dynamics of oscillations with emphasis on the existence, frequency and stability of periodic oscillations. Our main conclusion is that, in the autonomous case $F(x,t)\equiv F(x)$, for $F(x)<1$, there are periodic solutions with different frequencies of oscillations, though only slowlyoscillating solutions are (orbitally) stable. Under extra conditions we show the uniqueness of a periodic slowlyoscillating solution. We also give a criterion for the existence of bounded oscillations in the case of unbounded function $F(x,t)$. Our approach consists basically in reducing the problem to the study of dynamics of some discrete scalar system.
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