March  2003, 9(2): 339-358. doi: 10.3934/dcds.2003.9.339

Oscillations in a second-order discontinuous system with delay


School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel


Department of Electrical Engineering & Systems, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel


Division of Postgraduate and Investigation, Chihuahua Institute of Technology, Chihuahua, Chi, C.P. 31160, Mexico

Received  March 2001 Revised  February 2002 Published  December 2002

We consider the equation

$\alpha x''(t)=-x'(t)+F(x(t),t)-$sign$x(t-h),\quad\alpha=$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 slowly-oscillating solutions are (orbitally) stable. Under extra conditions we show the uniqueness of a periodic slowly-oscillating 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.

Citation: Eugenii Shustin, Emilia Fridman, Leonid Fridman. Oscillations in a second-order discontinuous system with delay. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 339-358. doi: 10.3934/dcds.2003.9.339

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