Oscillations in a second-order discontinuous system with delay

Abstract

We consider the equation αx″(t) = -x′(t) + F(x(t), t) - sign x(t - h), α = 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) = 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.