Oscillations in a Second-order Discontinuous System with Delay

We consider the equation x00(t) = x0(t) + F (x(t); t) signx(t h); = 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 jF (x)j < 1, there are periodic solutions with di erent 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. Introduction. Equations with discontinuity often appear in various control theory models [2, 4, 5, 10, 11, 15, 18]. Time delay, always existing in real systems, usually results in oscillations around the discontinuity surface. In the present paper we study various aspects of such oscillations for the system x(t) = x(t) + F (x(t); t) signx(t h) ; (0.1) where F is a smooth function and , h are positive constants (the positivity of means that signx(t h) provides a negative feedback). An important point is a 1991 Mathematics Subject Classi cation. 34K11, 34K13, 34K35.