Oscillations in a second-order discontinuous system with delay. (Q1865961)

The authors consider the equation \(\alpha x''(t)=-x'(t)+ F(x(t),t)-\text{sign\,}x(t-h)\), where \(F\) is a smooth function and \(\alpha, h\) are positive constants. They study the dynamics of oscillations with emphasis on the existence, frequency and stability of periodic oscillations. The main conclusion of the paper 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. Also, the uniqueness of a periodic slowly-oscillating solution is shown under extra conditions, and a criterion for the existence of bounded oscillations in the case of unbounded function \(F(x,t)\) is given.