Asymptotically stable almost-periodic oscillations in systems with hysteresis nonlinearities

We present some sufficient conditions for the asymptotic stability of forced almost-periodic oscillations in nonlinear systems subject to small hysteresis perturbations. The main technical restriction on hysteresis nonlinearity comes to a contraction-type property, which holds for some classical models of hysteresis. Also we require a special stability property of the unperturbed system in the sense of Lyapunov and the bounded input - bounded output.