LAUSR

Abstract Rapidly Oscillating Systems (ROS) is a class of perturbed dynamical systems with a small parameter that contains oscillating part with the period proportional to the parameter. These include a lot of classical perturbed systems that found wide applications in celestial mechanics, electronics, mechatronics, nanotechnology, etc. However, many important systems of the class cannot be treated classically. In recent years these non-classical ROS found important applications in control of bipedal walk, rapidly varying media, frequency demodulation, Atomic Force Microscopy, etc. A characteristic specific of the non-classical ROS is that oscillations depend not only on time, but also on phase variables. This allows to manipulate the oscillation functions and to use them for asymptotic control. The purpose of this paper is to illustrate non-classical ROS methods and their properties on simple, but representative, 2D examples without technical details of real applications. In some cases these also provide a new technique to the classical systems.