Fixed Points of a Mapping Generated by a System of Ordinary Differential Equations with Relay Hysteresis

We consider an n-dimensional system of ordinary differential equations with relay hysteresis on the right-hand side. Under certain conditions, the solution of the system defines a self-mapping of bounded sets lying in the discontinuity surfaces. We obtain conditions for the existence of fixed points of the mapping and the uniqueness of the fixed point as well as conditions under which fixed points for various types of mappings exist simultaneously. To each type of mapping there corresponds one type of periodic orbits: either with two switching points (the so-called unimodal orbits) or with an even number of switching points greater than two. In the case of unimodal orbits, examples of the existence of orbits of various configurations are given.