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