On limit cycles of feedback systems which contain a hysteresis nonlinearity

In this paper, we concern ourselves with the stability of limit cycles in feedback systems which have hysteresis nonlinearities. Although the quasistatic analysis of limit cycles (Loeb criterion) predicts, in most cases correctly, the stability properties of limit cycles, it is well known that analyses which are based on the method of describing functions may lead to erroneous conclusions. In this paper, we show to what extent the describing function method can be given a rigorous mathematical basis. We show that for a specific example, the main result of this paper predicts correctly the stability of a limit cycle while the Loeb criterion yields an incorrect result. Also, we show that our analysis explains to a certain extent the presence of distortions in solutions of the class of feedback systems considered herein. In arriving at the main result of this paper, use is made of several known facts for functional differential equations and of a result on integral manifolds.