Bifurcation of cycles of automatic control systems with ideal relay

The problem of existence and stability of the relay system cycles is discussed in numerous application-oriented publications (see, for example, the summarizing monograph [1]). However, there are only a few mathematically correct assertions and, in particular, the local methods of study which make an important part of our ideas of dynamics were not developed at all. The present paper partially filled this gap by introducing for the first time the cycle monodromy matrix, proving an analog of the Andronov–Hopf bifurcation theorem, and demonstrating existence of a countable number of stable cycles in the problem of temperature control.