The periodic orbits of some classes of three-dimensional hybrid relay systems

Over the last few decades, the interest in piecewise linear differential systems has increased strongly, mainly due to their many applications in various fields, such as mechanical problems, electrical circuits, and especially control theory. We study the periodic orbits of three classes of hybrid relay systems defined in R3. Each system is composed of two smooth vector fields separated by a switching surface and connected through a discrete reset mechanism. We show that these systems are completely integrable by explicitly constructing two functionally independent first integrals for each smooth subsystem. As a result, we prove that each system admits a one-parameter family of periodic orbits, providing its initial conditions. Consequently, none of the systems possesses limit cycles, because periodic orbits are not isolated.