Using the point mapping method, we obtained analytical expressions for the first return functions for determining simple and complex attractors in the stabilization mode by a general-purpose relay controller with the linear formation of the control signal. We investigated self-oscillations with account for the operating members’ aftereffect, the dead zone of the speed sensor, and the time-independent perturbation action. The study shows that the dead zone of the speed sensor introduces significant changes in the behavior of the system, giving it new properties. The analysis of dynamic processes on a three-sheet phase surface revealed a wide variety of limit cycles and their dependence on the system’s parameters. Complex limit cycles are represented by combining simple cycles of two types, which allowed for a simplifying approach to their search based on the theory of multidimensional transformations of Yu.I. Neymark. A more complete result was obtained in comparison with the well-known literary sources.