We consider in this paper an automatic control second-order discrete-time system whose nonlinearity satisfies the generalized Routh–Hurwitz conditions. Systems of this type are widely used in solving modern application problems that arise in engineering, theory of motion control, mechanics, physics, and robotics. Two constructed examples of discrete-time systems with nonlinearities that lie in a Hurwitz angle were presented in recent papers by W. Heath, J. Carrasco, and M. de la Sen. These examples demonstrate that in the discrete case, the Aizerman and Kalman conjectures are untrue even for second-order systems. One such system in these examples has a three-period cycle and the other system, a four-period cycle. We assume in the present paper that the nonlinearity is two-periodic and lies in a Hurwitz angle; here, we study a system for all possible parameter values. We explicitly present the conditions (for the parameters) under which it is possible to construct a two-periodic nonlinearity in such a way that a system with it is not globally asymptotically stable. Such a nonlinearity can be constructed in more than one way. We propose a method for constructing the nonlinearity in such a way that a family of four-period cycles is found in the system. The cycles are nonisolated; any solution of the system with the initial data, which lies on a certain specified ray, is a periodic solution.