In this paper, a second-order discrete-time automatic control system is studied. It is assumed that the nonlinearity of this system is 2-periodic and satisfies the generalized Routh-Hurwitz condition. Systems with nonlinearities of this type are used in solving various applied natural science and technical problems. This system is explored for all allowed values of the parameters. Conditions on the parameters under which it is possible to construct, the nonlinearity so that the system is not globally asymptotically stable are provided, hi this case the system has a family of non-isolated cycles of period four.