On the problem of Aizerman: coefficient conditions for an existence of four-period cycle in a second-order discrete-time system

In this paper, an automatic control discrete-time system of the second order is studied. Nonlinearity of this system satisfies the generalized Routh - Hurwitz conditions. Systems of this type are widely used in solving modern applied problems that arise in engineering, theory of motion control, mechanics, physics and robotics. In the recent papers published by W. Heath, J. Carrasco, and M. de la Sen, two examples of planar discrete-time systems with nonlinearity that lies in the Hurwitz angle are constructed. These examples demonstrate that discrete-time Aizerman and Kalman conjectures are false even for second-order systems. One of the systems constructed by the authors has a non-trivial periodic solution of period three, and the other one has a non-trivial periodic solution of period four. In this paper, we assume that the nonlinearity is two-periodic and lies in the Hurwitz angle, and we study the system for all possible values of the parameters. We explicitly indicate the conditions for the parameters under which it is possible to construct such a two-periodic nonlinearity that system is not globally asymptotically stable. Indicated nonlinearity can be constructed in more than one way. We provide a method for its construction. We prove that in a system with this nonlinearity a family of non-trivial periodic solutions of period four exists. Cycles are not isolated, any solution of the system with the initial data lying on some specified ray is periodic.