On the conditions for cycles existence in a second-order discrete-time system with sector-nonlinearity

In this paper, a second-order discrete-time automatic control system is studied. This work is a continuation of the research presented in the author’s papers “On the Aizerman problem: coefficient conditions for the existence of a four-period cycle in a second-order discrete-time system” and “On the Aizerman problem: coefficient conditions for the existence of threeand six-period cycles in a second-order discrete-time system”, where systems with two- and three-periodic nonlinearities lying in the Hurwitz angle were considered. The systems with nonlinearities subjected to stronger constraints are discussed in this paper. It is assumed that the nonlinearity not only lies in the Hurwitz angle, but also satisfies the additional sector-condition. This formulation of the problem is found in many works devoted to theoretical and applied questions of the automatic control theory. In this paper, a system with such nonlinearity is explored for all possible values of the parameters. It is shown that in this case there are parameter values for which a system with a two-periodic nonlinearity has a family of four-period cycles, and a system with a three-periodic nonlinearity has a family of three- or six-period cycles. The conditions on the parameters under which the system can have a family of periodic solutions are written out explicitly. The proofs of the theorems provide a method for constructing nonlinearity in such a way that any solution of the system with initial data lying on some definite ray is periodic.

Stability of solutions and the problem of Aizerman for sixth-order differential equations

This article is devoted to the investigation of stability of equilibrium of ordinary differential equations using the method of semi-definite Lyapunov’s functions. Types of scalar nonlinear sixth-order differential equations for which regular constant auxiliary functions are used are emphasized. Sufficient conditions of global asymptotic stability and instability of the zero solution have been obtained and it has been established that the Aizerman problem has a positive solution concerning the roots of the corresponding linear differential equation. Studies highlight the advantages of using semi-definite functions compared to definitely positive Lyapunov’s functions.