A Review of Research on the Qualitative Theory of Differential Equations at St. Petersburg University. III. Systems with Hysteresis Nonlinearities: The Aizerman Problem for Discrete-Time Systems

This paper is the third in a series of publications devoted to an overview of the scientific research results achieved by the employees of the Department of Differential Equations at St. Petersburg University over the past 3 decades. The paper is focused on the results achieved by studying systems with hysteretic and sector nonlinearities, with both continuous and discrete time. The first part of this research presents the results obtained for second-order automatic control systems with continuous time. The global stability and existence of limit cycles in a system with one hysteretic nonlinearity are investigated in this part. The second part considers a discrete-time system that consists of a linear scalar equation and a one-dimensional stop operator. This system can also be presented as a two-dimensional piecewise linear mapping. An analysis of global dynamics and bifurcations in the system depending on two parameters is presented. One-dimensional mappings arising when considering the Poincare map are studied. In particular, the dynamics of the so-called “skew tent map” are fully analyzed. In the third part of the paper, discrete second-order systems with nonlinearities under to the generalized Routh–Hurwitz conditions are considered (Aizerman problem). It is shown that a 2-periodic nonlinearity of the indicated type can be constructed in such a way that cycles of period four arise in the system. A 3-periodic nonlinearity can be constructed in such a way that cycles of period three or cycles of period six appear in the system.