Investigation of the stability of one difference equation with complex coefficients

Abstract

We study the stability of a linear autonomous difference equation with two (generally speaking, complex) coefficients. The starting point of the study is the Schur-Kohn theorem on the location of the roots of the characteristic equation with respect to the unit disk in the complex plane. To construct the domain of exponential stability in the parameter space, we use the D decompodition method, which consists in constructing curves (or surfaces) such that the number of roots of the characteristic equation outside the unit disk changes when passing through the curves; then the area is determined, which corresponds to the zero number of such roots; this is the area of stability. We implement this scheme for the above difference equation: geometric stability criteria are found and the domains of exponential stability in a four-dimensional space of coefficients are described, as well as their three-dimensional, two-dimensional and one-dimensional sections. The Lyapunov stability is studied separately, which is corresponded by the domain of exponential stability supplemented by a part of its boundary. To describe Lyapunov stability exactly we use a "multiplicity curve", which is a line such that all its points correspond to multiple roots of the characteristic equation. In addition, we find and construct a domain of absolute stability with respect to one of the parameters of the initial equation. For this domain, we formulate criteria of exponential stability and Lyapunov stability.The results obtained can be applied to the study of processes in physics, technology, economics, biology, which are modeled using discrete models in the form of difference equations.