Побудова областей стійкості лінійних автономних диференціальних рівнянь із запізненням

Abstract

The aim of the present article is to investigate of solutions stability of linear autonomous differential equations with retarded argument. The investigation of stability can be reduced to the root location problem for the characteristic equation. For the linear differential equation with several delays it is obtained the necessary and sufficient conditions, for all the roots of the characteristic equation to have negative real part (and hence the zero solution to be asymptotically stable). For the scalar delay differential equation stability domains in the parameter plane are obtained. We investigate the boundedness conditions and construct a domain of stability for linear autonomous differential equation with several delays. We use D-partition method, argument principle and numerical methods to construct of stability domains. In this article, we investigate the solutions stability of linear autonomous differential equations with several delays. It is obtained the necessary and sufficient conditions, for all the roots of the characteristic equation to have negative real part. We investigate the boundedness conditions with the help of argument principle and construct a domain of stability for linear autonomous differential equation with two delays. We use D-partition method, argument principle and numerical methods to construct of stability domains for linear autonomous differential equation with two delays. In the D-partition method, we look for parameter values for which the characteristic equation has at least one zero on the imaginary axis. We consider some examples of equations with two delays. In these cases, the stability domains are limited by two straight lines and a finite number of parametrically defined curves.