Устойчивость двупараметрических систем линейных автономных дифференциальных уравнений с ограниченным запаздыванием

Abstract

We consider a system of linear autonomous differential equations with bounded delay in the case where its characteristic function depends linearly on two scalar parameters. The development of the D-subdivision method is carried out in connection with the problem of constructing the stability domain of this system. Firstly, a complete classification of the points and lines of D-subdivision is carried out. Secondly, a complete classification of two-parameter characteristic equations by the type and structure of D-subdivision domains is carried out. All equations are divided into four types: D-subdivision domains of equations of the first type have curvilinear boundaries, D-subdivision domains of equations of the second and the third type have only rectilinear boundaries, equations of the fourth type are stable or unstable regardless of parameter values. Thirdly, for each type of equations, new methods of selecting the stability domain among regions of D-subdivision are developed. On the basis of the results obtained, stability domains are constructed for certain differential equations and systems of equations with concentrated and distributed delay.