Stability of three-parameter systems of two linear differential equations with delay. I (Q2285209)

The manuscript considers the stability problem of the second-order linear delay-differential equation of the form \[ \dot x(t)=A x(t) + B x(t-h) \] with delay \(h\), \(x\in \mathbb{R}^2\) and \(A,B\) are constant \(2\times 2\) matrices. The characteristic equation for this problem can be written in the form \[ a_1 + a_2 e^{-zh} + a_3 e^{-2zh} + a_4 z + a_5 z e^{-zh} + z^2 = 0 \] with five parameters \(a_i\), \(i=1,\dots,5\). The main result of this manuscript is to give conditions for the stability of 3-parameter subproblems. A classification of all possible 3-parameter problems is given. Such a problem is, for example, \[ \alpha\, \text{sh}\, z + \beta ze^z + \gamma z + z^2 e^z = 0 \] with three parameters \(\alpha\), \(\beta\), \(\gamma\). The main tool used is the \( D \)-subdivision method.