Stable periodic solutions for two-dimensional linear delay differential equations (Q1353723)

The author considers the equation \[ x'(t)= Ax(t- 1),\tag{1} \] where \(A\) is one of the following three matrices in Jordan normal form: \[ (i)-\alpha\begin{pmatrix} \cos\theta &-\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix},\;(ii)-\begin{pmatrix} a_1 & 0\\ 0 & a_2\end{pmatrix},\;(iii)-\begin{pmatrix} a & 1\\ 0 & a\end{pmatrix}.\tag{2} \] Here \(\alpha\), \(\theta\) \((0<|\theta|< \pi)\), \(a_1\), \(a_2\) and \(a\) are real numbers. Recently, \textit{T. Hara} and \textit{J. Sugie} [Funkc. Ekvacioj, Ser. Int. 39, No. 1, 69-86 (1996; Zbl 0860.34045)] have shown the following results: Theorem 1.1. Assume that \[ A= -\alpha\begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix},\quad \alpha>0,\;-\pi<\theta\leq\pi. \] Then the zero solution of (1) is asymptotic stable if and only if \[ 0<\alpha<{\pi\over 2}-|\theta|. \] Theorem 1.2. Assume that \[ A= -\begin{pmatrix} a_1 & b\\ 0 & a_2\end{pmatrix},\quad a_1,\;a_2,\;b\in\mathbb{R}. \] Then the zero solution of (1) is asymptotic stable if and only if \[ 0< a_1<{\pi\over 2},\quad 0<a_2<{\pi\over 2}. \] The purpose of this paper is to deal with the critical cases of the above results.