Similarity and structural stability with respect to delay of FDE phase portraits (Q1980775)

The relationship is considered between the linear system of differential equations with a small delay \(\tau >0\), \[ x'(t) = Ax(t)+A_{\tau}x(t-\tau),\;\; t\in [t_0, \theta], \] where \(A\) and \(A_{\tau}\) are \(n\times n\) constant matrices, and the system \(x'(t) = Ax(t)\). First, the similarity of phase portraits of the two systems is defined as any one solution of the delayed system on a finite interval can be approximated by a solution of the system without delay arbitrarily close. Then the structural stability of the delayed system on \([t_0, \theta]\) with respect to the delay is defined as \(\forall \epsilon >0\), \(\exists \delta>0\) such that \(\|A_{\tau}\| <\delta\) and \(\tau <\delta\) imply that the solutions of the two systems satisfy \(\|\tilde{x}(t)-x^*(t)\| <\epsilon\) for \(t\in [t_0, \theta]\) if \(\tilde{x}(t_0)= x^*(t_0)=x^0\). Then it follows from the definitions that structural stability of the delayed system with respect to delay implies the similarity of phase portraits of the two systems. But the main result is proved that the stability of \(x'(t) = Ax(t)\) implies the structural stability of the delayed systems with respect to delay. This seems a good piece of work though the result is natural and not surprising. For the entire collection see [Zbl 1467.34001].