The spectrum of delay differential equations with multiple hierarchical large delays

Abstract

<p style='text-indent:20px;'>We prove that the spectrum of the linear delay differential equation <inline-formula><tex-math id="M1">\begin{document}$ x'(t) = A_{0}x(t)+A_{1}x(t-\tau_{1})+\ldots+A_{n}x(t-\tau_{n}) $\end{document}</tex-math></inline-formula> with multiple hierarchical large delays <inline-formula><tex-math id="M2">\begin{document}$ 1\ll\tau_{1}\ll\tau_{2}\ll\ldots\ll\tau_{n} $\end{document}</tex-math></inline-formula> splits into two distinct parts: the strong spectrum and the pseudo-continuous spectrum. As the delays tend to infinity, the strong spectrum converges to specific eigenvalues of <inline-formula><tex-math id="M3">\begin{document}$ A_{0} $\end{document}</tex-math></inline-formula>, the so-called asymptotic strong spectrum. Eigenvalues in the pseudo-continuous spectrum however, converge to the imaginary axis. We show that after rescaling, the pseudo-continuous spectrum exhibits a hierarchical structure corresponding to the time-scales <inline-formula><tex-math id="M4">\begin{document}$ \tau_{1}, \tau_{2}, \ldots, \tau_{n}. $\end{document}</tex-math></inline-formula> Each level of this hierarchy is approximated by spectral manifolds that can be easily computed. The set of spectral manifolds comprises the so-called asymptotic continuous spectrum. It is shown that the position of the asymptotic strong spectrum and asymptotic continuous spectrum with respect to the imaginary axis completely determines stability. In particular, a generic destabilization is mediated by the crossing of an <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional spectral manifold corresponding to the timescale <inline-formula><tex-math id="M6">\begin{document}$ \tau_{n} $\end{document}</tex-math></inline-formula>.