The Spectrum of Delay Differential Equations with Multiple Hierarchical Large Delays

Abstract

We prove that the spectrum of the linear delay differential equation $x^{\prime }(t)=A_{0}x(t)+A_{1}x(t-{\tau }_1)+\dots +A_{n}x(t-{\tau }_n)$ with multiple hierarchical large delays $1\ll {\tau }_1\ll {\tau }_2\ll \dots \ll {\tau }_n$ 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 $A_0$, 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 ${\tau }_1,{\tau }_2,\dots ,{\tau }_n.$ 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 $n$-dimensional spectral manifold corresponding to the timescale ${\tau }_n$.

21 pages, 3 figures