The spectrum of delay differential equations with multiple hierarchical large delays

We prove that the spectrum of the linear delay differential equation x0(t) = A0x(t)+A1x(tô1)+: : :+Anx(tôn) with multiple hierarchical large delays 1-1-2: : :-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 speciffic eigenvalues of A0, 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-scales1;2 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 timescalefin.