We prove that the spectrum of the linear delay differential equation $ x'(t) = A_{0}x(t)+A_{1}x(t-\tau_{1})+\ldots+A_{n}x(t-\tau_{n}) $ with multiple hierarchical large delays $ 1\ll\tau_{1}\ll\tau_{2}\ll\ldots\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}, \ldots, \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} $.

中文翻译：

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具有多个分层大时滞的时滞微分方程的谱

我们证明线性延迟微分方程$ x'（t）= A_ {0} x（t）+ A_ {1} x（t- \ tau_ {1}）+ \ ldots + A_ {n} x的频谱（t- \ tau_ {n}）$具有多个较大的延迟$ 1 \ ll \ tau_ {1} \ ll \ tau_ {2} \ ll \ ldots \ ll \ tau_ {n} $分为两个不同的部分：强光谱和伪连续光谱。当延迟趋于无穷大时，强光谱会收敛到特定特征值A_ {0} $，即所谓的渐近强光谱。然而，伪连续谱中的特征值收敛于虚轴。我们表明，在重新缩放后，伪连续频谱展现出与时间标度$ \ tau_ {1}，\ tau_ {2}，\ ldots，\ tau_ {n}相对应的层次结构。$此层次结构的每个级别都可以通过易于计算的频谱流形进行近似。光谱流形的集合包括所谓的渐近连续光谱。结果表明，渐近强光谱和渐近连续光谱相对于虚轴的位置完全决定了稳定性。特别地，一般的不稳定是通过与时间尺度$ \ tau_ {n} $相对应的$ n维维谱流形的交叉来进行的。