arXiv 1902.004042019-02math.DS

具有多层大时滞的时滞微分方程的谱

The Spectrum of Delay Differential Equations with Multiple Hierarchical Large Delays

Stefan Ruschel, Serhiy Yanchuk

备注：: 21 pages, 3 figures

摘要：

证明了具有多阶大时滞的线性时滞微分方程$x'(t)=A_{0}x(t)+A_{1}x(t-τ_{1})+\ldots+A_{n}x(t-τ_{n})$的谱$1\llτ_{1}\llτ_{2}\ll\ldots\llτ_{n}$分裂为强谱和伪连续谱.当时滞趋于无穷大时，强谱收敛到$A_{0}$的特定特征值，即所谓的渐近强谱。然而，伪连续谱中的本征值收敛到虚轴。我们表明，重新标度后，伪连续谱表现出一个层次结构对应的时间尺度$τ_{1},τ_{2},\ldots,τ_{n}.$，这个层次的每一个层次是近似的频谱流形，可以很容易地计算。谱流形的集合包括所谓的渐近连续谱。证明了渐近强谱和渐近连续谱关于虚轴的位置完全决定稳定性。特别地，一般的不稳定是由对应于时间尺度$τ_{n}$的$n$维谱流形的交叉介导的。

英文摘要：

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

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