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The spectrum of delay differential equations with multiple hierarchical large delays

  • * Corresponding author: Stefan Ruschel

    * Corresponding author: Stefan Ruschel 

to A. Mielke on the occasion of his 60th birthday

The authors acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project 411803875 and SFB 910. The research was conducted while SR was doctoral student at Technische Universität Berlin

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  • 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} $.

    Mathematics Subject Classification: Primary: 34K06, 34K20; Secondary: 34K08.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Example of the numerically computed spectrum of eigenvalues for system (1) with $ n = 2 $, $ A_{0} = -0.4+0.5i $, $ A_{1} = 0.5 $, $ \tau_{1} = 100 $, and $ \tau_{2} = 10000 $. Panel (a): blue dots are numerically computed eigenvalues. Panel (b): zoom into panel (a)

    Figure 2.  Eigenvalues of the characteristic equation (26) corresponding to two hierarchical delays. Panels (a)-(f) show the destabilization of the spectrum varying parameter $ c $ (columns from left to right: $ c = 0.2 $ (stable), $ c = 0.3 $ (neutral), $ c = 0.4 $ (unstable)). Panels (a), (c), (e) show the spectrum (real part rescaled). Panels (b), (d), (f): approximation of the spectrum via the two-dimensional spectral manifold $ \gamma^{(2)} $ ($ S_{2}, $ colored surface). Other parameters are $ a = -0.4+0.5i $, $ b = 0.1 $, and $ \varepsilon = 0.01 $. $ S_{0}^{+} $ and $ S_{1}^{+} $ are not present. Blue dots are numerically computed eigenvalues

    Figure 3.  Eigenvalues of the characteristic equation (26) corresponding to two hierarchical delays. Two types of spectra coexisting: $ S_{1}^{+} $ (red) and $ S_{2} $ (blue). Panels (a)-(f) show the spectrum varying parameter $ \varepsilon $ (columns from left to right: $ \varepsilon = 0.01 $, $ \varepsilon = 0.003 $, $ \varepsilon = 0.003 $ (zoom)). Panels (a), (c), (e): approximation of the $ \tau_{1}- $spectrum (red) via spectral manifold $ \gamma^{(1)} $ (magenta dotted). Panels (b), (d), (f): approximation of the $ \tau_{2}- $spectrum (blue) via two-dimensional spectral manifolds $ \gamma^{(2)} $ (colored surface). Other parameters are $ a = -0.4+0.5i $, $ b = 0.5 $, and $ c = 0.3 $. $ S_{0}^{+} $ is not present. Blue dots are numerically computed eigenvalues

    Table 1.  Frequent notations

    Symbol Description Reference
    $\Sigma^\varepsilon$ Spectrum Eq. (5)
    $\Sigma_s^\varepsilon$ Strong spectrum Def. 2.3, Eq. (15)
    $\Sigma_c^\varepsilon$ Pseudo-continuous spectrum Def. 2.3, Eq. (16)
    $\tilde{\Sigma}_{k}^{\varepsilon}$ Truncated stable $\tau_k$-spectrum Def.2.1, Eq. (10)
    $\mathcal{A}_0$ Asymptotic strong spectrum Def. 2.3, Eq. (14)
    $S_{0}^{+}$ Asymptotic strong unstable spectrum Def. 2.3, Eq. (13)
    $\tilde{S}_{0}^{-}$ Asymptotic strong stable spectrum Def.2.1, Eq. (11)
    $\mathcal{A}_k$ Asymptotic continuous $\tau_k$-spectrum Def.2.4, Eq. (21)
    $S_{k}^{+}$ Asymptotic continuous stable $\tau_k$-spectrum Def.2.4, Eq. (19)
    $\tilde{S}_{k}^{-}$ Asymptotic continuous unstable $\tau_k$-spectrum Def.2.4, Eq. (20)
    $A_k$ Coefficient matrix corresponding to delay $\tau_k$ Eq. (1)
    $A_{j, 1}^{(k)}$ Projection of coefficient matrix $A_j$ to the cokernels of matrices $A_l$, $l=k, k+1, \ldots, n$ Eq. (9)
    $\chi^\varepsilon(\lambda)$ Characteristic function Eq. (6)
    $\tilde\chi^\varepsilon_k(\lambda)$ Projected characteristic equation, $0\leq k < n$ Def.2.1, Eq. (8)
    $\chi_k, \tilde{\chi}_k$ Truncated characteristic equation, $0\leq k < n$ Def. 2.4, Eqs. (17)–(18)
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    Table 2.  Summary of spectra and conditions for stability of Eq. (26)

    relevant asymptotic spectra parameters
    asymptotic strong unstable spectrum $S_{0}^{+}$ present (unstable) $\Re(a)>0$
    not present $\Re(a) < 0$
    asymptotic continuous spectrum $S_{1}^{+}$ present (unstable) $\left|b\right|>\left|\Re(a)\right|$
    not present $\left|b\right| < \left|\Re(a)\right|$
    singular points $\Re(a)=0$
    $S_{2}$ unstable $|c|>\left|\Re(a)\right|-|b|$
    stable $|c| < \left|\Re(a)\right|-|b|$
    singular points$\left|b\right|\geq\left|\Re(a)\right|$
     | Show Table
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