The spectrum of delay differential equations with multiple hierarchical large delays

Stefan Ruschel; Serhiy Yanchuk

doi:10.3934/dcdss.2020321

Article Contents

2021, Volume 14, Issue 1: 151-175. Doi: 10.3934/dcdss.2020321

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The spectrum of delay differential equations with multiple hierarchical large delays

Stefan Ruschel^1, , and
Serhiy Yanchuk^2,

1.
Department of Mathematics, University of Auckland, , Auckland 1142, New Zealand
2.
Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany

^* Corresponding author: Stefan Ruschel
^* Corresponding author: Stefan Ruschel

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

Received: January 31, 2019

Revised: September 30, 2019

Early access: April 2020

Published: January 2021

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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Abstract

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} $.

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

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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)

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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

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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

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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)	$\Re(a)>0$
asymptotic strong unstable spectrum	$S_{0}^{+}$	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	$\left\|b\right\|\geq\left\|\Re(a)\right\|$

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