# American Institute of Mathematical Sciences

January  2021, 14(1): 151-175. doi: 10.3934/dcdss.2020321

## The spectrum of delay differential equations with multiple hierarchical large delays

 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

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

Received  February 2019 Revised  October 2019 Published  April 2020

Fund Project: 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

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 $\tau_{n}$.

Citation: Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321
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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)
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
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
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)
 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)
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 $\left|b\right|\geq\left|\Re(a)\right|$
 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 $\left|b\right|\geq\left|\Re(a)\right|$
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