# American Institute of Mathematical Sciences

## 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, doi: 10.3934/dcdss.2020321
##### References:

show all references

##### References:
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 $\left|b\right|\geq\left|\Re(a)\right|$
 [1] A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701 [2] Serhiy Yanchuk, Leonhard Lücken, Matthias Wolfrum, Alexander Mielke. Spectrum and amplitude equations for scalar delay-differential equations with large delay. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 537-553. doi: 10.3934/dcds.2015.35.537 [3] Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361 [4] Qingwen Hu, Bernhard Lani-Wayda, Eugen Stumpf. Preface: Delay differential equations with state-dependent delays and their applications. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : ⅰ-ⅰ. doi: 10.3934/dcdss.20201i [5] Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109 [6] Samuel Bernard, Fabien Crauste. Optimal linear stability condition for scalar differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1855-1876. doi: 10.3934/dcdsb.2015.20.1855 [7] Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105 [8] David M. Bortz. Characteristic roots for two-lag linear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2409-2422. doi: 10.3934/dcdsb.2016053 [9] Samuel Bernard, Jacques Bélair, Michael C Mackey. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 233-256. doi: 10.3934/dcdsb.2001.1.233 [10] Zhen Wang, Xiong Li, Jinzhi Lei. Second moment boundedness of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2963-2991. doi: 10.3934/dcdsb.2014.19.2963 [11] Tomás Caraballo, Gábor Kiss. Attractors for differential equations with multiple variable delays. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1365-1374. doi: 10.3934/dcds.2013.33.1365 [12] Michael Dellnitz, Mirko Hessel-Von Molo, Adrian Ziessler. On the computation of attractors for delay differential equations. Journal of Computational Dynamics, 2016, 3 (1) : 93-112. doi: 10.3934/jcd.2016005 [13] Hermann Brunner, Stefano Maset. Time transformations for delay differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 751-775. doi: 10.3934/dcds.2009.25.751 [14] Klaudiusz Wójcik, Piotr Zgliczyński. Topological horseshoes and delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 827-852. doi: 10.3934/dcds.2005.12.827 [15] Hui Liang, Hermann Brunner. Collocation methods for differential equations with piecewise linear delays. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1839-1857. doi: 10.3934/cpaa.2012.11.1839 [16] Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577 [17] Jehad O. Alzabut. A necessary and sufficient condition for the existence of periodic solutions of linear impulsive differential equations with distributed delay. Conference Publications, 2007, 2007 (Special) : 35-43. doi: 10.3934/proc.2007.2007.35 [18] A. Domoshnitsky. About maximum principles for one of the components of solution vector and stability for systems of linear delay differential equations. Conference Publications, 2011, 2011 (Special) : 373-380. doi: 10.3934/proc.2011.2011.373 [19] Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521 [20] Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057

2018 Impact Factor: 0.545