# 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  January 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321
##### References:
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Giacomelli, Excitable wave patterns in temporal systems with two long delays and their observation in a semiconductor laser experiment, Phys. Rev. Lett., 122 (2019), 174102. doi: 10.1103/PhysRevLett.122.174102. [21] C. Otto, K. Lüdge, A. G. Vladimirov, M. Wolfrum and E. Schöll, Delay-induced dynamics and jitter reduction of passively mode-locked semiconductor lasers subject to optical feedback, New J. Phys., 14 (2012), 113033. [22] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discret. Impuls. Syst. Ser. A, 10 (2003), 863-874. [23] S. Ruschel, T. Pereira, S. Yanchuk and L.-S. Young, An SIQ delay differential equations model for disease control via isolation, J. Math. Biol., 79 (2019), 249-279.  doi: 10.1007/s00285-019-01356-1. [24] J. Sieber, Local bifurcations in differential equations with state-dependent delay, Chaos, 27 (2017), 114326, 12pp. doi: 10.1063/1.5011747. [25] M. C. Soriano, J. García-Ojalvo, C. Mirasso and I. Fischer, Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers, Rev. Mod. Phys., 85 (2013), 421-470.  doi: 10.1103/RevModPhys.85.421. [26] L. P. Shayer and S. A. Campbell, Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays, SIAM J. Appl. Math., 61 (2000), 673-700.  doi: 10.1137/S0036139998344015. [27] A. Saha and U. Feudel, Extreme events in FitzHugh-Nagumo oscillators coupled with two time delays, Phys. Rev. E, 95 (2017), 062219, 10pp. doi: 10.1103/physreve.95.062219. [28] J. Sieber, M. Wolfrum, M. Lichtner and S. Yanchuk, On the stability of periodic orbits in delay equations with large delay, Discrete Contin. Dyn. Syst., 33 (2013), 3109-3134.  doi: 10.3934/dcds.2013.33.3109. [29] S. Yanchuk, L. Lücken, M. Wolfrum and A. Mielke, Spectrum and amplitude equations for scalar delay-differential equations with large delay, Discret. Contin. Dyn. Syst. A, 35 (2015), 537-553.  doi: 10.3934/dcds.2015.35.537. [30] S. Yanchuk and G. Giacomelli, Spatio-temporal phenomena in complex systems with time delays, J. Phys. A Math. Theor., 50 (2017), 103001, 56pp. [31] S. Yanchuk and G. Giacomelli, Pattern formation in systems with multiple delayed feedbacks, Phys. Rev. Lett., 112 (2014), 174103. doi: 10.1103/PhysRevLett.112.174103. [32] S. Yanchuk and G. Giacomelli, Dynamical systems with multiple long-delayed feedbacks: Multiscale analysis and spatiotemporal equivalence, Phys. Rev. E, 92 (2015), 042903, 12pp. doi: 10.1103/PhysRevE.92.042903. [33] S. Yanchuk, S. Ruschel, J. Sieber and M. Wolfrum, Temporal dissipative solitons in time-delay feedback systems, Phys. Rev. Lett., 123 (2019), 053901, 6pp. doi: 10.1103/PhysRevLett.123.053901.

show all references

##### References:
 [1] A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera and K. A. Shore, Chaos-based communications at high bit rates using commercial fibre-optic links, Nature, 438 (2005), 343-346.  doi: 10.1038/nature04275. [2] L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso and I. Fischer, Information processing using a single dynamical node as complex system, Nat. Comm, 2 (2011), 468. doi: 10.1038/ncomms1476. [3] F. M. Atay, Complex Time-Delay Systems, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-02329-3. [4] C. Avellar and J. K. Hale, On the characterization of exponential polynomials, J. Math. Anal. Appl., 73 (1980), 434-452.  doi: 10.1016/0022-247X(80)90289-9. [5] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York-London, 1963. [6] J. Belair and S. A. Campbell, Stability and bifurcations of equilibria of multiple-delayed differential equations, SIAM J. Appl. Math., 54 (1994), 1402-1424.  doi: 10.1137/S0036139993248853. [7] K. L. Cooke and P. van den Driessche, On zeros of some transcendental functions, Funkcialaj Ekvacioj, 29 (1986), 77-90. [8] K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35 (1996), 240-260.  doi: 10.1007/s002850050051. [9] O. Diekmann, S. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations, Functional-, Complex, and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. [10] O. D'Huys, S. Zeeb, T. Jüngling, S. Heiligenthal, S. Yanchuk and W. Kinzel, Synchronisation and scaling properties of chaotic networks with multiple delays, Eur. Lett., 103 (2013), 10013. doi: 10.1209/0295-5075/103/10013. [11] T. Erneux, Applied Delay Differential Equations, vol. 3 of Surveys and Tutorials in the Applied Mathematical Sciences, Springer, New York, 2009. [12] A. L. Franz, R. Roy, L. B. Shaw and I. B. Schwartz, Effect of multiple time delays on intensity fluctuation dynamics in fiber ring lasers, Phys. Rev. E, 78 (2008), 016208. doi: 10.1103/PhysRevE.78.016208. [13] F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Chapter 5: Functional differential equations with state-dependent delays: Theory and applications, Handb. Differ. Equations Ordinary Differ. Equations, 3 (2006), 435-545.  doi: 10.1016/S1874-5725(06)80009-X. [14] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [15] N. D. Hayes, Roots of the transcendental equation associated with a certain differencedifferential equation, J. London Math. Soc. (1), 25 (1950), 226–232. doi: 10.1112/jlms/s1-25.3.226. [16] S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll and W. Kinzel, Strong and weak chaos in nonlinear networks with time-delayed couplings, Phys. Rev. Lett., 107 (2011), 234102. doi: 10.1103/PhysRevLett.107.234102. [17] L. Jaurigue, B. Krauskopf and K. Lüdge, Multipulse dynamics of a passively mode-locked semiconductor laser with delayed optical feedback, Chaos, 27 (2017), 114301, 12pp. doi: 10.1063/1.5006743. [18] M. Lichtner, M. Wolfrum and S. Yanchuk, The spectrum of delay differential equations with large delay, SIAM J. Math. Anal., 43 (2011), 788-802.  doi: 10.1137/090766796. [19] M. Marconi, J. Javaloyes, S. Barland, S. Balle and M. Giudici, Vectorial dissipative solitons in vertical-cavity surface-emitting lasers with delays, Nat. Photon., 9 (2015), 450-455.  doi: 10.1038/nphoton.2015.92. [20] F. Marino and G. Giacomelli, Excitable wave patterns in temporal systems with two long delays and their observation in a semiconductor laser experiment, Phys. Rev. Lett., 122 (2019), 174102. doi: 10.1103/PhysRevLett.122.174102. [21] C. Otto, K. Lüdge, A. G. Vladimirov, M. Wolfrum and E. Schöll, Delay-induced dynamics and jitter reduction of passively mode-locked semiconductor lasers subject to optical feedback, New J. Phys., 14 (2012), 113033. [22] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discret. Impuls. Syst. Ser. A, 10 (2003), 863-874. [23] S. Ruschel, T. Pereira, S. Yanchuk and L.-S. Young, An SIQ delay differential equations model for disease control via isolation, J. Math. Biol., 79 (2019), 249-279.  doi: 10.1007/s00285-019-01356-1. [24] J. Sieber, Local bifurcations in differential equations with state-dependent delay, Chaos, 27 (2017), 114326, 12pp. doi: 10.1063/1.5011747. [25] M. C. Soriano, J. García-Ojalvo, C. Mirasso and I. Fischer, Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers, Rev. Mod. Phys., 85 (2013), 421-470.  doi: 10.1103/RevModPhys.85.421. [26] L. P. Shayer and S. A. Campbell, Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays, SIAM J. Appl. Math., 61 (2000), 673-700.  doi: 10.1137/S0036139998344015. [27] A. Saha and U. Feudel, Extreme events in FitzHugh-Nagumo oscillators coupled with two time delays, Phys. Rev. E, 95 (2017), 062219, 10pp. doi: 10.1103/physreve.95.062219. [28] J. Sieber, M. Wolfrum, M. Lichtner and S. Yanchuk, On the stability of periodic orbits in delay equations with large delay, Discrete Contin. Dyn. Syst., 33 (2013), 3109-3134.  doi: 10.3934/dcds.2013.33.3109. [29] S. Yanchuk, L. Lücken, M. Wolfrum and A. Mielke, Spectrum and amplitude equations for scalar delay-differential equations with large delay, Discret. Contin. Dyn. Syst. A, 35 (2015), 537-553.  doi: 10.3934/dcds.2015.35.537. [30] S. Yanchuk and G. Giacomelli, Spatio-temporal phenomena in complex systems with time delays, J. Phys. A Math. Theor., 50 (2017), 103001, 56pp. [31] S. Yanchuk and G. Giacomelli, Pattern formation in systems with multiple delayed feedbacks, Phys. Rev. Lett., 112 (2014), 174103. doi: 10.1103/PhysRevLett.112.174103. [32] S. Yanchuk and G. Giacomelli, Dynamical systems with multiple long-delayed feedbacks: Multiscale analysis and spatiotemporal equivalence, Phys. Rev. E, 92 (2015), 042903, 12pp. doi: 10.1103/PhysRevE.92.042903. [33] S. Yanchuk, S. Ruschel, J. Sieber and M. Wolfrum, Temporal dissipative solitons in time-delay feedback systems, Phys. Rev. Lett., 123 (2019), 053901, 6pp. doi: 10.1103/PhysRevLett.123.053901.
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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