The spectrum of delay differential equations with multiple hierarchical large delays

Stefan Ruschel; Serhiy Yanchuk

doi:10.3934/dcdss.2020321

Previous Article
Global Hopf bifurcation in networks with fast feedback cycles
DCDS-S Home
This Issue
Next Article
Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads

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

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

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} $.

Keywords: Linear Delay Differential Equations, Large Delay, Multiple Delays.

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

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:

[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.

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.

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Figure Options

Download full-size image

Download as PowerPoint slide

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)

Table Options

Download as excel

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)

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	$\left\|b\right\|\geq\left\|\Re(a)\right\|$

Table Options

Download as excel

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	$\left\|b\right\|\geq\left\|\Re(a)\right\|$

[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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2009, 25 (3) : 751-775. doi: 10.3934/dcds.2009.25.751
[14]	Klaudiusz Wójcik, Piotr Zgliczyński. Topological horseshoes and delay differential equations. Discrete and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057

2021 Impact Factor: 1.865

Tools

Metrics

Tools

Article outline

Figures and Tables

2021 Impact Factor: 1.865

Tools

Figures and Tables

2021 Impact Factor: 1.865

American Institute of Mathematical Sciences

The spectrum of delay differential equations with multiple hierarchical large delays

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

The spectrum of delay differential equations with multiple hierarchical large delays

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors