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Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads
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 |
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} $.
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. Google Scholar |
[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. Google Scholar |
[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. |



Symbol Description | Reference | |
|
Spectrum | Eq. (5) |
Strong spectrum | Def. 2.3, Eq. (15) | |
Pseudo-continuous spectrum | Def. 2.3, Eq. (16) | |
Truncated stable |
Def.2.1, Eq. (10) | |
|
Asymptotic strong spectrum | Def. 2.3, Eq. (14) |
Asymptotic strong unstable spectrum | Def. 2.3, Eq. (13) | |
Asymptotic strong stable spectrum | Def.2.1, Eq. (11) | |
Asymptotic continuous |
Def.2.4, Eq. (21) | |
Asymptotic continuous stable |
Def.2.4, Eq. (19) | |
Asymptotic continuous unstable |
Def.2.4, Eq. (20) | |
|
Coefficient matrix corresponding to delay |
Eq. (1) |
Projection of coefficient matrix |
Eq. (9) | |
Characteristic function | Eq. (6) | |
Projected characteristic equation, |
Def.2.1, Eq. (8) | |
Truncated characteristic equation, |
Def. 2.4, Eqs. (17)–(18) |
Symbol Description | Reference | |
|
Spectrum | Eq. (5) |
Strong spectrum | Def. 2.3, Eq. (15) | |
Pseudo-continuous spectrum | Def. 2.3, Eq. (16) | |
Truncated stable |
Def.2.1, Eq. (10) | |
|
Asymptotic strong spectrum | Def. 2.3, Eq. (14) |
Asymptotic strong unstable spectrum | Def. 2.3, Eq. (13) | |
Asymptotic strong stable spectrum | Def.2.1, Eq. (11) | |
Asymptotic continuous |
Def.2.4, Eq. (21) | |
Asymptotic continuous stable |
Def.2.4, Eq. (19) | |
Asymptotic continuous unstable |
Def.2.4, Eq. (20) | |
|
Coefficient matrix corresponding to delay |
Eq. (1) |
Projection of coefficient matrix |
Eq. (9) | |
Characteristic function | Eq. (6) | |
Projected characteristic equation, |
Def.2.1, Eq. (8) | |
Truncated characteristic equation, |
Def. 2.4, Eqs. (17)–(18) |
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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