American Institute of Mathematical Sciences

Advanced
July  2013, 33(7): 3109-3134. doi: 10.3934/dcds.2013.33.3109

On the stability of periodic orbits in delay equations with large delay

Jan Sieber 1, , Matthias Wolfrum 2, , Mark Lichtner 2, and  Serhiy Yanchuk 3,

1. 

Harrison Building, North Park Road, CEMPS, University of Exeter, Exeter, EX4 4QF, United Kingdom
2. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
3. 

Institute of Mathematics, Humboldt University of Berlin, Rudower Chaussee 25, 12489, Berlin

Received  February 2012 Revised  December 2012 Published  January 2013

We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay.
Keywords: Periodic solutions, Floquet multipliers., stability, asymptotic continuous spectrum, large delay, strongly unstable spectrum.
Mathematics Subject Classification: 34K08, 34K13, 34K20, 37G15, 37L1.
Citation: Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109
References:
[1]

K. Engelborghs, T. Luzyanina and G. Samaey, "DDE-BIFTOOL v.2.00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations," Report TW 330, Katholieke Universiteit Leuven, 2001. Google Scholar

[2]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993.  Google Scholar

[3]

M. A. Kaashoek and S. M. Verduyn Lunel, Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc., 334 (1992), 479-517. doi: 10.2307/2154470.  Google Scholar

[4]

R. Lang and K. Kobayashi, External optical feedback effects on semiconductor injection properties, IEEE J. of Quant. El., 16 (1980), 347-355. Google Scholar

[5]

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.  Google Scholar

[6]

J. J. Loiseau, W. Michiels, S.-I. Niculescu and R. Sipahi, "Topics in Time Delay Systems: Analysis, Algorithms and Control," 388 of Lecture Notes in Control and Information Sciences. Springer, 2009. doi: 10.1007/978-3-642-02897-7.  Google Scholar

[7]

D. Roose and R. Szalai, Continuation and bifurcation analysis of delay differential equations, in " Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems" (eds. B Krauskopf, H M Osinga and J Galán-Vioque), Springer-Verlag, Dordrecht (2007), 51-75. doi: 10.1007/978-1-4020-6356-5_12.  Google Scholar

[8]

G. Samaey, K. Engelborghs and D. Roose, Numerical computation of connecting orbits in delay differential equations, Numer. Algorithms, 30 (2002), 335-352. doi: 10.1023/A:1020102317544.  Google Scholar

[9]

E. Schöll and H. Schuster, "Handbook of Chaos Control," Wiley, New York, 2 edition, 2008.  Google Scholar

[10]

J. Sieber and R. Szalai, Characteristic matrices for linear periodic delay differential equations, SIAM Journal on Applied Dynamical Systems, 10 (2011), 129-147. arXiv:1005.4522 doi: 10.1137/100796455.  Google Scholar

[11]

A. L. Skubachevskii and H.-O. Walther, On the Floquet multipliers of periodic solutions to nonlinear functional differential equations, J. Dynam. Diff. Eq., 18 (2006), 257-355. doi: 10.1007/s10884-006-9006-5.  Google Scholar

[12]

G. Stépán, "Retarded Dynamical Systems: Stability and Characteristic Functions," Longman Scientific and Technical, Harlow, Essex, 1989.  Google Scholar

[13]

R. Szalai, G. Stépán and S. J. Hogan, Continuation of bifurcations in periodic delay differential equations using characteristic matrices, SIAM Journal on Scientific Computing, 28 (2006), 1301-1317. doi: 10.1137/040618709.  Google Scholar

[14]

H.-O. Walther, Density of slowly oscillating solutions of $\dot x(t)=-f(x(t-1))$, Journal of Mathematical Analysis and Applications, 79 (1981), 127-140. doi: 10.1016/0022-247X(81)90014-7.  Google Scholar

[15]

M Wolfrum and S Yanchuk, Eckhaus instability in systems with large delay, Phys. Rev. Lett., 96 (2006), 220201. doi: 10.1103/PhysRevLett.96.220201.  Google Scholar

[16]

S Yanchuk and P Perlikowski, Delay and periodicity, Physical Review E., 79 (2009), 46221. doi: 10.1103/PhysRevE.79.046221.  Google Scholar

[17]

S Yanchuk and M Wolfrum, Stability of external cavity modes in the Lang-Kobayashi system with large delay, SIAM J. Appl. Dyn. Sys., 9 (2010), 519-535. doi: 10.1137/090751335.  Google Scholar

show all references

References:
[1]

K. Engelborghs, T. Luzyanina and G. Samaey, "DDE-BIFTOOL v.2.00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations," Report TW 330, Katholieke Universiteit Leuven, 2001. Google Scholar

[2]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993.  Google Scholar

[3]

M. A. Kaashoek and S. M. Verduyn Lunel, Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc., 334 (1992), 479-517. doi: 10.2307/2154470.  Google Scholar

[4]

R. Lang and K. Kobayashi, External optical feedback effects on semiconductor injection properties, IEEE J. of Quant. El., 16 (1980), 347-355. Google Scholar

[5]

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.  Google Scholar

[6]

J. J. Loiseau, W. Michiels, S.-I. Niculescu and R. Sipahi, "Topics in Time Delay Systems: Analysis, Algorithms and Control," 388 of Lecture Notes in Control and Information Sciences. Springer, 2009. doi: 10.1007/978-3-642-02897-7.  Google Scholar

[7]

D. Roose and R. Szalai, Continuation and bifurcation analysis of delay differential equations, in " Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems" (eds. B Krauskopf, H M Osinga and J Galán-Vioque), Springer-Verlag, Dordrecht (2007), 51-75. doi: 10.1007/978-1-4020-6356-5_12.  Google Scholar

[8]

G. Samaey, K. Engelborghs and D. Roose, Numerical computation of connecting orbits in delay differential equations, Numer. Algorithms, 30 (2002), 335-352. doi: 10.1023/A:1020102317544.  Google Scholar

[9]

E. Schöll and H. Schuster, "Handbook of Chaos Control," Wiley, New York, 2 edition, 2008.  Google Scholar

[10]

J. Sieber and R. Szalai, Characteristic matrices for linear periodic delay differential equations, SIAM Journal on Applied Dynamical Systems, 10 (2011), 129-147. arXiv:1005.4522 doi: 10.1137/100796455.  Google Scholar

[11]

A. L. Skubachevskii and H.-O. Walther, On the Floquet multipliers of periodic solutions to nonlinear functional differential equations, J. Dynam. Diff. Eq., 18 (2006), 257-355. doi: 10.1007/s10884-006-9006-5.  Google Scholar

[12]

G. Stépán, "Retarded Dynamical Systems: Stability and Characteristic Functions," Longman Scientific and Technical, Harlow, Essex, 1989.  Google Scholar

[13]

R. Szalai, G. Stépán and S. J. Hogan, Continuation of bifurcations in periodic delay differential equations using characteristic matrices, SIAM Journal on Scientific Computing, 28 (2006), 1301-1317. doi: 10.1137/040618709.  Google Scholar

[14]

H.-O. Walther, Density of slowly oscillating solutions of $\dot x(t)=-f(x(t-1))$, Journal of Mathematical Analysis and Applications, 79 (1981), 127-140. doi: 10.1016/0022-247X(81)90014-7.  Google Scholar

[15]

M Wolfrum and S Yanchuk, Eckhaus instability in systems with large delay, Phys. Rev. Lett., 96 (2006), 220201. doi: 10.1103/PhysRevLett.96.220201.  Google Scholar

[16]

S Yanchuk and P Perlikowski, Delay and periodicity, Physical Review E., 79 (2009), 46221. doi: 10.1103/PhysRevE.79.046221.  Google Scholar

[17]

S Yanchuk and M Wolfrum, Stability of external cavity modes in the Lang-Kobayashi system with large delay, SIAM J. Appl. Dyn. Sys., 9 (2010), 519-535. doi: 10.1137/090751335.  Google Scholar

[1]

Serhiy Yanchuk, Leonhard Lücken, Matthias Wolfrum, Alexander Mielke. Spectrum and amplitude equations for scalar delay-differential equations with large delay. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 537-553. doi: 10.3934/dcds.2015.35.537
[2]

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
[3]

Sergei A. Nazarov, Rafael Orive-Illera, María-Eugenia Pérez-Martínez. Asymptotic structure of the spectrum in a Dirichlet-strip with double periodic perforations. Networks & Heterogeneous Media, 2019, 14 (4) : 733-757. doi: 10.3934/nhm.2019029
[4]

Yves Guivarc'h. On the spectrum of a large subgroup of a semisimple group. Journal of Modern Dynamics, 2008, 2 (1) : 15-42. doi: 10.3934/jmd.2008.2.15
[5]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations & Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032
[6]

Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013
[7]

Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33
[8]

Fatih Bayazit, Ulrich Groh, Rainer Nagel. Floquet representations and asymptotic behavior of periodic evolution families. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4795-4810. doi: 10.3934/dcds.2013.33.4795
[9]

Dariusz Skrenty. Absolutely continuous spectrum of some group extensions of Gaussian actions. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 365-378. doi: 10.3934/dcds.2010.26.365
[10]

Bassam Fayad, A. Windsor. A dichotomy between discrete and continuous spectrum for a class of special flows over rotations. Journal of Modern Dynamics, 2007, 1 (1) : 107-122. doi: 10.3934/jmd.2007.1.107
[11]

Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633
[12]

C. T. Cremins, G. Infante. A semilinear $A$-spectrum. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 235-242. doi: 10.3934/dcdss.2008.1.235
[13]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
[14]

Fabien Durand, Alejandro Maass. A note on limit laws for minimal Cantor systems with infinite periodic spectrum. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 745-750. doi: 10.3934/dcds.2003.9.745
[15]

Günter Leugering, Sergei A. Nazarov, Jari Taskinen. The band-gap structure of the spectrum in a periodic medium of masonry type. Networks & Heterogeneous Media, 2020, 15 (4) : 555-580. doi: 10.3934/nhm.2020014
[16]

Said Boulite, S. Hadd, L. Maniar. Critical spectrum and stability for population equations with diffusion in unbounded domains. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 265-276. doi: 10.3934/dcdsb.2005.5.265
[17]

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
[18]

Ciprian Preda, Petre Preda, Adriana Petre. On the asymptotic behavior of an exponentially bounded, strongly continuous cocycle over a semiflow. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1637-1645. doi: 10.3934/cpaa.2009.8.1637
[19]

Eudes. M. Barboza, Olimpio H. Miyagaki, Fábio R. Pereira, Cláudia R. Santana. Radial solutions for a class of Hénon type systems with partial interference with the spectrum. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3159-3187. doi: 10.3934/cpaa.2020137
[20]

Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (120)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences