American Institute of Mathematical Sciences

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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

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