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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]

Report TW 330, Katholieke Universiteit Leuven, 2001. Google Scholar

[2]

99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993.  Google Scholar

[3]

Trans. Amer. Math. Soc., 334 (1992), 479-517. doi: 10.2307/2154470.  Google Scholar

[4]

IEEE J. of Quant. El., 16 (1980), 347-355. Google Scholar

[5]

SIAM J. Math. Anal., 43 (2011), 788-802. doi: 10.1137/090766796.  Google Scholar

[6]

388 of Lecture Notes in Control and Information Sciences. Springer, 2009. doi: 10.1007/978-3-642-02897-7.  Google Scholar

[7]

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]

Numer. Algorithms, 30 (2002), 335-352. doi: 10.1023/A:1020102317544.  Google Scholar

[9]

Wiley, New York, 2 edition, 2008.  Google Scholar

[10]

SIAM Journal on Applied Dynamical Systems, 10 (2011), 129-147. arXiv:1005.4522 doi: 10.1137/100796455.  Google Scholar

[11]

J. Dynam. Diff. Eq., 18 (2006), 257-355. doi: 10.1007/s10884-006-9006-5.  Google Scholar

[12]

Longman Scientific and Technical, Harlow, Essex, 1989.  Google Scholar

[13]

SIAM Journal on Scientific Computing, 28 (2006), 1301-1317. doi: 10.1137/040618709.  Google Scholar

[14]

Journal of Mathematical Analysis and Applications, 79 (1981), 127-140. doi: 10.1016/0022-247X(81)90014-7.  Google Scholar

[15]

Phys. Rev. Lett., 96 (2006), 220201. doi: 10.1103/PhysRevLett.96.220201.  Google Scholar

[16]

Physical Review E., 79 (2009), 46221. doi: 10.1103/PhysRevE.79.046221.  Google Scholar

[17]

SIAM J. Appl. Dyn. Sys., 9 (2010), 519-535. doi: 10.1137/090751335.  Google Scholar

show all references

References:
[1]

Report TW 330, Katholieke Universiteit Leuven, 2001. Google Scholar

[2]

99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993.  Google Scholar

[3]

Trans. Amer. Math. Soc., 334 (1992), 479-517. doi: 10.2307/2154470.  Google Scholar

[4]

IEEE J. of Quant. El., 16 (1980), 347-355. Google Scholar

[5]

SIAM J. Math. Anal., 43 (2011), 788-802. doi: 10.1137/090766796.  Google Scholar

[6]

388 of Lecture Notes in Control and Information Sciences. Springer, 2009. doi: 10.1007/978-3-642-02897-7.  Google Scholar

[7]

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]

Numer. Algorithms, 30 (2002), 335-352. doi: 10.1023/A:1020102317544.  Google Scholar

[9]

Wiley, New York, 2 edition, 2008.  Google Scholar

[10]

SIAM Journal on Applied Dynamical Systems, 10 (2011), 129-147. arXiv:1005.4522 doi: 10.1137/100796455.  Google Scholar

[11]

J. Dynam. Diff. Eq., 18 (2006), 257-355. doi: 10.1007/s10884-006-9006-5.  Google Scholar

[12]

Longman Scientific and Technical, Harlow, Essex, 1989.  Google Scholar

[13]

SIAM Journal on Scientific Computing, 28 (2006), 1301-1317. doi: 10.1137/040618709.  Google Scholar

[14]

Journal of Mathematical Analysis and Applications, 79 (1981), 127-140. doi: 10.1016/0022-247X(81)90014-7.  Google Scholar

[15]

Phys. Rev. Lett., 96 (2006), 220201. doi: 10.1103/PhysRevLett.96.220201.  Google Scholar

[16]

Physical Review E., 79 (2009), 46221. doi: 10.1103/PhysRevE.79.046221.  Google Scholar

[17]

SIAM J. Appl. Dyn. Sys., 9 (2010), 519-535. doi: 10.1137/090751335.  Google Scholar

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