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Complex dynamics in delay-differential equations with large delay

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Abstract.

We investigate the dynamical properties of delay differential equations with large delay. Starting from a mathematical discussion of the singular limit τ → ∞, we present a novel theoretical approach to the stability properties of stationary solutions in such systems. We introduce the notion of strong and weak instabilities and describe a method that allows us to calculate asymptotic approximations of the corresponding parts of the spectrum. The theoretical results are illustrated by several examples, including the control of unstable steady states of focus type by time delayed feedback control and the stability of external cavity modes in the Lang-Kobayashi system for semiconductor lasers with optical feedback.

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References

  1. T. Erneux, Applied delay differential equations (Springer, 2009)

  2. F.M. Atay (Ed.), Complex Time-Delay Systems (Springer, Berlin Heidelberg, 2010)

  3. W. Just, A. Pelster, M. Schanz, E. Schöll (eds.), Delayed Complex Systems, Theme Issue of Phil. Trans. R. Soc. A 368, 301 (2010)

    Google Scholar 

  4. P. Hövel, Control of Complex Nonlinear Systems with Delay (Springer, Heidelberg, 2010)

  5. K. Pyragas, Phys. Lett. A 170, 421 (1992)

    Article  ADS  Google Scholar 

  6. E. Schöll, H.G. Schuster (Eds.), Handbook of Chaos Control (Wiley-VCH, Weinheim, 2008), second completely revised and enlarged edition

  7. M.G. Rosenblum, A.S. Pikovsky, Phys. Rev. Lett. 92, 114102 (2004)

    Article  ADS  Google Scholar 

  8. A. Ahlborn, U. Parlitz, Phys. Rev. E 72, 016206 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  9. P. Hövel, E. Schöll, Phys. Rev. E 72, 046203 (2005)

    Article  Google Scholar 

  10. T. Dahms, P. Hövel, E. Schöll, Phys. Rev. E 76, 056201 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  11. A. Gjurchinovski, V. Urumov, Europhys. Lett. 84, 40013 (2008)

    Article  ADS  Google Scholar 

  12. A. Gjurchinovski, V. Urumov, Phys. Rev. E 81, 16209 (2010)

    Article  ADS  Google Scholar 

  13. S. Bielawski, M. Bouazaoui, D. Derozier, P. Glorieux, Phys. Rev. A 47, 3276 (1993)

    Article  ADS  Google Scholar 

  14. A. Chang, J.C. Bienfang, G.M. Hall, J.R. Gardner, D.J. Gauthier, Chaos 8, 782 (1998)

    Article  MATH  ADS  Google Scholar 

  15. K. Pyragas, V. Pyragas, I.Z. Kiss, J.L. Hudson, Phys. Rev. Lett. 89, 244103 (2002)

    Article  ADS  Google Scholar 

  16. K. Pyragas, V. Pyragas, I.Z. Kiss, J.L. Hudson, Phys. Rev. E 70, 026215 (2004)

    Article  ADS  Google Scholar 

  17. K.B. Blyuss, Y.N. Kyrychko, P. Hövel, E. Schöll, Eur. Phys. J. B 65, 571 (2008)

    Article  MATH  ADS  Google Scholar 

  18. Y.N. Kyrychko, K.B. Blyuss, P. Hövel, E. Schöll, Dyn. Sys. 24, 361 (2009)

    Article  MATH  Google Scholar 

  19. A. Ahlborn, U. Parlitz, Phys. Rev. Lett. 93, 264101 (2004)

    Article  ADS  Google Scholar 

  20. E. Schöll, in Nonlinear Dynamics of Nanosystems, edited by G. Radons, B. Rumpf, H.G. Schuster (Wiley-VCH, Weinheim, 2009), ISBN 978-3-527-40791-0

  21. L. Schimansky-Geier, B. Fiedler, J. Kurths, E. Schöll (Eds.), Analysis and control of complex nonlinear processes in physics, chemistry and biology (World Scientific, Singapore, 2007)

  22. S. Schikora, P. Hövel, H.J. Wünsche, E. Schöll, F. Henneberger, Phys. Rev. Lett. 97, 213902 (2006)

    Article  ADS  Google Scholar 

  23. R. Lang, K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980)

    Article  ADS  Google Scholar 

  24. J. Mørk, B. Tromborg, J. Mark, IEEE J. Quantum Electron. 28, 93 (1992)

    Article  ADS  Google Scholar 

  25. A.M. Levine, G.H.M. van Tartwijk, D. Lenstra, T. Erneux, Phys. Rev. A 52, R3436 (1995)

    Article  ADS  Google Scholar 

  26. P.M. Alsing, V. Kovanis, A. Gavrielides, T. Erneux, Phys. Rev. A 53, 4429 (1996)

    Article  ADS  Google Scholar 

  27. T. Dahms, P. Hövel, E. Schöll, Phys. Rev. E 78, 056213 (2008)

    Article  ADS  Google Scholar 

  28. J.E.S. Socolar, D.W. Sukow, D.J. Gauthier, Phys. Rev. E 50, 3245 (1994)

    Article  ADS  Google Scholar 

  29. M. Wolfrum, S. Yanchuk, Phys. Rev. Lett. 96, 220201 (2006)

    Article  ADS  Google Scholar 

  30. S. Yanchuk, M. Wolfrum, P. Hövel, E. Schöll, Phys. Rev. E 74, 026201 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  31. S. Yanchuk, M. Wolfrum, SIAM J. Appl. Dynam. Syst. 9, 519 (2010)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  32. M. Lichtner, M. Wolfrum, S. Yanchuk, Preprint 647, DFG Research Center Matheon, Berlin (2009)

  33. S. Yanchuk, P. Perlikowski, Phys. Rev. E 79, 046221 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  34. M. Nizette, Phys. Rev. E 70, 056204 (2004)

    Article  ADS  Google Scholar 

  35. S.A. Kashchenko, Siberian Math. J. 40, 483 (1999)

    Article  MathSciNet  Google Scholar 

  36. G. Giacomelli, A. Politi, Phys. Rev. Lett. 76, 2686 (1996)

    Article  ADS  Google Scholar 

  37. L.S. Tuckerman, D. Barkley, Physica D 46, 57 (1990)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  38. J. Mulet, C.R. Mirasso, Phys. Rev. E 59, 5400 (1999)

    Article  ADS  Google Scholar 

  39. M. Wolfrum, D. Turaev, Opt. Comm. 212, 127 (2002)

    Article  ADS  Google Scholar 

  40. M. Peil, T. Heil, I. Fischer, W. Elsäßer, Phys. Rev. Lett. 88, 174101 (2002)

    Article  ADS  Google Scholar 

  41. V.Z. Tronciu, H.-J. Wünsche, M. Wolfrum, M. Radziunas, Phys. Rev. E 73, 046205 (2006)

    Article  ADS  Google Scholar 

  42. A. Ritter, H. Haug, J. Opt. Soc. Am. B 10, 130 (1993)

    Article  ADS  Google Scholar 

  43. A. Ritter, H. Haug, J. Opt. Soc. Am. B 10, 145 (1993)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, 10117, Berlin, Germany

    M. Wolfrum

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    S. Yanchuk

  3. Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623, Berlin, Germany

    P. Hövel & E. Schöll

Authors
  1. M. Wolfrum
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  2. S. Yanchuk
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  3. P. Hövel
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  4. E. Schöll
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Wolfrum, M., Yanchuk, S., Hövel, P. et al. Complex dynamics in delay-differential equations with large delay. Eur. Phys. J. Spec. Top. 191, 91–103 (2010). https://doi.org/10.1140/epjst/e2010-01343-7

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  • DOI: https://doi.org/10.1140/epjst/e2010-01343-7

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