Noisy Dynamical Systems with Time Delay: Some Basic Analytical Perturbation Schemes with Applications

  • Wolfram JustEmail author
  • Paul M. Geffert
  • Anna Zakharova
  • Eckehard Schöll
Part of the Understanding Complex Systems book series (UCS)


Systems with time delay, a rather prominent branch in applied dynamical systems theory, constitute a special case of functional differential equations for which the general mathematical theory is fairly well developed and largely parallels the theory of ordinary differential equations. Hence analytic concepts like bifurcation theory, adiabatic elimination, global attractors, invariant manifolds and others can be used to study dynamical behaviour of systems with time delay if some care is applied to take special features of infinite dimensional phase spaces into account. Simple analytic perturbation schemes, frequently used to gain insight for ordinary differential equations, can be applied to time delay dynamics as well. However, such approaches seem to be used infrequently within the physics community, probably because of a lack of easily accessible expositions. Here we review some elementary and well established concepts for the analytical treatment of time delay dynamics, even when subjected to noise. We cover normal form reduction and adiabatic elimination, stochastic linearisation of time delay dynamics with noise, and some elements of weakly nonlinear- and bifurcation analysis. These tools will be illustrated with applications in control problems, time delay autosynchronisation, coherence resonance, and the computation and structure of power spectra in noisy time delay systems.


Periodic Orbit Time Delay System Dynamical System Theory Slow Manifold Coherence Resonance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was supported by DFG in the framework of SFB 910.


  1. 1.
    A. Einstein, Ann. Phys. 322, 549 (1905)CrossRefGoogle Scholar
  2. 2.
    H.A. Kramers, Physica 7, 284 (1940)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    P. Hänggi, P. Talkner, M. Borkovec, Rev. Mod. Phys. 62, 251 (1990)ADSCrossRefGoogle Scholar
  4. 4.
    A. Pikovsky, J. Kurths, Phys. Rev. Lett. 78, 775 (1997)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    P. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70(1), 223 (1998)ADSCrossRefGoogle Scholar
  6. 6.
    R. Klages, W. Just, C. Jarzynski (eds.), Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond (Wiley-VCH, Weinheim, 2013)zbMATHGoogle Scholar
  7. 7.
    W. Horsthemke, R. Lefever, Noise-Induced Transitions (Springer, Berlin, 1984)zbMATHGoogle Scholar
  8. 8.
    H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, Berlin, 1989)CrossRefzbMATHGoogle Scholar
  9. 9.
    L. Arnold, Random Dynamical Systems (Springer, Berlin, 2002)Google Scholar
  10. 10.
    R. Bellmann, K.L. Cooke, Differential-Difference Equations (Acad. Press, New York, 1963)zbMATHGoogle Scholar
  11. 11.
    E. Fick, G. Sauermann, The Quantum Statistics of Dynamic Processes (Springer, Berlin, 1990)CrossRefGoogle Scholar
  12. 12.
    R. Kubo, M. Toda, N. Hashitsume, Statistical Physics 2 (Nonequilibrium statistical mechanics) (Springer, Berlin, 1991)zbMATHGoogle Scholar
  13. 13.
    T. Erneux, Applied Delay Differential Equations (Springer, Berlin, 2009)zbMATHGoogle Scholar
  14. 14.
    F.M. Atay (ed.), Complex Time-Delay Systems (Springer, Berlin, Heidelberg, 2010)zbMATHGoogle Scholar
  15. 15.
    W. Just, A. Pelster, M. Schanz, E. Schöll (eds.), Delayed Complex Systems, Theme Issue of Phil. Trans. R. Soc. A 368, 301–513 (2010)Google Scholar
  16. 16.
    V. Flunkert, I. Fischer, E. Schöll (eds.), Dynamics, control and information in delay-coupled systems, Theme Issue of Phil. Trans. R. Soc. A 371, 20120465 (2013)Google Scholar
  17. 17.
    J.Q. Sun, G. Ding (eds.), Advances in Analysis and Control of Time-Delayed Dynamical Systems (World Scientific, Singapore, 2013)Google Scholar
  18. 18.
    M. Soriano, J. Garca-Ojalvo, C. Mirasso, I. Fischer, Rev. Mod. Phys. 85, 421 (2013)Google Scholar
  19. 19.
    J.K.  Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations (Springer, New York, 1993)Google Scholar
  20. 20.
    K. Engelborghs, DDE-BIFTOOL: a Mathlab package for bifurcation analysis of delay differential equations.
  21. 21.
    L. Pecora, T. Carroll, Phys. Rev. Lett. 80, 2109 (1998)ADSCrossRefGoogle Scholar
  22. 22.
    S. Yanchuk, M. Wolfrum, P. Hövel, E. Schöll, Phys. Rev. E 74, 026201 (2006)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    M. Wolfrum, S. Yanchuk, P. Hövel, E. Schöll, Eur. Phys. J. Special Topics 191, 91 (2010)ADSCrossRefGoogle Scholar
  24. 24.
    T.D. Frank, P.J. Beck, R. Friedrich, Phys. Rev. E 68, 021912 (2003)ADSCrossRefGoogle Scholar
  25. 25.
    T. Frank, Phys. Rev. E 72, 011112 (2005)ADSCrossRefGoogle Scholar
  26. 26.
    A. Nayfeh, Perturbation Methods (Wiley-VCH, Weinheim, 2000)CrossRefzbMATHGoogle Scholar
  27. 27.
    M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65(3), 851 (1993)ADSCrossRefGoogle Scholar
  28. 28.
    W. Just, H. Benner, C. v. Loewenich. Physica D 199, 33 (2004)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    P.M. Geffert, A. Zakharova, A. Vüllings, W. Just, E. Schöll, Eur. Phys. J. B 87, 291 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    H. Haken, Synergetics: Introduction and Advanced Topics (Springer, Berlin, 2004)CrossRefGoogle Scholar
  31. 31.
    A. Halanay, Differential Equations: Stability, Oscillations, Time Lags (Acad. Press, New York, 1966)zbMATHGoogle Scholar
  32. 32.
    A. Amann, E. Schöll, W. Just, Physica A 373, 191 (2007)ADSCrossRefGoogle Scholar
  33. 33.
    R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, Adv. Comp. Math. 5, 329 (1996)MathSciNetCrossRefGoogle Scholar
  34. 34.
    F. Giannakopulos, A. Zapp, Physica D 159, 215 (2001)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    B.F. Redmond, V.G. LeBlanc, A. Longtin, Physica D 166, 131 (2002)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    N.G. van Kampen, Phys. Rep. 124(2), 69 (1985)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    U. Küchler, B. Mensch, Stoch. Stoch. Rep. 40, 23 (1992)CrossRefGoogle Scholar
  38. 38.
    K. Pyragas, Phys. Lett. A 170, 421 (1992)ADSCrossRefGoogle Scholar
  39. 39.
    E. Schöll, H.G. Schuster (eds.), Handbook of Chaos Control (Wiley-VCH, Weinheim, 2008)zbMATHGoogle Scholar
  40. 40.
    B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, E. Schöll, Phys. Rev. Lett. 98, 114101 (2007)ADSCrossRefGoogle Scholar
  41. 41.
    W. Just, B. Fiedler, M. Georgi, V. Flunkert, P. Hövel, E. Schöll, Phys. Rev. E 76, 026210 (2007)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    G. Brown, C.M. Postlethwaite, M. Silber, Physica D 240, 859 (2011)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    G. Hu, T. Ditzinger, C.Z. Ning, H. Haken, Phys. Rev. Lett. 71, 807 (1993)ADSCrossRefGoogle Scholar
  44. 44.
    R. Aust, P. Hövel, J. Hizanidis, E. Schöll, Eur. Phys. J. Spec. Top. 187, 77 (2010)CrossRefGoogle Scholar
  45. 45.
    N.B. Janson, A.G. Balanov, E. Schöll, Phys. Rev. Lett. 93, 010601 (2004)ADSCrossRefGoogle Scholar
  46. 46.
    O.V. Ushakov, H.J. Wünsche, F. Henneberger, I.A. Khovanov, L. Schimansky-Geier, M.A. Zaks, Phys. Rev. Lett. 95, 123903 (2005)ADSCrossRefGoogle Scholar
  47. 47.
    A. Zakharova, T. Vadivasova, V. Anishchenko, A. Koseska, J. Kurths, Phys. Rev. E 81, 011106 (2010)ADSCrossRefGoogle Scholar
  48. 48.
    A. Zakharova, A. Feoktistov, T. Vadivasova, E. Schöll, Eur. Phys. J. Spec. Top. 222, 2481 (2013)CrossRefGoogle Scholar
  49. 49.
    V. Semenov, A. Feoktistov, T. Vadivasova, E. Schöll, A. Zakharova, Chaos 25, 033111 (2015)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    E. Schöll, A.G. Balanov, N.B. Janson, A.B. Neiman, Stoch. Dyn. 5, 281 (2005)MathSciNetCrossRefGoogle Scholar
  51. 51.
    J. Pomplun, A. Amann, E. Schöll, Europhys. Lett. 71, 366 (2005)ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    B. Dmitriev, Y. Zharkov, S. Sadovnikov, V. Skorokhodov, A. Stepanov, Tech. Phys. Lett. 37, 1082 (2011)ADSCrossRefGoogle Scholar
  53. 53.
    J.L.A. Dubbeldam, B. Krauskopf, D. Lenstra, Phys. Rev. E 60, 6580 (1999)ADSCrossRefGoogle Scholar
  54. 54.
    G. Giacomelli, M. Giudici, S. Balle, J.R. Tredicce, Phys. Rev. Lett. 84, 3298 (2000)ADSCrossRefGoogle Scholar
  55. 55.
    J.F.M. Avila, H.L.D. de S. Cavalcante, J.R.R. Leite. Phys. Rev. Lett. 93, 144101 (2004)Google Scholar
  56. 56.
    C. Otto, B. Lingnau, E. Schöll, K. Lüdge, Opt. Express 22, 13288 (2014)ADSCrossRefGoogle Scholar
  57. 57.
    D. Ziemann, R. Aust, B. Lingnau, E. Schöll, K. Lüdge, Europhys. Lett. 103, 14002 (2013)ADSCrossRefGoogle Scholar
  58. 58.
    J. Hizanidis, E. Schöll, Phys. Rev. E 78, 066205 (2008)ADSCrossRefGoogle Scholar
  59. 59.
    Y. Huang, H. Qin, W. Li, S. Lu, J. Dong, H.T. Grahn, Y. Zhang, Europhys. Lett. 105, 47005 (2014)ADSCrossRefGoogle Scholar
  60. 60.
    M.E. Bleich, J.E.S. Socolar, Phys. Lett. A 210, 87 (1996)ADSCrossRefGoogle Scholar
  61. 61.
    A. Vüllings, E. Schöll, B. Lindner, Eur. Phys. J. B 87, 31 (2014)ADSCrossRefGoogle Scholar
  62. 62.
    J. Kottalam, K. Lindenberg, B.J. West, J. Stat. Phys. 42, 979 (1986)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Wolfram Just
    • 1
    • 2
    Email author
  • Paul M. Geffert
    • 1
    • 2
  • Anna Zakharova
    • 2
  • Eckehard Schöll
    • 2
  1. 1.School of Mathematical SciencesQueen Mary University of LondonLondonUK
  2. 2.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany

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