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Time-Delayed Feedback Control of Spatio-Temporal Self-Organized Patterns in Dissipative Systems

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Control of Self-Organizing Nonlinear Systems

Part of the book series: Understanding Complex Systems ((UCS))

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Abstract

We are interested in the dynamical properties of spatio-temporal self-organized patterns in a Swift-Hohenberg equation subjected to time delayed feedback. We show that variation in the delay time and the feedback strength can lead to complex dynamical behavior of the system in question including formation of traveling hexagons, traveling zigzag patterns, or intricate oscillatory structures. Furthermore, we provide a bifurcation analysis of the system and derive a set of order parameter equations which allow us to analytically demonstrate how the time delayed feedback can change the stability of the homogeneous steady state as well as of periodic patterns. Direct numerical simulations are carried out, showing good agreement with analytical predictions based on linear stability analysis and bifurcation theory. The presented results are derived in general form and can be applied to a wide class of spatially extended systems.

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References

  1. E. Ott, C. Grebogi, J.A. Yorke, Phys. Rev. Lett. 64, 1196 (1990). doi:10.1103/PhysRevLett.64.1196

    Article  ADS  MathSciNet  Google Scholar 

  2. A.S. Mikhailov, K. Showalter, Phys. Rep. 425(2–3), 79 (2006). doi:10.1016/j.physrep.2005.11.003

    Article  ADS  MathSciNet  Google Scholar 

  3. E. Schöll, H.G. Schuster, Handbook of Chaos Control (Wiley, 2008)

    Google Scholar 

  4. K. Pyragas, Phys. Lett. A 170(6), 421 (1992). doi:10.1016/0375-9601(92)90745-8

    Article  ADS  Google Scholar 

  5. N. Baba, A. Amann, E. Schöll, W. Just, Phys. Rev. Lett. 89, 074101 (2002). doi:10.1103/PhysRevLett.89.074101

    Article  ADS  Google Scholar 

  6. J. Unkelbach, A. Amann, W. Just, E. Schöll, Phys. Rev. E 68, 026204 (2003). doi:10.1103/PhysRevE.68.026204

    Article  ADS  Google Scholar 

  7. T. Pierre, G. Bonhomme, A. Atipo, Phys. Rev. Lett. 76, 2290 (1996). doi:10.1103/PhysRevLett.76.2290

    Article  ADS  Google Scholar 

  8. H. Friedel, R. Grauer, K.H. Spatschek, Phys. Plasmas 5(9), 3187 (1998). doi:10.1063/1.872987

    Article  ADS  Google Scholar 

  9. P.V. Paulau, D. Gomila, T. Ackemann, N.A. Loiko, W.J. Firth, Phys. Rev. E 78, 016212 (2008)

    Article  ADS  Google Scholar 

  10. K. Green, B. Krauskopf, F. Marten, D. Lenstra, SIAM, J. Appl. Dyn. Syst. 8(1), 222 (2009). doi:10.1137/080712982

    Google Scholar 

  11. M. Tlidi, A.G. Vladimirov, D. Pieroux, D. Turaev, Phys. Rev. Lett. 103, 103904 (2009)

    Article  ADS  Google Scholar 

  12. A. Pimenov, A.G. Vladimirov, S.V. Gurevich, K. Panajotov, G. Huyet, M. Tlidi, Phys. Rev. A 88, 053830 (2013). doi:10.1103/PhysRevA.88.053830

    Article  ADS  Google Scholar 

  13. J. Javaloyes, M. Marconi, M. Giudici, Phys. Rev. A 90, 023838 (2014). doi:10.1103/PhysRevA.90.023838

    Article  ADS  Google Scholar 

  14. A.G. Vladimirov, A. Pimenov, S.V. Gurevich, K. Panajotov, E. Averlant, M. Tlidi, Phil. Trans. R. Soc. London A 372(2027) (2014). doi:10.1098/rsta.2014.0013

    Google Scholar 

  15. K. Panajotov, M. Tlidi, Opt. Lett. 39(16), 4739 (2014). doi:10.1364/OL.39.004739

    Article  ADS  Google Scholar 

  16. M. Kehrt, P. Hövel, V. Flunkert, M.A. Dahlem, P. Rodin, E. Schöll, EPJ B 68(4), 557 (2009). doi:10.1140/epjb/e2009-00132-5

    Google Scholar 

  17. M.A. Dahlem, F.M. Schneider, E. Schöll, Chaos 18(2), 026110 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  18. M.A. Dahlem, R. Graf, A.J. Strong, J.P. Dreier, Y.A. Dahlem, M. Sieber, W. Hanke, K. Podoll, E. Schöll, Phys. D 239(11), 889 (2010)

    Article  Google Scholar 

  19. T. Erneux, G. Kozyreff, M. Tlidi, Phil. Trans. R. Soc. London A 368(1911), 483 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  20. J. Fort, V. Méndez, Phys. Rev. Lett. 89, 178101 (2002)

    Article  ADS  Google Scholar 

  21. V. Ortega-Cejas, J. Fort, V. Méndez, Ecology 85(1), 258 (2004)

    Article  Google Scholar 

  22. M. Bestehorn, E.V. Grigorieva, S.A. Kaschenko, Phys. Rev. E 70, 026202 (2004). doi:10.1103/PhysRevE.70.026202

    Article  ADS  MathSciNet  Google Scholar 

  23. K.A. Montgomery, M. Silber, Nonlinearity 17(6), 2225 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  24. C.M. Postlethwaite, M. Silber, Phys. D 236(1), 65 (2007). doi:10.1016/j.physd.2007.07.011

    Article  MathSciNet  Google Scholar 

  25. M. Stich, A. Casal, C. Beta, Phys. Rev. E 88, 042910 (2013). doi:10.1103/PhysRevE.88.042910

    Article  ADS  Google Scholar 

  26. D. Puzyrev, S. Yanchuk, A. Vladimirov, S. Gurevich, SIAM, J. Appl. Dyn. Syst. 13(2), 986 (2014). doi:10.1137/130944643

    Google Scholar 

  27. C. Tian, L. Zhang, Phys. Rev. E 88, 012713 (2013). doi:10.1103/PhysRevE.88.012713

    Article  ADS  Google Scholar 

  28. Y.N. Kyrychko, K.B. Blyuss, S.J. Hogan, E. Schöll, Chaos 19(4), 043126 (2009). doi:10.1063/1.3270048

    Article  ADS  Google Scholar 

  29. Q.S. Li, H.X. Hu, J. Chem. Phys 127(15), 154510 (2007). doi:10.1063/1.2792877

    Article  ADS  Google Scholar 

  30. R. Iri, Y. Tonosaki, K. Shitara, T. Ohta, J. Phys. Soc. Jpn. 83(2) (2013)

    Google Scholar 

  31. M. Tlidi, A. Sonnino, G. Sonnino, Phys. Rev. E 87, 042918 (2013). doi:10.1103/PhysRevE.87.042918

    Article  ADS  Google Scholar 

  32. S.V. Gurevich, Phys. Rev. E 87, 052922 (2013). doi:10.1103/PhysRevE.87.052922

    Article  ADS  Google Scholar 

  33. S.V. Gurevich, Phil. Trans. R. Soc. London A 372(2027) (2014). doi:10.1098/rsta.2014.0014

    Google Scholar 

  34. J. Swift, P.C. Hohenberg, Phys. Rev. A 15, 319 (1977). doi:10.1103/PhysRevA.15.319

    Article  ADS  Google Scholar 

  35. M. Hilali, G. Dewel, P. Borckmans, Phys. Lett. A 217(45), 263 (1996). doi:10.1016/0375-9601(96)00344-1

    Article  ADS  Google Scholar 

  36. R. Lefever, N. Barbier, P. Couteron, O. Lejeune, J. Theor. Biol. 261(2), 194 (2009). doi:10.1016/j.jtbi.2009.07.030

    Article  MathSciNet  Google Scholar 

  37. M. Tlidi, P. Mandel, R. Lefever, Phys. Rev. Lett. 73, 640 (1994)

    Article  ADS  Google Scholar 

  38. G. Kozyreff, S. Chapman, M. Tlidi, Phys. Rev. E 68(12), 152011 (2003)

    Google Scholar 

  39. N. Stoop, R. Lagrange, D. Terwagne, P. Reis, J. Dunkel, Nat. Mater. 14(3), 337 (2015). doi:10.1038/nmat4202

    Article  ADS  Google Scholar 

  40. M. Tlidi, A.G. Vladimirov, D. Turaev, G. Kozyreff, D. Pieroux, T. Erneux, Eur. Phys. J. D 59(1), 59 (2010)

    Article  ADS  Google Scholar 

  41. S.V. Gurevich, R. Friedrich, Phys. Rev. Lett. 110, 014101 (2013). doi:10.1103/PhysRevLett.110.014101

    Article  ADS  Google Scholar 

  42. P. Hövel, E. Schöll, Phys. Rev. E 72, 046203 (2005). doi:10.1103/PhysRevE.72.046203

    Article  Google Scholar 

  43. R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, Adv. Comput. Math. 5(1), 329 (1996)

    Article  MathSciNet  Google Scholar 

  44. D.J.B. Lloyd, B. Sandstede, D. Avitabile, A.R. Champneys, SIAM, J. Appl. Dyn. Syst. 7(3), 1049 (2008). doi:10.1137/070707622

    Google Scholar 

  45. D. Avitabile, D.J.B. Lloyd, J. Burke, E. Knobloch, B. Sandstede, SIAM, J. Appl. Dyn. Syst. 9(3), 704 (2010). doi:10.1137/100782747

    Google Scholar 

  46. L.A. Segel, J. Fluid Mech. 38(01), 203 (1969)

    Article  ADS  Google Scholar 

  47. M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65(3), 851 (1993)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623, Berlin, Germany

    Alexander Kraft

  2. Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Str. 9, 48149, Münster, Germany

    Svetlana V. Gurevich

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  1. Alexander Kraft
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  2. Svetlana V. Gurevich
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Correspondence to Alexander Kraft .

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Editors and Affiliations

  1. Inst für Theoretische Physik, Technische Universität Berlin, Berlin, Berlin, Germany

    Eckehard Schöll

  2. Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany

    Sabine H. L. Klapp

  3. Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany

    Philipp Hövel

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Kraft, A., Gurevich, S.V. (2016). Time-Delayed Feedback Control of Spatio-Temporal Self-Organized Patterns in Dissipative Systems. In: Schöll, E., Klapp, S., Hövel, P. (eds) Control of Self-Organizing Nonlinear Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-28028-8_21

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