Skip to main content

Time-Delay Feedback Control of an Oscillatory Medium

  • 494 Accesses

Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI,volume 20)

Abstract

The supercritical Hopf bifurcation is one of the simplest ways in which a stationary state of a nonlinear system can undergo a transition to stable self-sustained oscillations. At the bifurcation point, a small-amplitude limit cycle is born, which already at onset displays a finite frequency. If we consider a reaction-diffusion system that undergoes a supercritical Hopf bifurcation, its dynamics is described by the complex Ginzburg-Landau equation (CGLE). Here, we study such a system in the parameter regime where the CGLE shows spatio-temporal chaos. We review a type of time-delay feedback methods which is suitable to suppress chaos and replace it by other spatio-temporal solutions such as uniform oscillations, plane waves, standing waves, and the stationary state.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-16585-7_1
  • Chapter length: 17 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   84.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-16585-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD   109.99
Price excludes VAT (USA)
Fig. 1

Figure from Ref. [71]

Fig. 2
Fig. 3

References

  1. Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851 (1993)

    CrossRef  Google Scholar 

  2. Kapral, R., Showalter, K. (eds.): Chemical Waves and Patterns. Kluwer Academic, Dordrecht (1995)

    MATH  Google Scholar 

  3. Walgraef, D.: Spatio-Temporal Pattern Formation. Springer, New York (1997)

    CrossRef  Google Scholar 

  4. Hoyle, R.: Pattern Formation: An Introduction to Methods. Cambridge University Press, Cambridge (2006)

    CrossRef  Google Scholar 

  5. Busse, F.H., Kramer, L. (eds.): Nonlinear Evolution of Spatio-Temporal Structures in Dissipative Continuous Systems. Plenum Press, New York (1990)

    Google Scholar 

  6. Li, Y.J., Oslonovitch, J., Mazouz, N., Plenge, F., Krischer, K., Ertl, G.: Turing-type patterns on electrode surfaces science. Science 291, 2395 (2001)

    CrossRef  Google Scholar 

  7. Schöll, E.: Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors. Cambridge University Press, Cambridge (2001)

    CrossRef  Google Scholar 

  8. Engel, H., Niedernostheide, F., Purwins, H., Schöll, E. (eds.): Self-Organization in Activator-Inhibitor-Systems: Semiconductors, Gas-Discharge and Chemical Active Media. Wissenschaft und Technik, Berlin (1996)

    Google Scholar 

  9. Ackemann, T., Lange, W.: Optical pattern formation in alkali metal vapors: Mechanisms, phenomena and use. Appl. Phys. B 72, 21 (2001)

    CrossRef  Google Scholar 

  10. Umbanhowar, P.B., Melo, F., Swinney, H.L.: Localized excitations in a vertically vibrated granular layer. Nature 382, 793 (1996)

    CrossRef  Google Scholar 

  11. Winfree, A.T.: Chemical waves and fibrillating hearts: discovery by computation. J. Biosci. 27, 465 (2002)

    CrossRef  Google Scholar 

  12. Haken, H.: Brain Dynamics. Springer, Berlin (2002)

    MATH  Google Scholar 

  13. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989)

    CrossRef  Google Scholar 

  14. Goldbeter, A.: Biochemical Oscillations and Cellular Rhythms. Cambridge University Press, Cambridge (1996)

    CrossRef  Google Scholar 

  15. Lechleiter, J., Girard, S., Peralta, E., Clapham, D.: Spiral calcium wave propagation and annihilation in Xenopus laevis oocytes. Science 252, 123 (1991)

    CrossRef  Google Scholar 

  16. Steinbock, O., Müller, S.C.: Spatial Attractors in Aggregation Patterns of Dictyostelium discoideum. Z. Naturforsch. C 50, 275 (1995)

    Google Scholar 

  17. Blasius, B., Huppert, A., Stone, L.: Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399, 354 (1999)

    CrossRef  Google Scholar 

  18. Beta, C., Kruse, K.: Intracellular Oscillations and Waves. Annu. Rev. Condens. Matter Phys. 8(1), 239 (2017)

    CrossRef  Google Scholar 

  19. Strogatz, S.H.: Nonlinear Dynamics and Chaos. Addison-Wesley, Reading (1994)

    Google Scholar 

  20. Belousov, B.P.: Sbornik referatov po Radiatsionnoi Meditsine, Medgiz, Moscow, p. 145. (Collections of abstracts on radiation medicine) (in Russian) (1959)

    Google Scholar 

  21. Zaikin, A.N., Zhabotinsky, A.M.: Concentration Wave Propagation in Two-dimensional Liquid-phase Self-oscillating System. Nature (London) 255, 535 (1970)

    CrossRef  Google Scholar 

  22. Winfree, A.T.: Spiral waves of chemical activity. Science 175, 634 (1972)

    CrossRef  Google Scholar 

  23. Winfree, A.T.: Scroll-shaped waves of chemical activity in three dimensions. Science 181, 937 (1973)

    CrossRef  Google Scholar 

  24. Vanag, V.K., Epstein, I.R.: Inwardly rotating spiral waves in a reaction-diffusion system. Science 294, 835 (2001)

    CrossRef  Google Scholar 

  25. Vanag, V.K., Zhabotinsky, A.M., Epstein, I.R.: Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction. Phys. Rev. Lett. 86, 552 (2001)

    CrossRef  Google Scholar 

  26. Jakubith, S., Rotermund, H.H., Engel, W., Von Oertzen, A., Ertl, G.: Spatiotemporal concentration patterns in a surface reaction: Propagating and standing waves, rotating spirals, and turbulence. Phys. Rev. Lett. 65, 3013 (1990)

    CrossRef  Google Scholar 

  27. Rotermund, H.H.: Imaging of dynamic processes on surfaces by light. Surf. Sci. Rep. 29, 265 (1997)

    CrossRef  Google Scholar 

  28. Kim, M., Bertram, M., Pollmann, M., von Oertzen, A., Mikhailov, A.S., Rotermund, H.H., Ertl, G.: Controlling chemical turbulence by global delayed feedback: pattern formation in catalytic co oxidation on Pt(110). Science 292, 1357 (2001)

    CrossRef  Google Scholar 

  29. Schenk, C.P., Or-Guil, M., Bode, M., Purwins, H.G.: Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains. Phys. Rev. Lett. 78, 3781 (1997)

    CrossRef  Google Scholar 

  30. Alonso, S., Sagués, F., Mikhailov, A.S.: Taming winfree turbulence of scroll waves in excitable media. Science 299, 1722 (2003)

    CrossRef  Google Scholar 

  31. Tyson, J.J., Alexander, K.A., Manoranjan, V.S., Murray, J.D.: Spiral waves of cyclic AMP in a model of slime mold aggregation. Physica D 34, 193 (1989)

    MathSciNet  CrossRef  Google Scholar 

  32. Mikhailov, A.S.: Foundations of Synergetics I, 2nd edn. Springer, Berlin (1994)

    CrossRef  Google Scholar 

  33. Aranson, I.S., Kramer, L.: The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 99 (2002)

    MathSciNet  CrossRef  Google Scholar 

  34. Chaté, H., Manneville, P.: Phase diagram of the two-dimensional complex Ginzburg-Landau equation. Physica A 224, 348 (1996)

    MathSciNet  CrossRef  Google Scholar 

  35. Ipsen, M., Kramer, L., Sørensen, P.G.: Amplitude equations for description of chemical reaction-diffusion systems. Phys. Rep. 337, 193 (2000)

    CrossRef  Google Scholar 

  36. Schöll, E., Schuster, H.G. (eds.): Handbook of Chaos Control. Wiley-VCH, Weinheim (2007)

    Google Scholar 

  37. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64(11), 1196 (1990)

    MathSciNet  CrossRef  Google Scholar 

  38. Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421 (1992)

    CrossRef  Google Scholar 

  39. Socolar, J.E.S., Sukow, D.W., Gauthier, D.J.: Stabilizing unstable periodic orbits in fast dynamical systems. Phys. Rev. E 50, 3245 (1994)

    CrossRef  Google Scholar 

  40. Erneux, T.: Applied Delay Differential Equations. Springer, New York (2009)

    MATH  Google Scholar 

  41. Lu, W., Yu, D., Harrison, R.G.: Control of patterns in spatiotemporal chaos in optics. Phys. Rev. Lett. 76, 3316 (1996)

    CrossRef  Google Scholar 

  42. Bleich, M.E., Hochheiser, D., Moloney, J.V., Socolar, J.E.S.: Controlling extended systems with spatially filtered, time-delayed feedback. Phys. Rev. E 55, 2119 (1997)

    CrossRef  Google Scholar 

  43. Beta, C., Mikhailov, A.S., Rotermund, H.H., Ertl, G.: Defect-mediated turbulence in a catalytic surface reaction. Europhys. Lett. 75, 868 (2006)

    CrossRef  Google Scholar 

  44. Krefting, D., Beta, C.: Theoretical analysis of defect-mediated turbulence in a catalytic surface reaction. Phys. Rev. E 81(3), 036209 (2010)

    Google Scholar 

  45. Bertram, M., Beta, C., Pollmann, M., Mikhailov, A.S., Rotermund, H.H., Ertl, G.: Pattern formation on the edge of chaos: experiments with CO oxidation on a Pt(110) surface under global delayed feedback. Phys. Rev. E 67, 036208 (2003)

    Google Scholar 

  46. Beta, C., Bertram, M., Mikhailov, A.S., Rotermund, H.H., Ertl, G.: Controlling turbulence in a surface chemical reaction by time-delay autosynchronization. Phys. Rev. E 67, 046224 (2003)

    Google Scholar 

  47. Bertram, M., Beta, C., Rotermund, H.H., Ertl, G.: Complex patterns in a periodically forced surface reaction. J. Phys. Chem. B 107(35), 9610 (2003)

    CrossRef  Google Scholar 

  48. Bodega, P.S., Kaira, P., Beta, C., Krefting, D., Bauer, D., Mirwald-Schulz, B., Punckt, C., Rotermund, H.H.: High frequency periodic forcing of the oscillatory catalytic CO oxidation on Pt (110). New J. Phys. 9, 61 (2007)

    CrossRef  Google Scholar 

  49. Beta, C., Moula, M.G., Mikhailov, A.S., Rotermund, H.H., Ertl, G.: Excitable CO oxidation on Pt(110) under nonuniform coupling. Phys. Rev. Lett. 93(18), 188302 (2004)

    Google Scholar 

  50. Wolff, J., Stich, M., Beta, C., Rotermund, H.H.: Laser-induced target patterns in the oscillatory CO oxidation on Pt(110). J. Phys. Chem. B 108(38), 14282 (2004)

    CrossRef  Google Scholar 

  51. Punckt, C., Stich, M., Beta, C., Rotermund, H.H.: Suppression of spatiotemporal chaos in the oscillatory CO oxidation on Pt(110) by focused laser light. Phys. Rev. E 77(4), 046222 (2008)

    Google Scholar 

  52. Stich, M., Punckt, C., Beta, C., Rotermund, H.H.: Control of spatiotemporal chaos in catalytic CO oxidation by laser-induced pacemakers. Phil. Trans. R. Soc. Lond. A 366, 419 (2008)

    CrossRef  Google Scholar 

  53. Dahlem, M.A., Schneider, F.M., Schöll, E.: Failure of feedback as a putative common mechanism of spreading depolarizations in migraine and stroke. Chaos 18, 026110 (2008)

    MathSciNet  CrossRef  Google Scholar 

  54. Schneider, F.W., Schöll, E., Dahlem, M.A.: Controlling the onset of traveling pulses in excitable media by nonlocal spatial coupling and time-delayed feedback. Chaos 19, 015110 (2009)

    MathSciNet  CrossRef  Google Scholar 

  55. Fenton, F.H., Cherry, E.M., Hastings, H.M., Evans, S.J.: Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity. Chaos 12, 852 (2002)

    CrossRef  Google Scholar 

  56. Christini, D.J., Glass, L.: Introduction: Mapping and control of complex cardiac arrhythmias. Chaos 12, 732 (2002)

    CrossRef  Google Scholar 

  57. Franceschini, G., Bose, S., Schöll, E.: Control of chaotic spatiotemporal spiking by time-delay autosynchronization. Phys. Rev. E 60(5), 5426 (1999)

    CrossRef  Google Scholar 

  58. Beck, O., Amann, A., Schöll, E., Socolar, J.E.S., Just, W.: Comparison of time-delayed feedback schemes for spatiotemporal control of chaos in a reaction-diffusion system with global coupling. Phys. Rev. E 66, 016213 (2002)

    Google Scholar 

  59. Baba, N., Amann, A., Schöll, E., Just, W.: Giant improvement of time-delayed feedback control by spatio-temporal filtering. Phys. Rev. Lett. 89, 074101 (2002)

    Google Scholar 

  60. Unkelbach, J., Amann, A., Just, W., Schöll, E.: Time-delay autosynchronization of the spatiotemporal dynamics in resonant tunneling diodes. Phys. Rev. E 68, 026204 (2003)

    Google Scholar 

  61. Battogtokh, D., Mikhailov, A.: Controlling turbulence in the complex Ginzburg-Landau equation. Physica D 90, 84 (1996)

    MathSciNet  CrossRef  Google Scholar 

  62. Battogtokh, D., Preusser, A., Mikhailov, A.: Controlling turbulence in the complex Ginzburg-Landau equation II. Two-dimensional systems. Phys. D 106, 327 (1997)

    MathSciNet  CrossRef  Google Scholar 

  63. Beta, C., Mikhailov, A.S.: Controlling spatiotemporal chaos in oscillatory reaction-diffusion systems by time-delay autosynchronization. Phys. D 199, 173 (2004)

    MathSciNet  CrossRef  Google Scholar 

  64. Bleich, M.E., Socolar, J.E.S.: Controlling spatiotemporal dynamics with time-delay feedback. Phys. Rev. E 54, R17 (1996)

    CrossRef  Google Scholar 

  65. Harrington, I., Socolar, J.E.S.: Limitation on stabilizing plane waves via time-delay feedback. Phys. Rev. E 64, 056206 (2001)

    Google Scholar 

  66. Montgomery, K.A., Silber, M.: Feedback control of travelling wave solutions of the complex Ginzburg-Landau equation. Nonlinearity 17, 2225 (2004)

    MathSciNet  CrossRef  Google Scholar 

  67. Postlethwaite, C.M., Silber, M.: Spatial and temporal feedback control of traveling wave solutions of the two-dimensional complex Ginzburg-Landau equation. Phys. D 236, 65 (2007)

    MathSciNet  CrossRef  Google Scholar 

  68. Stich, M., Casal, A.C., Díaz, J.I.: Control of turbulence in oscillatory reaction-diffusion systems through a combination of global and local feedback. Phys. Rev. E 76, 036209 (2007)

    Google Scholar 

  69. Stich, M., Beta, C.: Control of pattern formation by time-delay feedback with global and local contributions. Phys. D 239, 1681 (2010)

    MathSciNet  CrossRef  Google Scholar 

  70. Stich, M., Casal, A., Beta, C.: Stabilization of standing waves through time-delay feedback. Phys. Rev. E 88, 042910 (2013)

    Google Scholar 

  71. Stich, M., Elec, J.: Comments on multiple oscillatory solutions in systems with time-delay feedback. Diff. Eqs. Conf. 22, 99 (2015)

    Google Scholar 

  72. Stich, M., Chattopadhyay, A.K.: Noise-induced standing waves in oscillatory systems with time-delayed feedback. Phys. Rev. E 93, 052221 (2016)

    Google Scholar 

  73. Hövel, P., Schöll, E.: Control of unstable steady states by time-delayed feedback methods. Phys. Rev. E 72, 046203 (2005)

    Google Scholar 

  74. Koseska, A., Volkov, E., Kurths, J.: Oscillation quenching mechanisms: amplitude vs. oscillation death. Phys. Rep. 531, 173 (2013)

    MathSciNet  CrossRef  Google Scholar 

  75. Falcke, M., Engel, H., Neufeld, M.: Cluster formation, standing waves, and stripe patterns in oscillatory active media with local and global coupling. Phys. Rev. E 52, 763 (1995)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michael Stich .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Stich, M., Beta, C. (2019). Time-Delay Feedback Control of an Oscillatory Medium. In: Carballido-Landeira, J., Escribano, B. (eds) Biological Systems: Nonlinear Dynamics Approach. SEMA SIMAI Springer Series, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-030-16585-7_1

Download citation