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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851 (1993)
Kapral, R., Showalter, K. (eds.): Chemical Waves and Patterns. Kluwer Academic, Dordrecht (1995)
Walgraef, D.: Spatio-Temporal Pattern Formation. Springer, New York (1997)
Hoyle, R.: Pattern Formation: An Introduction to Methods. Cambridge University Press, Cambridge (2006)
Busse, F.H., Kramer, L. (eds.): Nonlinear Evolution of Spatio-Temporal Structures in Dissipative Continuous Systems. Plenum Press, New York (1990)
Li, Y.J., Oslonovitch, J., Mazouz, N., Plenge, F., Krischer, K., Ertl, G.: Turing-type patterns on electrode surfaces science. Science 291, 2395 (2001)
Schöll, E.: Nonlinear Spatio-Temporal Dynamics and Chaos in Semiconductors. Cambridge University Press, Cambridge (2001)
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)
Ackemann, T., Lange, W.: Optical pattern formation in alkali metal vapors: Mechanisms, phenomena and use. Appl. Phys. B 72, 21 (2001)
Umbanhowar, P.B., Melo, F., Swinney, H.L.: Localized excitations in a vertically vibrated granular layer. Nature 382, 793 (1996)
Winfree, A.T.: Chemical waves and fibrillating hearts: discovery by computation. J. Biosci. 27, 465 (2002)
Haken, H.: Brain Dynamics. Springer, Berlin (2002)
Murray, J.D.: Mathematical Biology. Springer, Berlin (1989)
Goldbeter, A.: Biochemical Oscillations and Cellular Rhythms. Cambridge University Press, Cambridge (1996)
Lechleiter, J., Girard, S., Peralta, E., Clapham, D.: Spiral calcium wave propagation and annihilation in Xenopus laevis oocytes. Science 252, 123 (1991)
Steinbock, O., Müller, S.C.: Spatial Attractors in Aggregation Patterns of Dictyostelium discoideum. Z. Naturforsch. C 50, 275 (1995)
Blasius, B., Huppert, A., Stone, L.: Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399, 354 (1999)
Beta, C., Kruse, K.: Intracellular Oscillations and Waves. Annu. Rev. Condens. Matter Phys. 8(1), 239 (2017)
Strogatz, S.H.: Nonlinear Dynamics and Chaos. Addison-Wesley, Reading (1994)
Belousov, B.P.: Sbornik referatov po Radiatsionnoi Meditsine, Medgiz, Moscow, p. 145. (Collections of abstracts on radiation medicine) (in Russian) (1959)
Zaikin, A.N., Zhabotinsky, A.M.: Concentration Wave Propagation in Two-dimensional Liquid-phase Self-oscillating System. Nature (London) 255, 535 (1970)
Winfree, A.T.: Spiral waves of chemical activity. Science 175, 634 (1972)
Winfree, A.T.: Scroll-shaped waves of chemical activity in three dimensions. Science 181, 937 (1973)
Vanag, V.K., Epstein, I.R.: Inwardly rotating spiral waves in a reaction-diffusion system. Science 294, 835 (2001)
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)
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)
Rotermund, H.H.: Imaging of dynamic processes on surfaces by light. Surf. Sci. Rep. 29, 265 (1997)
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)
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)
Alonso, S., Sagués, F., Mikhailov, A.S.: Taming winfree turbulence of scroll waves in excitable media. Science 299, 1722 (2003)
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)
Mikhailov, A.S.: Foundations of Synergetics I, 2nd edn. Springer, Berlin (1994)
Aranson, I.S., Kramer, L.: The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74, 99 (2002)
Chaté, H., Manneville, P.: Phase diagram of the two-dimensional complex Ginzburg-Landau equation. Physica A 224, 348 (1996)
Ipsen, M., Kramer, L., Sørensen, P.G.: Amplitude equations for description of chemical reaction-diffusion systems. Phys. Rep. 337, 193 (2000)
Schöll, E., Schuster, H.G. (eds.): Handbook of Chaos Control. Wiley-VCH, Weinheim (2007)
Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64(11), 1196 (1990)
Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421 (1992)
Socolar, J.E.S., Sukow, D.W., Gauthier, D.J.: Stabilizing unstable periodic orbits in fast dynamical systems. Phys. Rev. E 50, 3245 (1994)
Erneux, T.: Applied Delay Differential Equations. Springer, New York (2009)
Lu, W., Yu, D., Harrison, R.G.: Control of patterns in spatiotemporal chaos in optics. Phys. Rev. Lett. 76, 3316 (1996)
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)
Beta, C., Mikhailov, A.S., Rotermund, H.H., Ertl, G.: Defect-mediated turbulence in a catalytic surface reaction. Europhys. Lett. 75, 868 (2006)
Krefting, D., Beta, C.: Theoretical analysis of defect-mediated turbulence in a catalytic surface reaction. Phys. Rev. E 81(3), 036209 (2010)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Christini, D.J., Glass, L.: Introduction: Mapping and control of complex cardiac arrhythmias. Chaos 12, 732 (2002)
Franceschini, G., Bose, S., Schöll, E.: Control of chaotic spatiotemporal spiking by time-delay autosynchronization. Phys. Rev. E 60(5), 5426 (1999)
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)
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)
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)
Battogtokh, D., Mikhailov, A.: Controlling turbulence in the complex Ginzburg-Landau equation. Physica D 90, 84 (1996)
Battogtokh, D., Preusser, A., Mikhailov, A.: Controlling turbulence in the complex Ginzburg-Landau equation II. Two-dimensional systems. Phys. D 106, 327 (1997)
Beta, C., Mikhailov, A.S.: Controlling spatiotemporal chaos in oscillatory reaction-diffusion systems by time-delay autosynchronization. Phys. D 199, 173 (2004)
Bleich, M.E., Socolar, J.E.S.: Controlling spatiotemporal dynamics with time-delay feedback. Phys. Rev. E 54, R17 (1996)
Harrington, I., Socolar, J.E.S.: Limitation on stabilizing plane waves via time-delay feedback. Phys. Rev. E 64, 056206 (2001)
Montgomery, K.A., Silber, M.: Feedback control of travelling wave solutions of the complex Ginzburg-Landau equation. Nonlinearity 17, 2225 (2004)
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)
Stich, M., Casal, A.C., DÃaz, J.I.: Control of turbulence in oscillatory reaction-diffusion systems through a combination of global and local feedback. Phys. Rev. E 76, 036209 (2007)
Stich, M., Beta, C.: Control of pattern formation by time-delay feedback with global and local contributions. Phys. D 239, 1681 (2010)
Stich, M., Casal, A., Beta, C.: Stabilization of standing waves through time-delay feedback. Phys. Rev. E 88, 042910 (2013)
Stich, M., Elec, J.: Comments on multiple oscillatory solutions in systems with time-delay feedback. Diff. Eqs. Conf. 22, 99 (2015)
Stich, M., Chattopadhyay, A.K.: Noise-induced standing waves in oscillatory systems with time-delayed feedback. Phys. Rev. E 93, 052221 (2016)
Hövel, P., Schöll, E.: Control of unstable steady states by time-delayed feedback methods. Phys. Rev. E 72, 046203 (2005)
Koseska, A., Volkov, E., Kurths, J.: Oscillation quenching mechanisms: amplitude vs. oscillation death. Phys. Rep. 531, 173 (2013)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
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
DOI: https://doi.org/10.1007/978-3-030-16585-7_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-16584-0
Online ISBN: 978-3-030-16585-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)