Delay Stabilization of Rotating Waves without Odd Number Limitation

Reviews of Nonlinear Dynamics and Complexity

Volumn , Issue , 2009, Pages 53-68

  Fiedler, Bernold ^a   Flunkert, Valentin ^b   Georgi, Marc ^a   Hövel, Philipp ^b   Schöll, Eckehard ^b  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

^b TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

Stability considerations; Stabilization; Symmetry consideration; Time delayed feedback control; Unstable periodic orbits (UPOs)

Indexed keywords

EID: 76149125611     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9783527626359.ch2     Document Type: Chapter

References (44)

1. E. Schöll and H. G. Schuster (Eds.): Handbook of Chaos Control, second completely revised and enlarged edition. Wiley-VCH,Weinheim, 2007.
2. E. Ott, C. Grebogi, and J. A. Yorke: Controlling chaos, Phys. Rev. Lett. 64, 1196 (1990).
3. S. Boccaletti, C. Grebogi, Y. C. Lai, H. Mancini, and D. Maza: The control of chaos: theory and applications, Phys. Rep. 329,103, (2000).
4. D. J. Gauthier: Resource letter: Controlling chaos, Am. J. Phys. 71,750, (2003).
5. K. Pyragas: Continuous control of chaos by self-controlling feedback, Phys. Lett. A 170,421, (1992).
6. K. Pyragas and A. Tamaševicius: Experimental control of chaos by delayed self-controlling feedback, Phys. Lett. A 180,99, (1993).
7. S. Bielawski, D. Derozier, and P. Glorieux: Controlling unstable periodic orbits by a delayed continuous feedback, Phys. Rev. E 49, R971 (1994).
8. T. Pierre, G. Bonhomme, and A. Atipo: Controlling the chaotic regime of nonlinear ionization waves using time-delay autosynchronisation method, Phys. Rev. Lett. 76,2290, (1996).
9. K. Hall, D. J. Christini, M. Tremblay, J. J. Collins, L. Glass, and J. Billette: Dynamic control of cardiac alterans, Phys. Rev. Lett. 78,4518, (1997).
10. D.W. Sukow, M. E. Bleich, D. J. Gauthier, and J. E. S. Socolar: Controlling chaos in a fast diode resonator using time-delay autosynchronisation: Experimental observations and theoretical analysis, Chaos 7,560, (1997).
11. O. Lüthje, S.Wolff, and G. Pfister: Control of chaotic taylor-couette flow with time-delayed feedback, Phys. Rev. Lett. 86,1745, (2001).
12. P. Parmananda, R. Madrigal, M. Rivera, L.Nyikos, I. Z. Kiss, and V. Gáspár: Stabilization of unstable steady states and periodic orbits in an electrochemical system using delayed-feedback control, Phys. Rev. E 59,5266, (1999).
13. J. M. Krodkiewski and J. S. Faragher: Stabilization of motion of helicopter rotor blades using delayed feedback-modelling, computer simulation and experimental verification, J. Sound Vib. 234,591, (2000).
14. T. Fukuyama, H. Shirahama, and Y. Kawai: Dynamical control of the chaotic state of the current-driven ion acoustic instability in a laboratory plasma using delayed feedback, Phys. Plasmas 9,4525, (2002).
15. C. von Loewenich, H. Benner, and W. Just: Experimental relevance of global properties of time-delayed feedback control, Phys. Rev. Lett. 93,174101, (2004).
16. M. G. Rosenblum and A. Pikovsky: Controlling synchronization in an ensemble of globally coupled oscillators, Phys. Rev. Lett. 92,114102, (2004).
17. O. V. Popovych, C. Hauptmann, and P. A. Tass: Effective desynchronization by nonlinear delayed feedback, Phys. Rev. Lett. 94,164102, (2005).
18. S. Schikora, P.Hövel,H. J. Wünsche, E. Schöll, and F. Henneberger: All-optical noninvasive control of unstable steady states in a semiconductor laser, Phys. Rev. Lett. 97,213902, (2006).
19. E. Schöll, J. Hizanidis, P. Hövel, and G. Stegemann: Pattern formation in semiconductors under the influence of time-delayed feedback control and noise, in Analysis and control of complex nonlinear processes in physics, chemistry and biology, edited by L. Schimansky-Geier, B. Fiedler, J. Kurths, and E. Schöll. World Scientific, Singapore, 2007, pp. 135-183.
20. W. Just, T. Bernard, M. Ostheimer, E. Reibold, and H. Benner: Mechanism of time-delayed feedback control, Phys. Rev. Lett. 78,203, (1997).
21. H. Nakajima: On analytical properties of delayed feedback control of chaos, Phys. Lett. A 232,207, (1997).
22. H. Nakajima and Y. Ueda: Limitation of generalized delayed feedback control, Physica D 111,143, (1998).
23. I. Harrington and J. E. S. Socolar: Limitation on stabilizing plane waves via time-delay feedback, Phys. Rev. E 64,056206, (2001).
24. K. Pyragas, V. Pyragas, and H. Benner: Delayed feedback control of dynamical systems at subcritical Hopf bifurcation, Phys. Rev. E 70,056222, (2004).
25. V. Pyragas and K. Pyragas: Delayed feedback control of the Lorenz system: An analytical treatment at a subcritical Hopf bifurcation, Phys. Rev. E 73,036215, (2006).
26. J. E. S. Socolar, D. W. Sukow, and D. J. Gauthier: Stabilizing unstable periodic orbits in fast dynamical systems, Phys. Rev. E 50,3245, (1994).
27. H. G. Schuster and M. B. Stemmler: Control of chaos by oscillating feedback, Phys. Rev. E 56,6410, (1997).
28. H. Nakajima and Y. Ueda: Half-period delayed feedback control for dynamical systems with symmetries, Phys. Rev. E 58,1757, (1998).
29. K. Pyragas: Control of chaos via an unstable delayed feedback controller, Phys. Rev. Lett. 86,2265, (2001).
30. B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, and E. Schöll: Refuting the odd number limitation of time-delayed feedback control, Phys. Rev. Lett. 98,114101, (2007).
31. W. Just, B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, and E. Schöll: Beyond odd number limitation: a bifurcation analysis of time-delayed feedback control, Phys. Rev. E 76,02610, (2007).
32. O.Diekmann, S.A. van Gils, S. M. Verduyn Lunel, and H. O. Walther: Delay Equations. Springer-Verlag, New York, 1995.
33. A. Amann (private communication).
34. B. Fiedler: Global Bifurcation of Periodic Solutions with Symmetry. Springer-Verlag, Heidelberg, 1988.
35. A. Vanderbauwhede: Centre manifolds, normal forms and elementary bifurcations, Dynamics Reported 2,89, (1989).
36. B. Fiedler, B. Sandstede, A. Scheel, and C.Wulf: Bifurcation from Relative Equilibria of Noncompact Group Actions: Skew Products, Meanders, and Drifts, Documenta Mathematica 1,479, (1996).
37. B. Fiedler and A. Scheel: Dynamics of reaction-diffusion patterns, in Trends in Nonlinear Analysis, Festschrift dedicated to Willi Jäger for his 60th birthday, edited by M. Kirkilionis, R. Rannacher, and F. Tomi. Springer, Heidelberg, 2002.
38. B. Fiedler, M. Georgi, and N. Jangle: Spiral wave dynamics: Reaction and diffusion versus kinematics, in Analysis and Control of complex nonlinear Processes in Physics, Chemistry and Biology. World Scientific, Singapore, 2007.
39. M. Golubitsky and I. Stewart: Singularities and Groups in Bifurcation Theory. Volume 2, vol. 69. Springer-Verlag, New York, 1988.
40. S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radziunas, J. Sieber, H. J. Wünsche, and F. Henneberger: Nonlinear dynamics of semiconductor lasers with active optical feedback, Phys. Rev. E 69,016206, (2004).
41. S.Wieczorek, B. Krauskopf, and D. Lenstra: Unifying view of bifurcations in a semiconductor laser subject to optical injection, Optics Communications 172,279, (1999).
42. M. G. Rosenblum and A. Pikovsky: Delayed feedback control of collective synchrony: An approach to suppression of pathological brain rhythms, Phys. Rev. E 70,041904, (2004).
43. N. Baba, A. Amann, E. Schöll, and W. Just: Giant improvement of time-delayed feedback control by spatio-temporal filtering, Phys. Rev. Lett. 89,074101, (2002).
44. J. Sieber and B. Krauskopf: Control based bifurcation analysis for experiments, Nonlinear Dynamics, DOI: 10.1007/s11071-007-9217-2 (2007.