Summary

Proceedings of the 2012 International Symposium on Nonlinear Theory and its Applications

2012

Session Number:C2L-D

Session:

Number:683

A Twenty-Year Review of Time-Delay Feedback Control and Recent Developments

Kestutis Pyragas,  

pp.683-686

Publication Date:

Online ISSN:2188-5079

DOI:10.15248/proc.1.683

PDF download (279KB)

The delayed feedback control (DFC) method has been invented in 1992 (this year is a 20th anniversary). Following the original paper by Pyragas [Phys. Lett. A, 170, 421, 1992], more than 1500 papers devoted or related to the DFC have been published. Many different modifications of the algorithm have been proposed, and significant achievements are attained in the theory of the DFC. Although this theory is non-trivial, currently the mechanism of the DFC action is rather well understood, and the main limitations of the algorithm are established. The DFC has been successfully implemented in a number of experimental systems of different physical nature. The aim of this talk is to present a brief review of important modifications of the DFC algorithm, significant theoretical results and experimental implementations attained during the past twenty years. The recent results concerning adaptive modifications of the DFC and analytical achievements based on phase reduction of time-delay systems will be discussed as well.

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[2] E. Schöll and H. G. Shuster, Handbook of Chaos Control, (Wiley-VCH, Weinheim), 2008.

[3] K. Pyragas, “Delayed feedback control of chaos,” Philos. Trans. R. Soc. London, Ser. A, vol.364, pp.2309-2334, 2006.

[4] K. Yamasue, K. Kobayashib, H. Yamada, K. Matsushige, and T. Hikihara, “Controlling chaos in dynamic-mode atomic force microscope,” Phys. Lett. A, vol.373, pp.3140-3144, 2009.

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[7] K. Pyragas, “Control of chaos via extended delay feedback,” Phys. Lett. A, vol.206, pp.323-330, 1995.

[8] M. E. Bleich and J. E. S. Socolar, “Stability of periodic orbits controlled by time-delay feedback,” Phys. Lett. A, vol.210, pp.87-94, 1996.

[9] W. Just, T. Bernard, M. Ostheimer, E. Reibold, and H. Benner, “Mechanism of time-delayed feedback control,” Phys. Rev. Lett., vol.78, pp.203-206, 1997.

[10] K. Pyragas, “Analytical properties and optimization of time-delayed feedback control,” Phys. Rev. E., vol.66, p.026207, 2002.

[11] K. Pyragas, “Control of chaos via an unstable delayed feedback controller,” Phys. Rev. Lett., vol.86, pp.2265-2268, 2001.

[12] K. Pyragas, V. Pyragas, and H. Benner, “Delayed Feedback Control of Dynamical Systems at a Subcritical Hopf Bifurcation,” Phys. Rev. E., vol.70, p.056222, 2004.

[13] K. Pyragas and V. Pyragas, “Delayed feedback control of the Lorenz system: An analytical treatment at a subcritical Hopf bifurcation,” Phys. Rev. E., vol.73 p.036215, 2006.

[14] T. Pyragiene and K. Pyragas, “Delayed feedback control of forced self-sustained oscillations,” Phys. Rev. E., vol.72, p.026203, 2005.

[15] H. Nakajima, “On analytical properties of delayed feedback control of chaos,” Phys. Lett. A, vol.232, pp.207-210, 1997.

[16] B. Fiedler, V. Flunkert, M. Georgi, P. Hoevel, and E. Schöll, “Refuting the Odd-Number Limitation of Time-Delayed Feedback Control,” Phys. Rev. Lett., vol.98, p.114101, 2007.

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[18] D. Goldobin, M. Rosenblum, and A. Pikovsky, “Coherence of noisy oscillators with delayed feedback,” Physica A, vol.327, pp.124 128, 2003.

[19] D. Goldobin, M. Rosenblum, and A. Pikovsky, “Controlling oscillator coherence by delayed feedback,” Phys. Rev. E, vol.67, p.061119, 2003.

[20] N. B. Janson, A. G. Balanov, and E. Schöll, “Delayed Feedback as a Means of Control of Noise-Induced Motion,” Phys. Rev. Lett., vol.93, p.010601, 2004.

[21] A. G. Balanov, V. Beato, N. B. Janson, H. Engel, and E. Schöll, “Delayed feedback control of noiseinduced patterns in excitable media,” Phys. Rev. E, vol.74, p.016214, 2006.

[22] J. Pomplun, A. G. Balanov, and E. Schöll, “Longterm correlations in stochastic systems with extended time-delayed feedback,” Phys. Rev. E, vol.75, p.040101(R), 2007.

[23] S. Patidar, A. Pototsky, and N. B. Janson, “Controlling noise-induced behavior of excitable networks,” New Journal of Physics, vol.11, p.073001, 2009.

[24] D. S. Goldobin, “Anharmonic resonances with recursive delay feedback,” Phys. Lett. A, vol.375, pp.3410-3414, 2011.

[25] D. Hennig, “Current control in a tilted washboard potential via time-delayed feedback,” Phys. Rev. E, vol.79, p.041114, 2009.

[26] M. G. Rosenblum and A. S. Pikovsky, “Controlling Synchronization in an Ensemble of Globally Coupled Oscillators,” Phys. Rev. Lett., vol.92, p.114102, 2004.

[27] M. Rosenblum and A. Pikovsky, “Delayed feedback control of collective synchrony: An approach to suppression of pathological brain rhythms,” Phys. Rev. E, vol.70, p.041904, 2004.

[28] M. Rosenblum and A. Pikovsky, “Delayed feedback control of collective synchrony: An approach to suppression of pathological brain rhythms,” Phys. Rev. E, vol.70, p.041904, 2004.

[29] O. V. Popovych, C. Hauptmann, and P. A. Tass, “Effective Desynchronization by Nonlinear Delayed Feedback,” Phys. Rev. Lett., vol.94, p.164102, 2005.

[30] O. V. Popovych and P. A. Tass, “Synchronization control of interacting oscillatory ensembles by mixed nonlinear delayed feedback,” Phys. Rev. E, vol.82, p.026204, 2010.

[31] N. Baba, A. Amann, E. Schöll, and W. Just, “Giant Improvement of Time-Delayed Feedback Control by Spatio-Temporal Filtering,” Phys. Rev. Lett., vol.89, p.074101, 2002.

[32] O. Beck, A. Amann, E. Schöll, J. E. S. Socolar, and W. Just, “Comparison of time-delayed feedback schemes for spatiotemporal control of chaos in a reaction-diffusion system with global coupling,” Phys. Rev. E, vol.66, p.016213, 2002.

[33] A. G. Balanov, V. Beato, N. B. Janson, H. Engel, and E. Schöll, “Delayed feedback control of noiseinduced patterns in excitable media,” Phys. Rev. E, vol.74, p.016214, 2006.

[34] F. M. Schneider, E. Schöll, and M. A. Dahlem, “Controlling the onset of traveling pulses in excitable media by nonlocal spatial coupling and time-delayed feedback,” Chaos, vol.19, p.015110, 2009.

[35] A. Gjurchinovski and V. Urumov, “Stabilization of unstable steady states by variable-delay feedback control,” EPL, vol.84, p.40013, 2008.

[36] T. Jüngling, Gjurchinovski, and V. Urumov, “Experimental control of chaos by variable and distributed delay feedback,” http://arxiv.org/pdf/1202.0519.pdf, 2012.

[37] K. Höhne, H. Shirahama, C. U. Choe, H. Benner, K. Pyragas, and W. Just, “Global Properties in an Experimental Realization of Time-Delayed Feedback Control with an Unstable Control Loop,” Phys. Rev. Lett., vol.98, p.214102, 2007.

[38] A. Tamaševicius, G. Mikolaitis, V. Pyragas, and K. Pyragas, “Delayed feedback control of periodic orbits without torsion in nonautonomous chaotic systems: Theory and experiment,” Phys. Rev. E., vol.76, p.026203, 2007.

[39] V. Pyragas and K. Pyragas, “Using ergodicity of chaotic systems for improving the global properties of the delayed feedback control method,” Phys. Rev. E., vol.80, p.067201, 2009.

[40] V. Pyragas and K. Pyragas, “‘Adaptive modification of the delayed feedback control algorithm with a continuously varying time delay,” Phys. Lett. A, vol.375, pp.3866-3871, 2011.

[41] J. Lehnert, P. Hövel, V. Flunkert, P. Yu. Guzenko, A. L. Fradkov, and E. Schöll, “Adaptive tuning of feedback gain in time-delayed feedback control,” Chaos, vol.21, p.043111, 2011.

[42] V. Novicenko and K. Pyragas, “Phase reduction of weakly perturbrd limit cycle oscillations in time-delay systems,” Physica D, vol.241, pp.1090-1098, 2012.

[43] V. Novicenko and K. Pyragas, “Phase reduction theory-based treatment of extended delayed feedback control algorithm in the presence of a small time delay mismatch,” submitted to Phys. Rev. E, 2012.

[44] C. Loewenich, H. Benner, and W. Just “Experimental Relevance of Global Properties of Time-Delayed Feedback Control,” Phys. Rev. Lett., vol.93, p.174101, 2004.

[45] A. Kittel, J. Parisi, and K. Pyragas “Delayed feedback control of chaos by self-adapted delay time,” Phys. Lett. A, vol.198, pp.433-436, 1995.

[46] W. Just, D. Reckwerth, J. Möckel, E. Reibold, and H. Benner “Delayed Feedback Control of Periodic Orbits in Autonomous Systems,” Phys. Rev. Lett., vol.81, p.562, 1998.