Summary

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

2012

Session Number:C2L-D

Session:

Number:679

Control by Pyragas method with variable delay: from simple models to experiments

Aleksandar Gjurchinovski,  Thomas Jüngling,  Viktor Urumov,  

pp.679-682

Publication Date:

Online ISSN:2188-5079

DOI:10.15248/proc.1.679

PDF download (956.6KB)

We will report on our findings concerning generalization of the Pyragas feedback technique by introducing variability in the delays. In addition to the theoretical basis for the method, several examples will be provided starting from the simplest case of an unstable steady state of focus type, to stabilization of unstable points and orbits in several standard systems, and finally an experimental realization with an electronic circuit. Variability of the delay, whether temporal or distributed, deterministic or not, leads to significant extension of the domain of successful stabilization and increased robustness.

[1] K. Pyragas,“Continuous control of chaos by selfcontrolling feedback”, Phys. Lett. A, vol.170, pp.421-428, 1992.

[2] E. Schöll and H. G. Schuster (eds.),“Handbook of Chaos Control”, Wiley-VCH, Weinheim, 2008, 2nd edition.

[3] P. Hövel and E. Schöll, “Control of unstable steady states by time-delayed feedback methods”, Phys. Rev. E, vol.72, 046203-1-7, 2005.

[4] W. Michiels, V. Van Assche and S. Niculescu, “Stabilization of time-delay systems with a controlled time-varying delay and applications”, IEEE Trans. Autom. Control, vol.50, pp.493-504, 2005.

[5] A. Gjurchinovski and V. Urumov,“Stabilization of unstable steady states by variable-delay feedback control”, Europhys. Lett., vol.84, 40013-1-6, 2008.

[6] A. Ahlborn and U. Parlitz, “Stabilizing unstable steady states using multiple delay feedback control”, Phys. Rev. Lett., vol.93, pp.264101-1-4, 2004.

[7] A. Gjurchinovski and V. Urumov, “Stabilization of unstable steady states and unstable periodic orbits by feedback with variable delay”, to appear in Rom. J. Phys., 2013.

[8] K. Konishi, H. Kokame and N. Hara, “Stability analysis and design of amplitude death induced by a timevarying delay connection”, Phys. Lett. A, vol.374, pp.733-738, 2010.

[9] Y. Kyrychko, K. Blyuss and E. Schöll, “Amplitude death in systems of coupled oscillators with distributed-delay coupling”, Eur. J. Phys. B, vol.84, pp.307-315, 2011.

[10] O. E. Rössler, “An equation for continuous chaos”, Phys. Lett. A, vol.57, pp.397-398, 1976.

[11] H. G. Schuster and M. B. Stemmler, “Control of chaos by oscillating feedback”, Phys. Rev. E, vol.56, pp.6410-6417, 2005.

[12] J. L. Hindmarsh and R. M. Rose, “A model of neuronal bursting using three coupled first order differential equations”, Proc. R. Soc. London B, vol.221, pp.87-102, 1984.

[13] M. Rosenblum and A. Pikovsky, “Delayed feedback control of collective synchrony: An approach to suppression of pathological brain rhythms”, Phys. Rev. E, vol.70, 041904-1-11, 2004.

[14] A. Gjurchinovski, V. Urumov and Z. Vasilkoski, “Desynchronization of systems of coupled Hindmarsh-Rose oscillators”, Bulg. J. Phys., vol.38, pp.303-308, 2011.

[15] T. Jüngling, H. Benner, H. Shirahama and K. Fukushima, “Complete chaotic synchronization and exclusion of mutual Pyragas control in two delaycoupled Rössler-type oscillators”, Phys. Rev. E, vol.84, 056208-1-9, 2011.

[16] T. Jüngling, A. Gjurchinovski and V. Urumov, “Experimental control of chaos by variable and distributed delay feedback”, arXiv:1202.0519.

[17] J. E. S. Socolar, D. W. Sukow and D. J. Gauthier, “Stabilizing unstable periodic orbits in fast dynamical systems”, Phys. Rev. E, vol.50, 3245-3248, 1994.