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January  2016, 36(1): 451-467. doi: 10.3934/dcds.2016.36.451

Eliminating restrictions of time-delayed feedback control using equivariance

Isabelle Schneider 1, and  Matthias Bosewitz 1,

1. 

Institut für Mathematik, Freie Universität Berlin, 14195 Berlin, Germany, Germany

Received  June 2014 Revised  May 2015 Published  June 2015

Pyragas control is a widely used time-delayed feedback control for the stabilization of periodic orbits in dynamical systems. In this paper we investigate how we can use equivariance to eliminate restrictions of Pyragas control, both to select periodic orbits for stabilization by their spatio-temporal pattern and to render Pyragas control possible at all for those orbits. Another important aspect is the optimization of equivariant Pyragas control, i.e. to construct larger control regions. The ring of $n$ identical Stuart-Landau oscillators coupled diffusively in a bidirectional ring serves as our model.
Keywords: equivariant Hopf bifurcation, spatio-temporal patterns, coupled oscillators., Time-delayed feedback, Pyragas control, networks, pattern-selective.
Mathematics Subject Classification: Primary: 34H15; Secondary: 34K2.
Citation: Isabelle Schneider, Matthias Bosewitz. Eliminating restrictions of time-delayed feedback control using equivariance. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 451-467. doi: 10.3934/dcds.2016.36.451
References:
[1]

M. Bosewitz, Stabilisierung gekoppelter Oszillatoren durch verzögerte Rückkopplungs-kontrolle, (German) [Stabilization of coupled oscillators by delayed feedback-control] Bachelor Thesis, Freie Universität Berlin, 2013. Google Scholar

[2]

K. Bubolz, Stabilisierung periodischer Orbits im System zweier gekoppelter Oszillatoren durch zeitverzögerte Rückkopplungskontrolle, (German) [Stabilization of periodic orbits in a system of two coupled oscillators by time-delayed feedback-control] Bachelor Thesis, Freie Universität Berlin, 2013. Google Scholar

[3]

B. Fiedler, Global Bifurcation of Periodic Solutions with Symmetry, Lect. Notes Math. 1309, Springer-Verlag, Heidelberg, 1988.  Google Scholar

[4]

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 (2007), 114101. doi: 10.1103/PhysRevLett.98.114101.  Google Scholar

[5]

B. Fiedler, V. Flunkert, P. Hövel and E. Schöll, Delay stabilization of periodic orbits in coupled oscillator systems, Phil. Trans. R. Soc. A, 368 (2010), 319-341. doi: 10.1098/rsta.2009.0232.  Google Scholar

[6]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Applied Mathematical Sciences Vol. 110, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar

[7]

M. Golubitsky, I. Stewart and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. 2. AMS 69, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-4574-2.  Google Scholar

[8]

M. Golubitsky and I. Stewart, The Symmetry Perspective. From Equilibrium to Chaos in Phase Space and Physical Space, Progress in Mathematics, Volume 200, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8167-8.  Google Scholar

[9]

C. M. Postlethwaite, G. Brown and M. Silber, Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems, Phil. Trans. R. Soc. A, 371 (2013), 20120467, 20pp. doi: 10.1098/rsta.2012.0467.  Google Scholar

[10]

K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170 (1992), 421-428. Google Scholar

[11]

K. Pyragas, A twenty-year review of time-delay feedback control and recent developments, in International Symposium on Nonlinear Theory and its Applications, 1 (2012), 683-686. doi: 10.15248/proc.1.683.  Google Scholar

[12]

I. Schneider, Delayed feedback control of three diffusively coupled Stuart-Landau oscillators: a case study in equivariant Hopf bifurcation, Phil. Trans. R. Soc. A, 371 (2013), 20120472, 10pp. doi: 10.1098/rsta.2012.0472.  Google Scholar

[13]

I. Schneider, Equivariant Pyragas Control, Master Thesis, Freie Universität Berlin, 2014. Google Scholar

show all references

References:
[1]

M. Bosewitz, Stabilisierung gekoppelter Oszillatoren durch verzögerte Rückkopplungs-kontrolle, (German) [Stabilization of coupled oscillators by delayed feedback-control] Bachelor Thesis, Freie Universität Berlin, 2013. Google Scholar

[2]

K. Bubolz, Stabilisierung periodischer Orbits im System zweier gekoppelter Oszillatoren durch zeitverzögerte Rückkopplungskontrolle, (German) [Stabilization of periodic orbits in a system of two coupled oscillators by time-delayed feedback-control] Bachelor Thesis, Freie Universität Berlin, 2013. Google Scholar

[3]

B. Fiedler, Global Bifurcation of Periodic Solutions with Symmetry, Lect. Notes Math. 1309, Springer-Verlag, Heidelberg, 1988.  Google Scholar

[4]

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 (2007), 114101. doi: 10.1103/PhysRevLett.98.114101.  Google Scholar

[5]

B. Fiedler, V. Flunkert, P. Hövel and E. Schöll, Delay stabilization of periodic orbits in coupled oscillator systems, Phil. Trans. R. Soc. A, 368 (2010), 319-341. doi: 10.1098/rsta.2009.0232.  Google Scholar

[6]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Applied Mathematical Sciences Vol. 110, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar

[7]

M. Golubitsky, I. Stewart and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. 2. AMS 69, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-4574-2.  Google Scholar

[8]

M. Golubitsky and I. Stewart, The Symmetry Perspective. From Equilibrium to Chaos in Phase Space and Physical Space, Progress in Mathematics, Volume 200, Birkhäuser, Basel, 2002. doi: 10.1007/978-3-0348-8167-8.  Google Scholar

[9]

C. M. Postlethwaite, G. Brown and M. Silber, Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems, Phil. Trans. R. Soc. A, 371 (2013), 20120467, 20pp. doi: 10.1098/rsta.2012.0467.  Google Scholar

[10]

K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170 (1992), 421-428. Google Scholar

[11]

K. Pyragas, A twenty-year review of time-delay feedback control and recent developments, in International Symposium on Nonlinear Theory and its Applications, 1 (2012), 683-686. doi: 10.15248/proc.1.683.  Google Scholar

[12]

I. Schneider, Delayed feedback control of three diffusively coupled Stuart-Landau oscillators: a case study in equivariant Hopf bifurcation, Phil. Trans. R. Soc. A, 371 (2013), 20120472, 10pp. doi: 10.1098/rsta.2012.0472.  Google Scholar

[13]

I. Schneider, Equivariant Pyragas Control, Master Thesis, Freie Universität Berlin, 2014. Google Scholar

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