American Institute of Mathematical Sciences

Advanced
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

[1]

Cicely K. Macnamara, Mark A. J. Chaplain. Spatio-temporal models of synthetic genetic oscillators. Mathematical Biosciences & Engineering, 2017, 14 (1) : 249-262. doi: 10.3934/mbe.2017016
[2]

Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197
[3]

Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021151
[4]

Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2019-2034. doi: 10.3934/dcdss.2019130
[5]

Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii. Restrictions to the use of time-delayed feedback control in symmetric settings. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 543-556. doi: 10.3934/dcdsb.2017207
[6]

Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183
[7]

Lin Wang, James Watmough, Fang Yu. Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with Dirichlet boundary conditions. Mathematical Biosciences & Engineering, 2015, 12 (4) : 699-715. doi: 10.3934/mbe.2015.12.699
[8]

Francesca Sapuppo, Elena Umana, Mattia Frasca, Manuela La Rosa, David Shannahoff-Khalsa, Luigi Fortuna, Maide Bucolo. Complex spatio-temporal features in meg data. Mathematical Biosciences & Engineering, 2006, 3 (4) : 697-716. doi: 10.3934/mbe.2006.3.697
[9]

Noura Azzabou, Nikos Paragios. Spatio-temporal speckle reduction in ultrasound sequences. Inverse Problems & Imaging, 2010, 4 (2) : 211-222. doi: 10.3934/ipi.2010.4.211
[10]

Xiaoying Chen, Chong Zhang, Zonglin Shi, Weidong Xiao. Spatio-temporal keywords queries in HBase. Big Data & Information Analytics, 2016, 1 (1) : 81-91. doi: 10.3934/bdia.2016.1.81
[11]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344
[12]

Wenjia Jing, Panagiotis E. Souganidis, Hung V. Tran. Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 915-939. doi: 10.3934/dcdss.2018055
[13]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182
[14]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71
[15]

Martin Gugat, Markus Dick. Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Mathematical Control & Related Fields, 2011, 1 (4) : 469-491. doi: 10.3934/mcrf.2011.1.469
[16]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228
[17]

Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017
[18]

Buddhi Pantha, Judy Day, Suzanne Lenhart. Investigating the effects of intervention strategies in a spatio-temporal anthrax model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1607-1622. doi: 10.3934/dcdsb.2019242
[19]

Jingdong Wei, Jiangbo Zhou, Wenxia Chen, Zaili Zhen, Lixin Tian. Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2853-2886. doi: 10.3934/cpaa.2020125
[20]

Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (105)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences