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March  2018, 23(2): 543-556. doi: 10.3934/dcdsb.2017207

Restrictions to the use of time-delayed feedback control in symmetric settings

Edward Hooton , Pavel Kravetc and  Dmitrii Rachinskii ,

Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA

* Corresponding author

Received  February 2017 Revised  June 2017 Published  December 2017

Fund Project: Authors are supported by NSF grant DMS-1413223.

We consider the problem of stabilization of unstable periodic solutions to autonomous systems by the non-invasive delayed feedback control known as Pyragas control method. The Odd Number Theorem imposes an important restriction upon the choice of the gain matrix by stating a necessary condition for stabilization. In this paper, the Odd Number Theorem is extended to equivariant systems. We assume that both the uncontrolled and controlled systems respect a group of symmetries. Two types of results are discussed. First, we consider rotationally symmetric systems for which the control stabilizes the whole orbit of relative periodic solutions that form an invariant two-dimensional torus in the phase space. Second, we consider a modification of the Pyragas control method that has been recently proposed for systems with a finite symmetry group. This control acts non-invasively on one selected periodic solution from the orbit and targets to stabilize this particular solution. Variants of the Odd Number Limitation Theorem are proposed for both above types of systems. The results are illustrated with examples that have been previously studied in the literature on Pyragas control including a system of two symmetrically coupled Stewart-Landau oscillators and a system of two coupled lasers.

Keywords: Stabilization of periodic orbits, Pyragas control, delayed feedback, S1-equivariance, finite symmetry group.
Mathematics Subject Classification: 34H15, 93B52, 37C80.
Citation: 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
References:
[1]

G. Brown, C. Postlethwaite and M. Silber, Time-delayed feedback control of unstable periodic orbits near a subcritical hopf bifurcation, Physica D: Nonlinear Phenomena, 240 (2011), 859-871. doi: 10.1016/j.physd.2010.12.011. Google Scholar

[2]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using dde-biftool, ACM Trans. Math. Softw., 28 (2002), 1-21. doi: 10.1145/513001.513002. Google Scholar

[3]

K. Engelborghs, T. Luzyanina and G. Samaey, DDE-BIFTOOL v. 2. 00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations. Department of Computer Science, Technical report, KU Leuven, Technical Report TW-330, Leuven, Belgium, 2001.Google Scholar

[4]

B. Fiedler, V. Flunkert, P. Hövel and E. Schöll, Delay stabilization of periodic orbits in coupled oscillator systems, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 368 (2010), 319-341. doi: 10.1098/rsta.2009.0232. Google Scholar

[5]

B. Fiedler, S. Yanchuk, V. Flunkert, P. Hövel, H. -J. Wünsche and E. Schöll, Delay stabilization of rotating waves near fold bifurcation and application to all-optical control of a semiconductor laser, Physical Review E, 77 (2008), 066207, 9pp doi: 10.1103/PhysRevE.77.066207. Google Scholar

[6]

E. Hooton, Z. Balanov, W. Krawcewicz and D. Rachinskii, Noninvasive Stabilization of Periodic Orbits in O4-Symmetrically Coupled Systems Near a Hopf Bifurcation Point, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750087, 18 doi: 10.1142/S0218127417500870. Google Scholar

[7]

E. Hooton and A. Amann, Analytical limitation for time-delayed feedback control in autonomous systems, Physical Review Letters, 109 (2012), 154101.Google Scholar

[8]

H. Nakajima, On analytical properties of delayed feedback control of chaos, Physics Letters A, 232 (1997), 207-210. Google Scholar

[9]

C. 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, Physics Letters A, 170 (1992), 421-428. Google Scholar

[11]

S. Schikora, P. Hövel, H. -J. Wünsche, E. Schöll and F. Henneberger, All-optical noninvasive control of unstable steady states in a semiconductor laser, Physical Review Letters, 97(2006), 213902.Google Scholar

[12]

I. Schneider, Delayed feedback control of three diffusively coupled Stuart–Landau oscillators: a case study in equivariant Hopf bifurcation, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 371 (2013), 20120472, 10pp. doi: 10.1098/rsta.2012.0472. Google Scholar

[13]

I. Schneider, Equivariant Pyragas Control, Master's thesis, Freie Universität Berlin, 2014.Google Scholar

[14]

I. Schneider and M. Bosewitz, Eliminating restrictions of time-delayed feedback control using equivariance, Discrete and Continuous Dynamical Systems Series A, 36 (2016), 451-467. doi: 10.3934/dcds.2016.36.451. Google Scholar

[15]

I. Schneider and B. Fiedler, Symmetry-breaking control of rotating waves, in Control of SelfOrganizing Nonlinear Systems, Springer, 2016,105–126. Google Scholar

[16]

J. Sieber, Generic stabilizability for time-delayed feedback control, in Proc. R. Soc. A, vol. 472, The Royal Society, 2016,20150593.Google Scholar

[17]

J. Sieber, A. Gonzalez-Buelga, S. Neild, D. Wagg and B. Krauskopf, Experimental continuation of periodic orbits through a fold, Physical Review Letters, 100 (2008), 244101.Google Scholar

[18]

M. Tlidi, A. Vladimirov, D. Pieroux and D. Turaev, Spontaneous motion of cavity solitons induced by a delayed feedback, Physical Review Letters, 103 (2009), 103904.Google Scholar

[19]

S. Yanchuk, K. R. Schneider and L. Recke, Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit, Physical Review E, 69 (2004), 056221, URL http://link.aps.org/doi/10.1103/PhysRevE.69.056221.Google Scholar

show all references

References:
[1]

G. Brown, C. Postlethwaite and M. Silber, Time-delayed feedback control of unstable periodic orbits near a subcritical hopf bifurcation, Physica D: Nonlinear Phenomena, 240 (2011), 859-871. doi: 10.1016/j.physd.2010.12.011. Google Scholar

[2]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using dde-biftool, ACM Trans. Math. Softw., 28 (2002), 1-21. doi: 10.1145/513001.513002. Google Scholar

[3]

K. Engelborghs, T. Luzyanina and G. Samaey, DDE-BIFTOOL v. 2. 00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations. Department of Computer Science, Technical report, KU Leuven, Technical Report TW-330, Leuven, Belgium, 2001.Google Scholar

[4]

B. Fiedler, V. Flunkert, P. Hövel and E. Schöll, Delay stabilization of periodic orbits in coupled oscillator systems, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 368 (2010), 319-341. doi: 10.1098/rsta.2009.0232. Google Scholar

[5]

B. Fiedler, S. Yanchuk, V. Flunkert, P. Hövel, H. -J. Wünsche and E. Schöll, Delay stabilization of rotating waves near fold bifurcation and application to all-optical control of a semiconductor laser, Physical Review E, 77 (2008), 066207, 9pp doi: 10.1103/PhysRevE.77.066207. Google Scholar

[6]

E. Hooton, Z. Balanov, W. Krawcewicz and D. Rachinskii, Noninvasive Stabilization of Periodic Orbits in O4-Symmetrically Coupled Systems Near a Hopf Bifurcation Point, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750087, 18 doi: 10.1142/S0218127417500870. Google Scholar

[7]

E. Hooton and A. Amann, Analytical limitation for time-delayed feedback control in autonomous systems, Physical Review Letters, 109 (2012), 154101.Google Scholar

[8]

H. Nakajima, On analytical properties of delayed feedback control of chaos, Physics Letters A, 232 (1997), 207-210. Google Scholar

[9]

C. 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, Physics Letters A, 170 (1992), 421-428. Google Scholar

[11]

S. Schikora, P. Hövel, H. -J. Wünsche, E. Schöll and F. Henneberger, All-optical noninvasive control of unstable steady states in a semiconductor laser, Physical Review Letters, 97(2006), 213902.Google Scholar

[12]

I. Schneider, Delayed feedback control of three diffusively coupled Stuart–Landau oscillators: a case study in equivariant Hopf bifurcation, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 371 (2013), 20120472, 10pp. doi: 10.1098/rsta.2012.0472. Google Scholar

[13]

I. Schneider, Equivariant Pyragas Control, Master's thesis, Freie Universität Berlin, 2014.Google Scholar

[14]

I. Schneider and M. Bosewitz, Eliminating restrictions of time-delayed feedback control using equivariance, Discrete and Continuous Dynamical Systems Series A, 36 (2016), 451-467. doi: 10.3934/dcds.2016.36.451. Google Scholar

[15]

I. Schneider and B. Fiedler, Symmetry-breaking control of rotating waves, in Control of SelfOrganizing Nonlinear Systems, Springer, 2016,105–126. Google Scholar

[16]

J. Sieber, Generic stabilizability for time-delayed feedback control, in Proc. R. Soc. A, vol. 472, The Royal Society, 2016,20150593.Google Scholar

[17]

J. Sieber, A. Gonzalez-Buelga, S. Neild, D. Wagg and B. Krauskopf, Experimental continuation of periodic orbits through a fold, Physical Review Letters, 100 (2008), 244101.Google Scholar

[18]

M. Tlidi, A. Vladimirov, D. Pieroux and D. Turaev, Spontaneous motion of cavity solitons induced by a delayed feedback, Physical Review Letters, 103 (2009), 103904.Google Scholar

[19]

S. Yanchuk, K. R. Schneider and L. Recke, Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit, Physical Review E, 69 (2004), 056221, URL http://link.aps.org/doi/10.1103/PhysRevE.69.056221.Google Scholar

Figure 1.  Bifurcation diagram obtained with numerical package DDE-BIFTOOL [2,3] for system (34)-(37). Thin 'eight'-shaped line: relative equilibria; thick line: relative periodic solutions. Solid and dashed lines represent stable and unstable parts of the branches, respectively. H: subcritical Hopf bifurcation point; gray dot: unstable periodic orbit targeted for stabilization by Pyragas control. Parameters are $\varepsilon = 0.03$, $J = 1$, $\eta = 0.2$, $\delta = 0.3$, $\alpha = 2$.
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Figure 2.  Domains of stability of the target relative periodic solution. Parameters correspond to the gray dot in Figure 1. Black region: sufficient condition (29) for instability is satisfied; white region: relative periodic solution is stable; gray region: relative periodic solution is unstable.
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Figure 3.  Panel (a): Floquet multipliers of the target relative periodic orbit in the uncontrolled system (34)-(37). Panel (b): Floquet multipliers of the same relative periodic orbit in the controlled system with the parameters $b_0 =0.3036$ and $\beta =6$ of control (40).
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