# American Institute of Mathematical Sciences

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

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

 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.

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. [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. [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. [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. [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. [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. [7] E. Hooton and A. Amann, Analytical limitation for time-delayed feedback control in autonomous systems, Physical Review Letters, 109 (2012), 154101. [8] H. Nakajima, On analytical properties of delayed feedback control of chaos, Physics Letters A, 232 (1997), 207-210. [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. [10] K. Pyragas, Continuous control of chaos by self-controlling feedback, Physics Letters A, 170 (1992), 421-428. [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. [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. [13] I. Schneider, Equivariant Pyragas Control, Master's thesis, Freie Universität Berlin, 2014. [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. [15] I. Schneider and B. Fiedler, Symmetry-breaking control of rotating waves, in Control of SelfOrganizing Nonlinear Systems, Springer, 2016,105–126. [16] J. Sieber, Generic stabilizability for time-delayed feedback control, in Proc. R. Soc. A, vol. 472, The Royal Society, 2016,20150593. [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. [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. [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.

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. [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. [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. [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. [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. [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. [7] E. Hooton and A. Amann, Analytical limitation for time-delayed feedback control in autonomous systems, Physical Review Letters, 109 (2012), 154101. [8] H. Nakajima, On analytical properties of delayed feedback control of chaos, Physics Letters A, 232 (1997), 207-210. [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. [10] K. Pyragas, Continuous control of chaos by self-controlling feedback, Physics Letters A, 170 (1992), 421-428. [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. [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. [13] I. Schneider, Equivariant Pyragas Control, Master's thesis, Freie Universität Berlin, 2014. [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. [15] I. Schneider and B. Fiedler, Symmetry-breaking control of rotating waves, in Control of SelfOrganizing Nonlinear Systems, Springer, 2016,105–126. [16] J. Sieber, Generic stabilizability for time-delayed feedback control, in Proc. R. Soc. A, vol. 472, The Royal Society, 2016,20150593. [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. [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. [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.
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 $\varepsilon = 0.03$, $J = 1$, $\eta = 0.2$, $\delta = 0.3$, $\alpha = 2$.
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.
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).
 [1] Isabelle Schneider, Matthias Bosewitz. Eliminating restrictions of time-delayed feedback control using equivariance. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 451-467. doi: 10.3934/dcds.2016.36.451 [2] Gonzalo Robledo. Feedback stabilization for a chemostat with delayed output. Mathematical Biosciences & Engineering, 2009, 6 (3) : 629-647. doi: 10.3934/mbe.2009.6.629 [3] Alexander Moreto. Complex group algebras of finite groups: Brauer's Problem 1. Electronic Research Announcements, 2005, 11: 34-39. [4] Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial & Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303 [5] 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 [6] Sanling Yuan, Yongli Song, Junhui Li. Oscillations in a plasmid turbidostat model with delayed feedback control. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 893-914. doi: 10.3934/dcdsb.2011.15.893 [7] 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 [8] Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085 [9] John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 5-7. [10] Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653 [11] Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin, Bing Chen. Stability and output feedback control for singular Markovian jump delayed systems. Mathematical Control & Related Fields, 2018, 8 (2) : 475-490. doi: 10.3934/mcrf.2018019 [12] Daniel Wilczak, Piotr Zgliczyński. Topological method for symmetric periodic orbits for maps with a reversing symmetry. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 629-652. doi: 10.3934/dcds.2007.17.629 [13] Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369 [14] Motserrat Corbera, Jaume Llibre, Claudia Valls. Periodic orbits of perturbed non-axially symmetric potentials in 1:1:1 and 1:1:2 resonances. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-39. doi: 10.3934/dcdsb.2018101 [15] Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109 [16] Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619 [17] Fabio Bagagiolo. Optimal control of finite horizon type for a multidimensional delayed switching system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 239-264. doi: 10.3934/dcdsb.2005.5.239 [18] Dario Bambusi, Simone Paleari. Families of periodic orbits for some PDE’s in higher dimensions. Communications on Pure & Applied Analysis, 2002, 1 (2) : 269-279. doi: 10.3934/cpaa.2002.1.269 [19] Markus Dick, Martin Gugat, Günter Leugering. A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 225-244. doi: 10.3934/naco.2011.1.225 [20] Jean-Pierre Raymond, Laetitia Thevenet. Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1159-1187. doi: 10.3934/dcds.2010.27.1159

2016 Impact Factor: 0.994

## Tools

Article outline

Figures and Tables