# 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
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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).
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