Such a concept is simple to implement in quite different experimental contexts like magnetic, optical or chemical systems. But a deeper theoretical understanding of such control schemes requires a sound analysis of the corresponding delay-differential equations which constitute an inherently infinite-dimensional dynamical system. The talk presents an overview of recent theoretical results for time-delayed feedback control and of corresponding illustrations in different experimental contexts.
[1] C.v. Loewenich et al., Phys. Rev. Lett. 93, 174101 (2004).
[1] K. Pyragas, Phys. Rev. Lett. 86, 2265 (2001).
[2] K. Pyragas et al. Phys. Rev. E 70,026412 (2004).
TDFC is an interesting and challenging topic even from the theoretical perspective since one is dealing from the very beginning with infinite-dimensional delay dynamics. It has been shown that only a certain class of periodic orbits characterized by a finite torsion can be stabilized by TDFC. Such a topological constraint means that any unstable periodic orbit with an odd number of real Floquet multipliers larger than unity can never be stabilized by TDFC.
Different strategies have been applied to overcome this constraint. The so-called rhythmic control is based on the periodic modulation of the control parameters with a period different from that of the orbit. Another way suggested recently [2] is based on the counter-intuitive idea of introducing an unstable degree of freedom into the control device. The key idea is to provide an even number of real Floquet multipliers by including an unstable degree of freedom in the feedback loop and to overcome the limitation mentioned above. Both methods were successfully applied to control torsion-free unstable periodic orbits in numerical simulations as well as in real experiments on nonlinear electronic circuits.
While chaos control by time-delayed feedback is meanwhile quite well understood from the point of view of linear stability analysis, very little is known about global features of the control problem. From an experimental perspective the question which type of initial condition yields successful stabilisation, i.e. in theoretical terms the basin of attraction, is of utmost importance. Applying general arguments from bifurcation theory we propose that discontinuous transitions at the control boundaries, i.e. subcritical behaviour, severely limit such basins. We illustrate our theoretical concept by numerical simulations and electronic circuit experiments [3].
[1] Handbook of Chaos Control, ed. H.G. Schuster (Wiley-VCH,
Berlin, 1999).
[2] K. Pyragas, Phys. Rev. Lett. 86, 2265 (2001).
[3] C.v. Loewenich, H. Benner and W. Just, Phys. Rev. Lett. 93, 174101
(2004).