Time-delayed feedback

Pattern formation in semiconductors under the influence of time-delayed feedback control and noise... 

Time-delayed feedback control has also been applied to purely noise-induced oscillations in a regime where the deterministic system rests in a steady state. It has been shown that in this way both the coherence and the mean frequency of the oscillations can be controlled in simple models [7-9, 49, 50] as well as in spatially extended systems [51-53]. 

We shall now introduce a time-delayed feedback loop to control the chaotic front motion and stabilize a periodic oscillation mode which is inherent in the chaotic attractor [48, 73]. As a global output signal which is coupled back in the feedback loop, it is natural to use the total current density J = jyq-j- Jm-tm+i- For the uncontrolled chaotic oscillations, J... 

Fig. 5.5. a) Control circuit including the low-pass filter with cut-off frequency ot and the time-delayed feedback loop (K) and its extension to multiple time delays (R). b) Control domain for global voltage control. Pull circles denote successful control, small dots denote no control. Parameters as in Fig. 5.4. [73]... 

To conclude, noise-induced front motion and oscillations have been observed in a spatially extended system. The former are induced in the vicinity of a global saddle-node bifurcation on a limit cycle where noise uncovers a mechanism of excitability responsible also for coherence resonance. In another dynamical regime, namely below a Hopf bifurcation, noise induces oscillations of decreasing regularity but with almost constant basic time scales. Applying time-delayed feedback enhances the regularity of those oscillations and allows to manipulate the time scales of the system by varying the time delay t. 

In order to control the noised-induced patterns, we will now use the method of time-delayed feedback which was previously applied successfully in deterministic chaos control of this particular system [47] as well as for control of noise-induced oscillations in simple models [7-9] without spatial degrees of freedom. 

The voltage u is easily accessible in a real experiment. Therefore, as a simple and adaptive method of control we add the time-delayed feedback only to the voltage variable u in eq. (5.26) ... 

While these investigations have enlightened our basic understanding of nonlinear, spatially extended systems under the influence of time-delayed feedback and noise, they may also open up relevant applications as nano-electronic devices like oscillators and sensors. 

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H. Benner and W. Just Control of chaos by time delayed feedback in high power ferromagnetic resonance experiments, J. Kor. Phys. Soc. 40, 1046 (2002). 

C. von Loewenich, H. Benner, and W. Just Experimental relevance of global properties of time-delayed feedback control, Phys. Rev. Lett. 93, 174101 (2004). 

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W. Just, H. Benner, and E. Scholl Control of chaos by time-delayed feedback a survey of theoretical and experimental aspects, in Advances in Solid State Phyics, edited by B. Kramer (Springer, Berlin, 2003), vol. 43, pp. 589-603. 

P. Hovel and J. E. S. Socolar Stability domains for time-delay feedback control with latency, Phys. Rev. E 68, 036206 (2003). 

P. Hovel and E. Scholl Control of unstable steady states by time-delayed feedback methods, Phys. Rev. E 72, 046203 (2005). 

B. Fiedler, V. Flunkert, M. Georgi, P. Hovel, and E. Scholl Refuting the odd number limitation of time-delayed feedback control, http //arxiv.org/abs/nlin.CD/0609056 (2006). 

E. Scholl, A. Balanov, N. B. Janson, and A. Neiman Controlling stochastic oscillations close to a Hopf bifurcation by time-delayed feedback, Stoch. Dyn. 5, 281 (2005). 

J. Schlesner, A. Amann, N. B. Janson, W. Just, and E. Scholl Selfstabilization of chaotic domain oscillations in superlattices by time-delayed feedback control, Semicond. Sci. Technol. 19, S34 (2004). 

W. Just, E. Reibold, K. Kacperski, P. Fronczak, J. A. Holyst, and H. Benner Influence of stable floquet exponents on time-delayed feedback control, Phys. Rev. E 61, 5045 (2000). 

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