Global delayed feedback

M. Bertram and A. Mikhailov. Pattern formation on the edge of chaos mathematical modeling of CO oxidation on a Pt(llO) surface under global delayed feedback. Phys. Rev. E, 67 1-9, 2003. 

M. Kim, M. Bertram, M. Pollmann, A. v. Oertzen, A. S. Mikhailov, H. H. Rotermund, and G. Ertl. Controlling chemical turbulence by global delayed feedback pattern formation in catalytic co oxidation reaction on pt(llO). Science, 292 1357-1360, 2001. 

FIGURE 8.16. Principle of controlling pattern formation by global delayed feedback [42],... 

CO oxidation experiments with global delayed feedback... 

FIGURES. 17. Summary of controlled formation of patterns in CO oxidation on Pt(l 10) by global delayed feedback with varying delay times and strengths of feedback [42]. (See color insert.)... 

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. 

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

O. Beck, A. Amann, E. Scholl, J. E. S. Socolar, and W. Just Comparison of time-delayed feedback schemes for spatio-temporal control of chaos in a reaction-diffusion system with global coupling, Phys. Rev. E 66, 016213 (2002). 

In this chapter, we present our recent findings related to synchronization in coupled nonlinear oscillator systems with and without time delay as specific representation of certain neuronal behaviors. In the following section, we discuss about the ubiquitous nature of synchronization in dynamical systems. In Section 3, we explain the structure of neurons and how neurons function as a collective system. Section 4 is dedicated to the discussion of the occurrence of oscillations and synchronization in neuronal networks. In Section 5, we demonstrate the occurrence of event related desynchronization in a system of coupled nonlinear oscillators in the presence of an external field. In Section 6, we discuss the effect of time delay in coupled populations of nonlinear oscillators and demonstrate the occurrence of globally clustered chimera (GCC) states. In Section 7, we present a demand controlled delayed feedback mechanism for controlling the occurrence of mass pathological synchronization in the brain. Finally, in Section 8, we present our conclusions and summarize the chapter. 

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