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Stabilization of deterministic spatio-temporal patterns

A further active application field of the deterministic Pyragas control is the stabilization of unstable spatio-temporal patterns. Besides classical pattern-forming systems, semiconductors have been established as models for complex spatio-temporal dynamics. A series of concepts of pattern formation have been successfully tested in nonlinear semiconductor transport during the last years.  In this field, research is focused mainly on semiconductor nanostructures like, for example, superlattices and resonant tunneling diodes.

We have described the double barrier resonant tunneling diode (DBRT) by a nonlinear reaction-diffusion system of activator-inhibitor type with global coupling. Unstable periodic spatio-temporal orbits can be stabilized through time-delayed feedback, i. e. regular spatio-temporal patterns like spiking or breathing current filaments are stabilized in a regime of the control parameters. Here, we have realized the feedback in various ways (diagonal, local, global or just in the voltage variable) and have compared the stabilization regimes and the Floquet spectra.

As further nanostructure, we have considered a semiconductor superlattice based on the microscopic model developed and  intensively studied in our group. It exhibits complex spatio-temporal patterns which are generated through the interaction and collision of travelling charge carrier fronts and chaotic bifurcation scenarios typical for a large class of front systems. Here as well, we have investigated chaos control via time-delayed feedback and have developed an easy to implement global control method which stabilizes a periodic front pattern which is embedded in the chaotic spatio-temporal attractor. This way, self-generated high-frequency current oscillations can be stabilized over a wide range of the applied voltage. The results of these two classes of nanostructures are summarized  in a review article.

Furthermore, we have studied the stabilization of the rotation of spiral waves in excitable media with local diffusive coupling, via time-delayed feedback, in collaboration with Subdivision A6. A reaction-diffusion system (2-variable Oregonator model) served as a model describing the light-sensitive Belousov-Zhabotinski(BZ) reaction. The feedback-based control in this easy-to-handle classical system was experimentally realized in Subdivision B6. As a result, we could show that both methods could stabilize the rigid rotation of the spiral core.

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