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## Aims:

Object of this project is the influence and control of complex spatiotemporal structures through time delayed feedback methods (time delay autosynchronization). Attention is given on the influence of stochastic noise. We are especially interested in analyzing noise induced patterns in systems which in the deterministic case, in the lack of noise and external stimulus, do not exhibit autonomous oscillations. Such systems are for instance systems below a Hopf bifurcation, in excitable or bistable (multistable) systems. In this case one can control the coherence of the noise induced oscillations as well as the time scales of the system by feedback of the difference between a signal and its time-delayed version (Pyragas-control). Furthermore the chaotic scenarios of front dynamics under the influence of global coupling should be investigated. The basic effects should be numerically and analytically investigated by means of simpler generic models, which serve as prototypes for oscillatory, excitable and bistable behaviour respectively. A detailed analysis of the bifurcation mechanisms of the time delayed feedback should be made. Based on our previous work on chaos control through time delayed feedback and the investigation of various control mechanisms as well as on chaotic front dynamics in concrete systems, we should expand our research investigating noise induced effects and pattern selection

As application examples two semiconductor nanostructures are used, which are of great interest in the field of modern semiconductor physics due to the fact that they can be used as electronic and optoelctronic devices in the future. These are the resonant tunneling diode and the superlattice. Through a feedback loop, the time delayed output signal (e. g. the global current) can be used as a feedback input signal , applying the Pyragas control techniques. In a double barrier tunneling structure the resonant tunneling of electrons leads to a Z-shaped current density-voltage characteristic. The transverse pattern formation in the form of spatiotemporal spiking and breathing can be described, as we have shown, by a generic reaction-diffusion model with global coupling through the load resistance. Depending on the model and the electric circuit parameters, this model exhibits both excitable and oscillatory behaviour with a Hopf bifurcation of a breathing limit cycle. In this case control of noise induced patterns should be applied. Semiconductor superlattices consist of a periodic sequence of alternating layers of two different materials with different band gaps thus forming potential wells and barriers. Applying a voltage, inhomogeneous field distributions (stationary or traveling field domains) arise, which are constrained through a global condition (constant total voltage). As we have shown, traveling domains are linked to complex front dynamics which exhibits universal chaotic behaviour and can be controlled through a time delayed global feedback of the current. We would like to expand these investigations including Low pass filters and latency effects of the signal processing. Here as well, control of noise induced oscillations below the Hopf bifurcation of the front oscillations should be examined.