Inhalt des Dokuments
Summary of research programme
- © SFB 910
Nonlinear dynamical systems driven far from
thermodynamic equilibrium often exhibit self-organization, i.e.,
the spontaneous emergence of temporal, spatial, or spatio-temporal
structures from the inherent nonlinear cooperative dynamics. In
dissipative systems such patterns can be maintained by a continuous
influx of energy or matter. Dissipative structures in
self-organizing nonlinear systems are widespread in physics,
chemistry, and biology.
It is the purpose of this proposed Collaborative Research Center (CRC) to go beyond merely describing such dissipative structures to actively controlling and designing them. By combining an interdisciplinary team of applied mathematicians, theoretical physicists, and computational neuroscientists, we aim at developing novel theoretical approaches and methods of control, and demonstrating the application of these concepts to a selection of innovative systems ranging from condensed matter to biological systems. These systems involve multiple space scales, typically from nanometer to micrometer and millimeter, and they involve quantum as well as classical dynamics. This diversity requires to develop control concepts bridging length and time scales from the nanoscale to the meso- and macroscale. Our focus is on theoretical and methodological developments, rather than on a description of a menagerie of specific phenomena or diverse experimental systems, although the application of our concepts to concrete experiments will be fostered by specific external collaborations of the individual projects. Thus the main methodological thrust of our CRC, which is based upon a coherent conceptual approach, will be enriched by diverse application aspects. We aim to bring together different control communities which have so far been separate, and have had very little interaction: (i) the nonlinear dynamics control community, which has originally emerged from chaos control, but has now a much broader scope, (ii) the classical mathematical control and optimization community, (iii) the coherent quantum control community. From this synergy we expect a cross-fertilization with a long-term perspective of completely new concepts and insights. In particular, novel areas of applications, e.g., to quantum nanostructures, to soft matter films, and to neurosystems are anticipated.
The field of control of nonlinear systems has various aspects, comprising stabilization of unstable steady states, periodic oscillations, or spatio-temporal patterns, suppression of chaos (chaos control), design of dynamics of complex networks, and control of the coherence and timescales of noise-mediated motion. Feedback control loops represent an important concept (closed-loop control) to stabilize unstable states adaptively by using the internal dynamics of the system to adjust the control force, rather than externally imposing a fixed value. A versatile example is provided by time-delayed feedback control, where the control signal is constructed from some time-delayed output variable of the system. Using algorithms of optimal control, the proposed control methods can be optimized with respect to the forcing or feedback protocol in order to minimize, for example, the energy and the time needed to achieve control. With research towards control of networks and quantum control we plan to open up two promising novel fields of application for control algorithms which have hitherto been mainly confined to classical macroscopic systems. This includes adaptable networks whose architecture can change as a result of the feedback, which amounts to engineering of dynamical systems. Feedback control of quantum systems will yield a qualitatively new approach to control, since quantum coherence and dominating quantum fluctuations will require new control strategies.
Another focus concerns applications to soft condensed matter, ranging from driven colloidal systems to active biomembranes, and to excitable heterogeneous and multi-scale systems which can give rise to self-organized pattern formation on strongly varying spatial and temporal scales. Besides their importance in a variety of technological and biological contexts, soft-matter systems constitute excellent, tunable model systems which can display a variety of instabilities and dynamic patterns far from equilibrium. Developing and applying feedback control strategies to stabilize specific nonequilibrium states in soft matter systems is a novel and innovative issue. The third focus of application of control concepts will be on neural dynamics, where inherent time-delayed and nonlocal feedbacks play an important role. Understanding the intrinsic control mechanism might eventually lead to new therapeutic measures for treating pathological states as well as improving our understanding of learning and memory.