This section contains a brief overview over the planned CATS research activities. More details are given in section .
Classical and Quantum Chaos:
The goal of this research is the development of a classical and quantum description of chaotic, non-integrable systems, of analytic and numerical quality comparable to those obtainable by the traditional techniques for nearly integrable systems. The NBI/Nordita group has established that strongly chaotic systems can be described surprisingly accurately by means of unstable periodic orbits. So far, the most exciting application of the theory has been to quantum chaos, where one can now experimentally measure the energy levels that the chaos theory predicts from the classical unstable orbits; the theory is under rapid development, with recent and highly impressive advances such as the first periodic-orbit calculation of the Helium spectrum. In the coming years we plan to apply this theory to a variety of solid state, mesoscopic, and other experimental systems. Some of the important open problems of this field that we intend to confront are: How can the quantum-mechanical unitarity be built into the periodic orbit theory? Can the periodic orbit quantization be made exact? How can the theory be extended to generic systems which exhibit a mixture of chaotic and integrable behavior?
While today we know much about what deterministic chaos is and how to characterize it, our understanding of turbulence is still rudimentary. Progress in this field is difficult, but a serious theoretical advance is crucial in its implications for classical hydrodynamics, statistical mechanics, and strongly nonlinear field theories. The CATS groups study turbulence as chaotic phenomena in spatially extended systems; the main question is to what extent the methods developed in the context of low-dimensional chaotic dynamics can be applied to dynamical systems of high intrinsic dimensions. The groups plan to investigate transitions from coherent to incoherent motion, the turbulent motion of vortices in reaction-diffusion systems, the instability of boundary layers in convective systems, intermittency, and scaling exponents of turbulence. This involves extensive numerical investigations of spatio-temporal chaos: coupled-map lattices, cellular automata, neural networks, and a variety of discretized models of chemical and hydrodynamical turbulent phenomena. Such models have enabled us to study the onset of turbulent vortex structures in models of inhomogeneous oscillatory chemical reactions, as well as to discover new types of phase transitions, subsequently observed experimentally in the Belousov-Zhabotinsky reaction and in charge-density waves. A common thread of all CATS projects is extensive computer experimentation; chaos researchers were among the first to utilize interactive graphical computation, and our future research will to a high degree rely on the advent of parallelism and other advances in computation.
Experimental investigations are crucial to the development of our understanding of complex systems. Carefully designed experiments help developing intuition and stimulate new theoretical advances. Some of the first experiments in chaotic systems were carried out by CATS groups, and it is of paramount importance for the Center to build a strong experimental program. Currently the CATS-groups have ongoing experiments in chemical reactions (quenching analysis, invariant manifolds, chemical waves), fluid dynamics (turbulence in pipe flows, soap film flows capillary waves, hydraulic jump), acoustics (spectral statistics of high Q resonators), biology (yeast colony growth), and solid state phenomena (relaxation oscillators, neural network electronics, dynamical friction, granular flows).
Chemical Reaction-Diffusion Systems:
Aspects of nonlinear dynamics, such as chaos and the theory of bifurcations, play an important role in chemical reaction-diffusion systems where chaotic behavior is by now an established experimental fact. The HCØ-KI group was one of the first to initiate quantitative experimental measurements of such dynamical systems. The group has shown that the Turing structures formed in these systems are relevant for biological pattern formation; the ability to describe complex chemical systems is essential for understanding key phenomena in biological control processes. The goal of the HCØ-KI/NBI collaboration, combining the experimental and theoretical strengths of the two groups, is to describe chemical turbulence in spatially extended systems.
One of the key notions in nonlinear science is the concept of fractals. Dendritic growth, dielectric breakdown, electrochemical deposition, viscous fingering of a fluid penetrating another fluid are some of the fractals generated in nature. The goal of the NBI experimental investigations of fractal structures and their theoretical descriptions is to discover, by techniques closely related to statistical mechanics, new observable consequences of chaoticity, such as universal scalings, phase transitions, fractal dimensions, and self-organizing critical phenomena.
One of the most obvious areas in which to apply and further develop the tools of nonlinear dynamics is biology. Unstable phenomena are increasingly being recognized as significant in the regulation and function of living systems. Investigations performed during the last decade have revealed the existence of a large variety of biological rhythms with periods ranging from fractions of a second to hours or even days. Other investigations have demonstrated the occurrence of deterministic chaos in physiological systems. These observations pose a number of fundamental questions concerning possible advantages of complex behavior and the mechanisms of evolutionary processes. A particularly interesting topic is the evolution and development of biological forms. All of these closely related areas are experiencing dramatic breakthroughs, obtained through crossfertilization of ideas from physics, mathematics, computer science and biology. The computer simulation of self-organization in realistic 3-dimensional systems based on bifurcations in reaction-diffusion systems was pioneered by the HCØ-KI group. Much of the DTU group research in the coming years will focus on collaborations with biologists, medical doctors and chemists in order to study oscillations, chaos and biological form formation in a number of concrete examples.