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. Recently the group has extended the theory by inclusion of the diffraction contributions to quantal spectra; also the theory has found several unanticipated applications to the experiments performed at CATS, in particular to the statistics of acoustic spectra, and the discovery of a classical orbit modulation of the surface waves power spectrum in the hydraulic jump.
While today we know much about what is deterministic chaos 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. We 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, 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.
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 and/or disorder, such as universal scalings, phase transitions, fractal dimensions, and critical phenomena.
Dynamics of biological systems:
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, and 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 cross-fertilization of ideas from physics, mathematics, computer science and biology. CATS groups have collaborated with biologists, medical doctors and chemists in order to study oscillations, chaos and biological form formation in a number of concrete examples, and modeled the punctuated-equilibria hypothesis of evolution.