Dynamics and Kinetics

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

The project addresses several central problems in the analysis and prediction of the evolution of complex systems, i.e. the systems with complex dynamics and/or systems with a large number of interacting components (networks). Real world and engineering networks often have tens of thousands of components. Theory, developed by the principal investigator, allows one to compress networks of any type into much smaller networks while preserving important information about the initial large network. In the proposed project, this theory will be further developed. And, in collaboration with Center for Disease Control (CDC) researchers, applied to an analysis of transmission and cross-immunoreactivity networks for Hepatitis C. Cross-immunoreactivity means that antibodies produced by the immune system to fight some specific type of viruses sometimes fights other types of viruses as well. Cross-immunoreactivity is found for Hepatitis C, influenza, dengue, and is a basis for a mathematical model developed by the author with CDC colleagues. Traditional views on evolution of systems with complex dynamics suggest that only long term predictions for such systems are possible, since one should wait until the system stabilizes. A new approach, proposed by the principal investigator, allows one to make short term predictions on the evolution of chaotic systems. It will be possible to answer questions such as, "which state of the system is most likely?". This research will extend this approach to a larger class of systems with complex dynamics and also address problem of recurrence. Understanding of a fundamental mechanism of chaos, called defocusing, will be essentially extended. Discovery of this mechanism by the principal investigator has already found numerous applications in theoretical and experimental physics. New visual models of systems with various types of chaotic behavior can be included into basic and advanced courses on complex systems and chaos. The proposed research addresses some long standing and technically challenging, as well as new natural problems, in the theory of dynamical systems and statistical mechanics. A new area of finite time qualitative properties of dynamical and stochastic systems, pioneered by the principal investigator, will be further developed. This will make use of combinations of methods and ideas of symbolic dynamics, discrete mathematics and probability. The theory of isospectral transformation of networks will be further developed with a special emphasis on analysis of attractors of sequences of such transformations in the space of networks. Understanding the mechanism of defocusing, a fundamental mechanism of hyperbolicity (chaos), will be advanced. This uses new classes of chaotic billiards where focusing components are not forced to be far from each other. Classical Ehrenfests' gas will be shown to have unexpected properties thanks to a new observation that billiards with a point and with a finite size particle can have quite different dynamics.