Dynamics and Kinetics

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

The proposed research addresses some fundamental problems in the theory of dynamical systems and develops further a new promising direction, pioneered by the PI, which is concerned with a finite time properties of dynamics. A classical approach to studies of chaotic dynamics deals only with the asymptotic in time properties. A new approach to the design of hyperbolic (chaotic) elastic impact systems will be developed which will allow to essentially extend this class. It is based on a new general characterization of absolutely focusing mirrors in terms of continued fractions. A 40-years old problem on smoothening of stadium elastic impact systems will be resolved, which will allow to advance in understanding dynamics of Hamiltonian systems with divided into regular islands and chaotic seas phase spaces. Dynamics of a finite size particles in nonconvex polygons will be shown to be chaotic on contrary to a general view that impact systems with point and finite size particles have similar dynamics. This will be applied to classical Ehrenfest's wind-tree model in statistical mechanics to demonstrate that this model may even surpass the richness of dynamics in the Lorentz gas. A new area of finite time qualitative properties of dynamical and stochastic systems will be advanced into studies of open systems and multidimensional systems. The proposed research will make an essential impact in science and technology. A new area of finite time qualitative properties of chaotic and of complex systems, pioneered by the PI, opens new promising avenues in the studies of transport in chaotic systems and dynamical networks. It can be, in particular, directly applied to real world networks in order to detect e.g. the most efficient receptors and transmitters among the elements nodes) and among the links of networks as well as to analysis and design of networks. A new approach to study closed systems by making "holes" at different places of their spaces of states and observing how position of such "hole" influences dynamics has already been taken up by physicists in numerical experiments and will be a useful and easily implemented tool in laboratory experiments as well in physics, chemistry and technology. Simple visual models of systems with impacts proposed by the PI will continue to be built as experimental devices in physics labs all over the world. As clear visual examples they play an important role in teaching nonlinear dynamics and complex systems to students and young scientists in all areas of science and engineering. The proposed research will have impact not only within the theory of dynamical systems but in other areas of Mathematics as well, especially in Mathematical Physics. Analysis of chaotic motion of finite size particles in nonconvex polygons may have applications for transport in nanotubes where particles propagate one after another and do not interact. The proposed research will enhance national and international collaborations with Mexico, Canada, Brasil, UK, Germany and Italy. Among the PI's collaborators are several females and Hispanics. Graduate and undergraduate students will be involved in the proposed research. It will also serve to a broad dissemination to enhance scientific and technological understanding via participation of the PI, his collaborators and students in interdisciplinary conferences as well as through special lectures for students and young researchers.