Mathematical Sciences: Dynamics and Kinetics of Spatially Extended Systems

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Abstract Bunimovich The proposed research deals with the dynamics of spatially extended systems. The systems under study include nonlinear hyperbolic partial differential equations, lattice dynamical systems (in particular, coupled map lattices), lattice gas cellular automata and the systems of interacting particles. The pattern formation problem will be studied as well as order-chaos transitions. Most of the questions that will be addressed have their origin in physics and hydrodynamics, and were transformed into the mathematical form by the principal investigator with collaborators during the last fifteen years. A broad approach will be used that combines methods of the ergodic theory, the theory of nonuniformly hyperbolic dynamical systems, stability theory of infinite dimensional dynamical systemq and the theory of bifurcations. The recent breakthrough in the understanding of a complex (chaotic) dynamics of systems evolving in time (dynamical systems) was connected with the studies of nonextended (point) dynamical systems whose dynamics can be completely described by measurements performed at a single point of a real (physical) space. Evolution of extended systems such as the flow of fluids and gases, large systems of interacting elements in electronics (e.g. Josephson junctions) needs to be studied in space and time simultaneously. The purpose of this research is to pursue the adequate approaches to study such space-time evolution and to develop algorithms to evaluate quantities that characterize space-time dynamics. The other novelty is the study of systems with dynamics which have both random and deterministic components (but not small, random perturbations of deterministic systems or vise versa). Most of systems in applications are of this type, e.g. in chemistry, artificial intelligence, in the management of warehouses etc.