Dynamics and Kinetics

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PI: Leonid BunimovichProposal Number: 0140165 ABSTRACTThe proposed research deals with the studies of finite- andinfinite-dimensional dynamical systems as well as of somehybrid (intermediate between dynamical and stochastic) systems.The class of hybrid systems which will be specifically addressedis formed by deterministic walks in random environments. It has been shownrecently that a large subclass of such systems (walksin rigid environments) is completely solvable in 1D. The proposedstudy deals with the case of environments with variable rigidity.Another part of the proposal deals with the farther development ofrecently constructed first natural examples of Hamiltonian systemswith divided phase space. The goal now is to show that a rather general class of such billiards has positive metric entropy in any (finite) dimension. The considered problems of statistical mechanics deal with the rigorous derivation of formulas for kinetic coefficients and for escape rate in open systems. Among the problems which deal with infinite-dymensional systems are chaos-order transitions in Lattice Dynamical Systems and chaos in nonlinear wave equations. Other problems deal with various systems ranging from the Ergodic Theory (Benford's law)to some simple models of brain dynamics and logistics. Traditionally the mathematical models of real systems are either purely deterministic or purely stochastic. These two classes of models enjoy having a rich theory, and researchers have quite good intuition on their behavior, i.e. they basically know what to expect. This intuition is essentially based on a rich collection of completely solvable (i.e. completely understood) models. However, a majority of real systems are neither purely deterministic nor purely stochastic. Instead they have both (deterministic and stochastic) features. This project deals with a big class of such models which were independently introduced in communication theory, statistical physics,chemical kinetics, theory of artificiall intellect, etc.