Topics in Dynamical Systems: Attractors, Dimension, Lattice Model

Abstract.

The project outlines scientific activity in three directions. The first

is the theory of coupled map lattices (CML) where research is focused

on the thermodynamical limit for infinite-dimensional SRB measures

(in the sense of Sinai); stability of traveling wave solutions, and

transition from CMLs to PDEs via traveling waves.

It also includes the study of some well-known PDEs (for example,

Swift-Hohenberg equation which is of great interest in neurobiology).

The second direction is concerned with the dimension theory of dynamical

systems and includes the multifractal analysis of Birkhoff averages

with applications to some problems in number theory as well as

the multifractal rigidity phenomenon for conformal expanding maps.

The third direction of research deals with some recent advantages

in smooth ergodic theory. In particular, it is proposed to construct

some new examples of volume preserving diffeomorphisms

with non-zero Lyapunov exponents (on any Riemannian manifold and

with countable number of ergodic components). It is also proposed

to effect a multifractal formalism for two-dimensional hyperbolic

measures.

The main goal of the proposed research is to develop further the

mathematical theory of chaos. It deals with various aspects of

this theory including the appearance of chaotic motions in dynamical

systems (which are mathematical models of various phenomena in

physics, biology, economics, etc. ) and the interplay between chaotic

regimes and fractal geometric structure of the space. It is proposed

to study some macro-characteristics of chaotic behavior (such as

entropy and Lyapunov exponents) and relate them to fractal dimension

of the space. It is also proposed to construct some new examples of

chaotic systems both conservative and dissipative, finite-dimensional

as well as infinite-dimensional. These examples may serve as models

of such extremely complicated phenomena as turbulence in hydrodynamics,

neuron and memory activity in neuroscience, plant growth in plant biology, etc.