Continuous Complexity and Dynamics

Ibm Thomas J Watson Research Center, Yorktown Heights NY

Investigators

Abstract

Abstract : The Principal Investigator is Michael Shub. Shub and co-workers Pugh, Wilkinson and others have established the stable ergodicity of a wide range of partially hyperbolic dynamical systems which preserve a smooth volume element, thus establishing the statistical robustness of these chaotic systems. The thrust of the work proposed is to extend these results to include almost all partially hyperbolic systems and to eliminate the hypothesis of volume preservation. This is an ambitious goal, but the results obtained included the development of new tools, namely julienne quasi-conformality and julienne density point preservation of the stable holonomy maps, whose power has not been fully explored. A second theme is a study of the complexity of continuous problems centered around the analysis and application of variants of Newton's method. Chaotic dynamical systems are pervasive in scientific, engineering and numerical simulation applications. The proposed work would lay foundations for the statistical analysis of these systems, whose deterministic behavior is unpredictable due to the presence of sensitive dependence on initial conditions. Newton's method for solving systems of equations is one of the main algorithms of numerical analysis. A better understanding of its properties and those of variants can make the solution of problems of scientific and engineering origin more efficient.