Masters Theses

Date of Award

8-1989

Degree Type

Thesis

Degree Name

Master of Science

Major

Mathematics

Major Professor

Mark Kot

Committee Members

Lawrence Husch, Michael Thomason

Abstract

This thesis details the development of two efficient algorithms for the detection and characterization of scale-invariant structure in strange attractors of nonlinear dynamical systems. Its first chapter is preliminary exposition, the object of which is the self-contained development, in an historical context, of the necessary background and nomenclature of nonlinear science and geometric measure theory; the principal topics are strange attractors and their fractal dimensions. The subsequent two chapters describe relevant structures and results of set theory, discrete mathematics and computer science, and their application in the estimation of fractal dimensions of strange attractors. Chapter 2 introduces an abstract structure of discrete mathematics, the tree, and its application in an unconventional algorithm for the estimation of one fractal dimension, capacity. The collating function, a one-to-one correspondence defined by Georg Cantor, is exploited to represent a bounded, finite set of points as an inherently tree-like list of paths. It is shown that, given a lexicographically-ordered list of paths and a comparison operation defined on its elements, one may count, simultaneously for all lengths 5 in some geometrically decreasing sequence, the number of cubes of edge length 5 required to cover the corresponding set of points. Given this information, one may then proceed in some conventional manner to estimate the capacity of the set. Chapter 3 describes conventional methods and data structures of non-numeric programming, namely range searching and the multidimensional binary tree, and their application in one algorithm for the estimation of correlation dimension. Empirical studies of the computational complexity of this algorithm are presented; they indicate that significant reductions in execution times may be achieved by avoiding computations that involve the large-scale spatial structure of an attractor. Multifractal characterizations of spatial structure are, by choice, not within the scope of this thesis. However, proponents of the multifractal approach will find that the methods described herein may readily be incorporated in efficient algorithms for the estimation of spectra of generalized dimensions; the manner in which this may be done is briefly indicated in the introduction.

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