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https://hdl.handle.net/2142/21923
Description
Title
Two problems in iterative nonlinear maps
Author(s)
Perez, Gabriel
Issue Date
1990
Doctoral Committee Chair(s)
Chang, Shau-Jin
Department of Study
Physics
Discipline
Physics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Statistics
Physics, General
Language
eng
Abstract
This thesis consists of two parts. In the first part we study the dynamics and the boundaries between basins of attraction of a logistic map defined in real 2 x 2 matrices. We show how these boundaries are of a mixed type, combining fractal and smooth regions. We find in these boundaries a very complex distribution of transient scaling behaviors that can mask in some regions the fractal nature of the set. We propose a mechanism for the origin of these arbitrarily large transients in the scaling laws.
In the second part we consider the classical Chirikov map as an eigenvalue problem and analyze the implications of its translational symmetry in the momentum coordinate p. The map is then formulated on a discrete lattice of size N x N, and numerical solutions are obtained for the eigenstates of the classical evolution operator, and for the spectrum of eigenvalues as a function of the lattice momentum. A comparison is made with the analogous system defined on a discrete finite quantum lattice. We obtain the semiclassical limit on this lattice and show how it corresponds to a modified classical evolution operator, where delta functions are relaxed into Gaussians and some phases are introduced. This modifications reduce the number of participating eigenstates from $N\sp2$ to N. The possibility is raised that an arbitrary area-preserving map could be quantized, in an approximated way, by a careful smearing of the map equations and the introduction of well-chosen phases.
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