How well can one resolve the state space of a chaotic map?

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Title: How well can one resolve the state space of a chaotic map?
Author: Lippolis, Domenico
Abstract: All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching/contraction and the smearing due to noise. My goal is to determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additive white noise. That is achieved by computing the local eigenfunctions of the Fokker-Planck evolution operator in linearized neighborhoods of the periodic orbits of the corresponding deterministic system, and using overlaps of their widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables. The method applies in principle to both continuous- and discrete-time dynamical systems. Numerical tests of such optimal partitions on unimodal maps support my hypothesis.
Type: Dissertation
Date: 2010-04-06
Publisher: Georgia Institute of Technology
Subject: Symbolic dynamics
Periodic orbits.
State-space methods
Partitions (Mathematics)
Stochastic analysis
Department: Physics
Advisor: Committee Chair: Cvitanovic, Predrag; Committee Member: Bellissard, Jean; Committee Member: Weiss, Howard; Committee Member: Wiesenfeld, Kurt; Committee Member: Zhou, Hao Min
Degree: Ph.D.

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