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A qualitative numerical study of hig...

Albers, David James.

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A qualitative numerical study of high dimensional dynamical systems.

Record Type:	Electronic resources : Monograph/item
Title/Author:	A qualitative numerical study of high dimensional dynamical systems./
Author:	Albers, David James.
Description:	184 p.
Notes:	Source: Dissertation Abstracts International, Volume: 65-08, Section: B, page: 4062.
Contained By:	Dissertation Abstracts International65-08B.
Subject:	Physics, General. -
Online resource:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3143090
ISBN:	0496010638

A qualitative numerical study of high dimensional dynamical systems.
Albers, David James.

A qualitative numerical study of high dimensional dynamical systems. - 184 p.

Source: Dissertation Abstracts International, Volume: 65-08, Section: B, page: 4062.

Thesis (Ph.D.)--The University of Wisconsin - Madison, 2004.

Since Poincare, the father of modern mathematical dynamical systems, much effort has been exerted to achieve a qualitative understanding of the physical world via a qualitative understanding of the functions we use to model the physical world. In this thesis, we construct a numerical framework suitable for a qualitative, statistical study of dynamical systems using the space of artificial neural networks. We analyze the dynamics along intervals in parameter space, separating the set of neural networks into roughly four regions: the fixed point to the first bifurcation; the route to chaos; the chaotic region; and a transition region between chaos and finite-state neural networks. The study is primarily with respect to high-dimensional dynamical systems. We make the following general conclusions as the dimension of the dynamical system is increased: the probability of the first bifurcation being of type Neimark-Sacker is greater than ninety-percent; the most probable route to chaos is via a cascade of bifurcations of high-period periodic orbits, quasi-periodic orbits, and 2-tori; there exists an interval of parameter space such that hyperbolicity is violated on a countable, Lebesgue measure 0, "increasingly dense" subset; chaos is much more likely to persist with respect to parameter perturbation in the chaotic region of parameter space as the dimension is increased; moreover, as the number of positive Lyapunov exponents is increased, the likelihood that any significant portion of these positive exponents can be perturbed away decreases with increasing dimension. The maximum Kaplan-Yorke dimension and the maximum number of positive Lyapunov exponents increases linearly with dimension. The probability of a dynamical system being chaotic increases exponentially with dimension. The results with respect to the first bifurcation and the route to chaos comment on previous results of Newhouse, Ruelle, Takens, Broer, Chenciner, and Iooss. Moreover, results regarding the high-dimensional chaotic region of parameter space is interpreted and related to the closing lemma of Pugh, the windows conjecture of Barreto, the stable ergodicity theorem of Pugh and Shub, and structural stability theorem of Robbin, Robinson, and Mane.

ISBN: 0496010638Subjects--Topical Terms:

1018488
Physics, General.

A qualitative numerical study of high dimensional dynamical systems.
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035 $a (UnM)AAI3143090
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100 1 $a Albers, David James. $3 1926880
245 1 2 $a A qualitative numerical study of high dimensional dynamical systems.
300 $a 184 p.
500 $a Source: Dissertation Abstracts International, Volume: 65-08, Section: B, page: 4062.
500 $a Supervisor: J. Clint Sprott.
502 $a Thesis (Ph.D.)--The University of Wisconsin - Madison, 2004.
520 $a Since Poincare, the father of modern mathematical dynamical systems, much effort has been exerted to achieve a qualitative understanding of the physical world via a qualitative understanding of the functions we use to model the physical world. In this thesis, we construct a numerical framework suitable for a qualitative, statistical study of dynamical systems using the space of artificial neural networks. We analyze the dynamics along intervals in parameter space, separating the set of neural networks into roughly four regions: the fixed point to the first bifurcation; the route to chaos; the chaotic region; and a transition region between chaos and finite-state neural networks. The study is primarily with respect to high-dimensional dynamical systems. We make the following general conclusions as the dimension of the dynamical system is increased: the probability of the first bifurcation being of type Neimark-Sacker is greater than ninety-percent; the most probable route to chaos is via a cascade of bifurcations of high-period periodic orbits, quasi-periodic orbits, and 2-tori; there exists an interval of parameter space such that hyperbolicity is violated on a countable, Lebesgue measure 0, "increasingly dense" subset; chaos is much more likely to persist with respect to parameter perturbation in the chaotic region of parameter space as the dimension is increased; moreover, as the number of positive Lyapunov exponents is increased, the likelihood that any significant portion of these positive exponents can be perturbed away decreases with increasing dimension. The maximum Kaplan-Yorke dimension and the maximum number of positive Lyapunov exponents increases linearly with dimension. The probability of a dynamical system being chaotic increases exponentially with dimension. The results with respect to the first bifurcation and the route to chaos comment on previous results of Newhouse, Ruelle, Takens, Broer, Chenciner, and Iooss. Moreover, results regarding the high-dimensional chaotic region of parameter space is interpreted and related to the closing lemma of Pugh, the windows conjecture of Barreto, the stable ergodicity theorem of Pugh and Shub, and structural stability theorem of Robbin, Robinson, and Mane.
590 $a School code: 0262.
650 4 $a Physics, General. $3 1018488
650 4 $a Mathematics. $3 515831
690 $a 0605
690 $a 0405
710 2 0 $a The University of Wisconsin - Madison. $3 626640
773 0 $t Dissertation Abstracts International $g 65-08B.
790 1 0 $a Sprott, J. Clint, $e advisor
790 $a 0262
791 $a Ph.D.
792 $a 2004
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3143090