Open Access Theses and Dissertations

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You searched for subject:(area preserving maps). Showing records 1 – 3 of 3 total matches.

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University of Colorado

1. Mitchell, Rebecca Amelia. Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps.

Degree: PhD, Applied Mathematics, 2017, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/94 

Mixing of a passive scalar in a fluid flow results from a two part process in which large gradients are first created by advection and then smoothed by diffusion. We investigate methods of designing efficient stirrers to optimize mixing of a passive scalar in a two-dimensional nonautonomous, incompressible flow over a finite time interval. The flow is modeled by a sequence of area-preserving maps whose parameters change in time, defining a mixing protocol. Stirring efficiency is measured by the mix norm, a negative Sobolev seminorm; its decrease implies creation of fine-scale structure. A Perron-Frobenius operator is used to numerically advect the scalar for three examples: compositions of Chirikov standard maps, of Harper maps, and of blinking vortex maps. In the case of the standard maps, we find that a protocol corresponding to a single vertical shear composed with horizontal shearing at all other steps is nearly optimal. For the Harper maps, we devise a predictive, one-step scheme to choose appropriate fixed point stabilities and to control the Fourier spectrum evolution to obtain a near optimal protocol. For the blinking vortex model, we devise two schemes: A one-step predictive scheme to determine a vortex location, which has modest success in producing an efficient stirring protocol, and a scheme that finds the true optimal choice of vortex positions and directions of rotation given four possible fixed vortex locations. The results from the numerical experiments suggest that an effective stirring protocol must include not only steps devoted to decreasing the mix norm, but also steps devoted to preparing the density profile for future steps of mixing. Advisors/Committee Members: James D. Meiss, Keith Julien, Juan Restrepo, John Crimaldi, Roseanna Neupauer.

Subjects/Keywords: area-preserving maps; chaotic mixing; dynamical sytems; Perron-Frobenius; Applied Mathematics

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APA (6th Edition):

Mitchell, R. A. (2017). Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/94

Chicago Manual of Style (16th Edition):

Mitchell, Rebecca Amelia. “Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps.” 2017. Doctoral Dissertation, University of Colorado. Accessed January 18, 2021. https://scholar.colorado.edu/appm_gradetds/94.

MLA Handbook (7th Edition):

Mitchell, Rebecca Amelia. “Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps.” 2017. Web. 18 Jan 2021.

Vancouver:

Mitchell RA. Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps. [Internet] [Doctoral dissertation]. University of Colorado; 2017. [cited 2021 Jan 18]. Available from: https://scholar.colorado.edu/appm_gradetds/94.

Council of Science Editors:

Mitchell RA. Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps. [Doctoral Dissertation]. University of Colorado; 2017. Available from: https://scholar.colorado.edu/appm_gradetds/94


Queens University

2. Jensen, Erik. Homoclinic Points in the Composition of Two Reflections .

Degree: Mathematics and Statistics, 2013, Queens University

URL: http://hdl.handle.net/1974/8288 

We examine a class of planar area preserving mappings and give a geometric condition that guarantees the existence of homoclinic points. To be more precise, let f,g:R  →  R be C1 functions with domain all of R. Let F:R2  →  R2 denote a horizontal reflection in the curve x=-f(y), and let G:R2  →  R2 denote a vertical reflection in the curve y=g(x). We consider maps of the form T=G ∘ F and show that a simple geometric condition on the fixed point sets of F and G leads to the existence of a homoclinic point for T.

Subjects/Keywords: Area Preserving Maps ; Homoclinic Points ; Mathematics ; Dynamical Systems

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Jensen, E. (2013). Homoclinic Points in the Composition of Two Reflections . (Thesis). Queens University. Retrieved from http://hdl.handle.net/1974/8288

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Jensen, Erik. “Homoclinic Points in the Composition of Two Reflections .” 2013. Thesis, Queens University. Accessed January 18, 2021. http://hdl.handle.net/1974/8288.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Jensen, Erik. “Homoclinic Points in the Composition of Two Reflections .” 2013. Web. 18 Jan 2021.

Vancouver:

Jensen E. Homoclinic Points in the Composition of Two Reflections . [Internet] [Thesis]. Queens University; 2013. [cited 2021 Jan 18]. Available from: http://hdl.handle.net/1974/8288.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Jensen E. Homoclinic Points in the Composition of Two Reflections . [Thesis]. Queens University; 2013. Available from: http://hdl.handle.net/1974/8288

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation


University of Texas – Austin

3. Eschbacher, Peter Andrew. Quantifying stickiness in 2D area-preserving maps by means of recurrence plots.

Degree: MA, Physics, 2009, University of Texas – Austin

URL: http://hdl.handle.net/2152/ETD-UT-2009-05-158 

Stickiness is a ubiquitous property of dynamical systems. However, recognizing whether an orbit is temporarily `stuck' (and therefore very nearly quasiperiodic) is hard to detect. Outlined in this thesis is an approach to quantifying stickiness in area-preserving maps based on a tool called recurrence plots that is not very commonly used. With the analyses presented herein it is shown that recurrence plot methods can give very close estimates to stickiness exponents that were previously calculated using Poincare recurrence and other methods. To capture the dynamics, RP methods require shorter data series than more conventional methods and are able to represent a more-global analysis of recurrence. A description of stickiness of the standard map for a wide array of parameter strengths is presented and a start at analyzing the standard nontwist map is presented. Advisors/Committee Members: Morrison, Philip J. (advisor), Hazeltine, R. D. (Richard D.) (committee member).

Subjects/Keywords: Physics; Area-Preserving Maps; Stickiness; Recurrence Plots

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Eschbacher, P. A. (2009). Quantifying stickiness in 2D area-preserving maps by means of recurrence plots. (Masters Thesis). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/ETD-UT-2009-05-158

Chicago Manual of Style (16th Edition):

Eschbacher, Peter Andrew. “Quantifying stickiness in 2D area-preserving maps by means of recurrence plots.” 2009. Masters Thesis, University of Texas – Austin. Accessed January 18, 2021. http://hdl.handle.net/2152/ETD-UT-2009-05-158.

MLA Handbook (7th Edition):

Eschbacher, Peter Andrew. “Quantifying stickiness in 2D area-preserving maps by means of recurrence plots.” 2009. Web. 18 Jan 2021.

Vancouver:

Eschbacher PA. Quantifying stickiness in 2D area-preserving maps by means of recurrence plots. [Internet] [Masters thesis]. University of Texas – Austin; 2009. [cited 2021 Jan 18]. Available from: http://hdl.handle.net/2152/ETD-UT-2009-05-158.

Council of Science Editors:

Eschbacher PA. Quantifying stickiness in 2D area-preserving maps by means of recurrence plots. [Masters Thesis]. University of Texas – Austin; 2009. Available from: http://hdl.handle.net/2152/ETD-UT-2009-05-158

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