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ACCURATE HIGH ORDER COMPUTATION OF INVARIANT MANIFOLDS FOR LONG PERIODIC ORBITS OF MAPS AND EQUILIBRIUM STATES OF PDE
Title
ACCURATE HIGH ORDER COMPUTATION OF INVARIANT MANIFOLDS FOR LONG PERIODIC ORBITS OF MAPS AND EQUILIBRIUM STATES OF PDE.
Creator
Gonzalez, Jorge L., Mireles-James, Jason, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description
The study of the long time behavior of nonlinear systems is not effortless, but it is very rewarding. The computation of invariant objects, in particular manifolds provide the scientist with the ability to make predictions at the frontiers of science. However, due to the presence of strong nonlinearities in many important applications, understanding the propagation of errors becomes necessary in order to quantify the reliability of these predictions, and to build sound foundations for future...
Show moreThe study of the long time behavior of nonlinear systems is not effortless, but it is very rewarding. The computation of invariant objects, in particular manifolds provide the scientist with the ability to make predictions at the frontiers of science. However, due to the presence of strong nonlinearities in many important applications, understanding the propagation of errors becomes necessary in order to quantify the reliability of these predictions, and to build sound foundations for future discoveries. This dissertation develops methods for the accurate computation of high-order polynomial approximations of stable/unstable manifolds attached to long periodic orbits in discrete time dynamical systems. For this purpose a multiple shooting scheme is applied to invariance equations for the manifolds obtained using the Parameterization Method developed by Xavier Cabre, Ernest Fontich and Rafael De La Llave in [CFdlL03a, CFdlL03b, CFdlL05].
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Date Issued
2020
PURL
http://purl.flvc.org/fau/fd/FA00013468
Subject Headings
Invariant manifolds, Nonlinear systems, Diffeomorphisms, Parabolic partial differential equations, Differential equations, Partial
Format
Document (PDF)
Spectral decomposition of grid data
Title
Spectral decomposition of grid data.
Creator
Donovan, Andrew., Harriet L. Wilkes Honors College
Abstract/Description
Spectral decomposition is a method of expressing functions as a harmonic series, and can be used for the simplification of complicated physical problems. This type of analysis requires knowledge of the function at all points on a circle or sphere. In problems where the function is known only at discreet points, regular intervals in a rectangular grid, for example, numerical methods must be employed to compute approximate coefficients for the harmonic expansion. In this paper, we investigate...
Show moreSpectral decomposition is a method of expressing functions as a harmonic series, and can be used for the simplification of complicated physical problems. This type of analysis requires knowledge of the function at all points on a circle or sphere. In problems where the function is known only at discreet points, regular intervals in a rectangular grid, for example, numerical methods must be employed to compute approximate coefficients for the harmonic expansion. In this paper, we investigate numerical methods for computing Fourier coefficients of a two dimensional function at a fixed radius, and spherical harmonic coefficients in three dimensions on a sphere of fixed radius.
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Date Issued
2005
PURL
http://purl.flvc.org/FAU/11572
Subject Headings
Inverse problems (Differential equations), Boundary value problems, Differential equations, Partial, Mathematical physics, Harmonic analysis
Format
Document (PDF)
General relativistic quasi-local angular momentum continuity and the stability of strongly elliptic eigenvalue problems
Title
General relativistic quasi-local angular momentum continuity and the stability of strongly elliptic eigenvalue problems.
Creator
Wilder, Shawn M., Beetle, Christopher, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Physics
Abstract/Description
In general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the...
Show moreIn general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the continuity of an angular momentum definition that employs an approximate Killing field that is an eigenvector of a particular second-order differential operator. We find that the eigenvector varies continuously in Hilbert space under smooth perturbations of a smooth boundary geometry. Furthermore, we find that not only is the approximate Killing field continuous but that the eigenvalue problem which defines it is stable in the sense that all of its eigenvalues and eigenvectors are continuous in Hilbert space. We conclude that the stability follows because the eigenvalue problem is strongly elliptic. Additionally, we provide a practical introduction to the mathematical theory of strongly elliptic operators and generalize the above stability results for a large class of such operators.
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Date Issued
2014
PURL
http://purl.flvc.org/fau/fd/FA00004235
Subject Headings
Boundary element methods, Boundary value problems, Differential equations, Elliptic -- Numerical solutions, Differential equations, Partial -- Numerical solutions, Eigenvalues, Spectral theory (Mathematics)
Format
Document (PDF)
Derivation of planar diffeomorphisms from Hamiltonians with a kick
Title
Derivation of planar diffeomorphisms from Hamiltonians with a kick.
Creator
Barney, Zalmond C., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
In this thesis we will discuss connections between Hamiltonian systems with a periodic kick and planar diffeomorphisms. After a brief overview of Hamiltonian theory we will focus, as an example, on derivations of the Hâenon map that can be obtained by considering kicked Hamiltonian systems. We will conclude with examples of Hâenon maps of interest.
Date Issued
2011
PURL
http://purl.flvc.org/FAU/3329833
Subject Headings
Mathematical physics, Differential equations, Partial, Hamiltonian systems, Algebra, Linear, Chaotic behavior in systems
Format
Document (PDF)
On the Laplacian and fractional Laplacian in exterior domains, and applications to the dissipative quasi-geostrophic equation
Title
On the Laplacian and fractional Laplacian in exterior domains, and applications to the dissipative quasi-geostrophic equation.
Creator
Kosloff, Leonardo., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
In this work, we develop an extension of the generalized Fourier transform for exterior domains due to T. Ikebe and A. Ramm for all dimensions n>2 to study the Laplacian, and fractional Laplacian operators in such a domain. Using the harmonic extension approach due to L. Caffarelli and L. Silvestre, we can obtain a localized version of the operator, so that it is precisely the square root of the Laplacian as a self-adjoint operator in L2 with DIrichlet boundary conditions. In turn, this...
Show moreIn this work, we develop an extension of the generalized Fourier transform for exterior domains due to T. Ikebe and A. Ramm for all dimensions n>2 to study the Laplacian, and fractional Laplacian operators in such a domain. Using the harmonic extension approach due to L. Caffarelli and L. Silvestre, we can obtain a localized version of the operator, so that it is precisely the square root of the Laplacian as a self-adjoint operator in L2 with DIrichlet boundary conditions. In turn, this allowed us to obtain a maximum principle for solutions of the dissipative two-dimensional quasi-geostrophic equation the exterior domain, which we apply to prove decay results using an adaptation of the Fourier Splitting method of M.E. Schonbek.
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Date Issued
2012
PURL
http://purl.flvc.org/FAU/3355570
Subject Headings
Fluid dynamics, Data processing, Laplacian matrices, Attractors (Mathematics), Differential equations, Partial
Format
Document (PDF)
Stability analysis for singularly perturbed systems with time-delays
Title
Stability analysis for singularly perturbed systems with time-delays.
Creator
Yang, Yang, Wang, Yuan, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
Singularly perturbed systems with or without delays commonly appear in mathematical modeling of physical and chemical processes, engineering applications, and increasingly, in mathematical biology. There has been intensive work for singularly perturbed systems, yet most of the work so far focused on systems without delays. In this thesis, we provide a new set of tools for the stability analysis for singularly perturbed control systems with time delays.
Date Issued
2015
PURL
http://purl.flvc.org/fau/fd/FA00004423, http://purl.flvc.org/fau/fd/FA00004423
Subject Headings
Biology -- Mathematical models, Biomathematics, Differentiable dynamical systems, Differential equations, Partial -- Numerical solutions, Global analysis (Mathematics), Lyapunov functions, Nonlinear theories
Format
Document (PDF)