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 Title
 An Algorithmic Approach to The Lattice Structures of Attractors and Lyapunov functions.
 Creator
 Kasti, Dinesh, Kalies, William D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Ban and Kalies [3] proposed an algorithmic approach to compute attractor repeller pairs and weak Lyapunov functions based on a combinatorial multivalued mapping derived from an underlying dynamical system generated by a continuous map. We propose a more e cient way of computing a Lyapunov function for a Morse decomposition. This combined work with other authors, including Shaun Harker, Arnoud Goulet, and Konstantin Mischaikow, implements a few techniques that makes the process of nding a...
Show moreBan and Kalies [3] proposed an algorithmic approach to compute attractor repeller pairs and weak Lyapunov functions based on a combinatorial multivalued mapping derived from an underlying dynamical system generated by a continuous map. We propose a more e cient way of computing a Lyapunov function for a Morse decomposition. This combined work with other authors, including Shaun Harker, Arnoud Goulet, and Konstantin Mischaikow, implements a few techniques that makes the process of nding a global Lyapunov function for Morse decomposition very e  cient. One of the them is to utilize highly memorye cient data structures: succinct grid data structure and pointer grid data structures. Another technique is to utilize Dijkstra algorithm and Manhattan distance to calculate a distance potential, which is an essential step to compute a Lyapunov function. Finally, another major technique in achieving a signi cant improvement in e ciency is the utilization of the lattice structures of the attractors and attracting neighborhoods, as explained in [32]. The lattice structures have made it possible to let us incorporate only the joinirreducible attractorrepeller pairs in computing a Lyapunov function, rather than having to use all possible attractorrepeller pairs as was originally done in [3]. The distributive lattice structures of attractors and repellers in a dynamical system allow for general algebraic treatment of global gradientlike dynamics. The separation of these algebraic structures from underlying topological structure is the basis for the development of algorithms to manipulate those structures, [32, 31]. There has been much recent work on developing and implementing general compu tational algorithms for global dynamics which are capable of computing attracting neighborhoods e ciently. We describe the lifting of sublattices of attractors, which are computationally less accessible, to lattices of forward invariant sets and attract ing neighborhoods, which are computationally accessible. We provide necessary and su cient conditions for such a lift to exist, in a general setting. We also provide the algorithms to check whether such conditions are met or not and to construct the lift when they met. We illustrate the algorithms with some examples. For this, we have checked and veri ed these algorithms by implementing on some noninvertible dynamical systems including a nonlinear Leslie model.
Show less  Date Issued
 2016
 PURL
 http://purl.flvc.org/fau/fd/FA00004668
 Subject Headings
 Differential equations  Numerical solutions., Differentiable dynamical systems., Algorithms.
 Format
 Document (PDF)
 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.
Show less  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)
 Title
 ACCURATE HIGH ORDER COMPUTATION OF INVARIANT MANIFOLDS FOR LONG PERIODIC ORBITS OF MAPS AND EQUILIBRIUM STATES OF PDE.
 Creator
 Gonzalez, Jorge L., MirelesJames, 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 highorder 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].
Show less  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)
 Title
 General relativistic quasilocal 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 welldefined 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 welldefined 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 secondorder 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.
Show less  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)
 Title
 Stability analysis for singularly perturbed systems with timedelays.
 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)
 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)
 Title
 Modeling and simulating interest rates via timedependent mean reversion.
 Creator
 Dweck, Andrew Jason, Long, Hongwei, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

The purpose of this thesis is to compare the effectiveness of several interest rate models in fitting the true value of interest rates. Up until 1990, the universally accepted models were the equilibrium models, namely the RendlemanBartter model, the Vasicek model, and the CoxIngersollRoss (CIR) model. While these models were probably considered relatively accurate around the time of their discovery, they do not provide a good fit to the initial term structure of interest rates, making...
Show moreThe purpose of this thesis is to compare the effectiveness of several interest rate models in fitting the true value of interest rates. Up until 1990, the universally accepted models were the equilibrium models, namely the RendlemanBartter model, the Vasicek model, and the CoxIngersollRoss (CIR) model. While these models were probably considered relatively accurate around the time of their discovery, they do not provide a good fit to the initial term structure of interest rates, making them substandard for use by traders in pricing interest rate options. The fourth model we consider is the HullWhite onefactor model, which does provide this fit. After calibrating, simulating, and comparing these four models, we find that the HullWhite model gives the best fit to our data sets.
Show less  Date Issued
 2014
 PURL
 http://purl.flvc.org/fau/fd/FA00004103, http://purl.flvc.org/fau/fd/FA00004103
 Subject Headings
 Game theory, Investment analysis, Options (Finance), Recursive functions, Stochastic differential equations
 Format
 Document (PDF)
 Title
 On the Laplacian and fractional Laplacian in exterior domains, and applications to the dissipative quasigeostrophic 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 selfadjoint 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 selfadjoint operator in L2 with DIrichlet boundary conditions. In turn, this allowed us to obtain a maximum principle for solutions of the dissipative twodimensional quasigeostrophic equation the exterior domain, which we apply to prove decay results using an adaptation of the Fourier Splitting method of M.E. Schonbek.
Show less  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)
 Title
 A general pressure based NavierStokes solver in arbitrary configurations.
 Creator
 Ke, Zhao Ping., Florida Atlantic University, Chow, Wen L., College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

A pressurebased computer program for a general NavierStokes equations has been developed. Bodyfitted coordinate system is employed to handle flows with complex geometry. Nonstaggered grid is used while the pressure oscillation is eliminated by a special pressure interpolation scheme. The hybrid algorithm is adopted to discretize the equations and the finitedifference equations are solved by TDMA, while the whole solution is obtained through an underrelaxed iterative process. The...
Show moreA pressurebased computer program for a general NavierStokes equations has been developed. Bodyfitted coordinate system is employed to handle flows with complex geometry. Nonstaggered grid is used while the pressure oscillation is eliminated by a special pressure interpolation scheme. The hybrid algorithm is adopted to discretize the equations and the finitedifference equations are solved by TDMA, while the whole solution is obtained through an underrelaxed iterative process. The pressure field is evaluated using the compressible from of the SIMPLE algorithm., To test the accuracy and efficiency of the computer program, problems of incompressible and compressible flows are calculated. As examples of inviscid compressible flow problems, flows over a bump with 10% and 4% thickness are computed with the incoming Mach numbers of M[infinity] = 0.5 (subsonic flow), M[infinity] = 0.675 (transonic flow and M[infinity] = 1.65 (supersonic flow). One laminar subsonic flow over a bump with 5% thickness at M[infinity] = 0.5 is also calculated with the consideration of the full energy equation. With the help of the kepsilon model incorporating the wall function, the computations of two turbulent incompressible flows are carried out. One is the flow past a flat plate and the other over a flame holder. As an application to the threedimensional flow, a laminar flow in a driven cubic cavity is calculated. All the numerical results obtained here are compared with experimental data or other numerical results available in the literature.
Show less  Date Issued
 1993
 PURL
 http://purl.flvc.org/fcla/dt/12330
 Subject Headings
 NavierStokes equationsNumerical solutionsData processing, Algorithms, Flows (Differential dynamical systems)
 Format
 Document (PDF)
 Title
 KarhunenLoeve decomposition for non stationary propulsor flow noise.
 Creator
 Kersulec, JeanLuc., Florida Atlantic University, Glegg, Stewart A. L., College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

The aim of this thesis is to develop a theory for non stationary propulsor flow noise. The model which is proposed is based on Amiet's paper "Acoustic Radiation from an Airfoil in a Turbulent Stream" [1], which describes broad band noise when a simple model of airfoil interacts with a turbulent flow, under the assumption of stationarity. The KarhunenLoeve method provides a set of modes which describe the turbulent flow without the assumption of stationarity. A method is described to obtain...
Show moreThe aim of this thesis is to develop a theory for non stationary propulsor flow noise. The model which is proposed is based on Amiet's paper "Acoustic Radiation from an Airfoil in a Turbulent Stream" [1], which describes broad band noise when a simple model of airfoil interacts with a turbulent flow, under the assumption of stationarity. The KarhunenLoeve method provides a set of modes which describe the turbulent flow without the assumption of stationarity. A method is described to obtain broad band noise calculations when the mean turbulent flow varies with time and produces non stationary turbulence. A comparison of the numerical results obtained with the results from the paper of reference [1] shows the characteristics of time varying sound radiation. The various mathematical formulae will give a starting point to the analysis of real time varying flows, which are not considered in this thesis.
Show less  Date Issued
 2005
 PURL
 http://purl.flvc.org/fcla/dt/13233
 Subject Headings
 Aerodynamic noise, Turbulence, Aerofoils, Unsteady flow (Aerodynamics), Nonlinear control theory, Differential equations, Nonlinear
 Format
 Document (PDF)
 Title
 Curve shortening in secondorder lagrangian.
 Creator
 Adams, Ronald Edward, Kalies, William D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

A secondorder Lagrangian system is a generalization of a classical mechanical system for which the Lagrangian action depends on the second derivative of the state variable. Recent work has shown that the dynamics of such systems c:an be substantially richer than for classical Lagrangian systems. In particular, topological properties of the planar curves obtained by projection onto the lowerorder derivatives play a key role in forcing certain types of dynamics. However, the application of...
Show moreA secondorder Lagrangian system is a generalization of a classical mechanical system for which the Lagrangian action depends on the second derivative of the state variable. Recent work has shown that the dynamics of such systems c:an be substantially richer than for classical Lagrangian systems. In particular, topological properties of the planar curves obtained by projection onto the lowerorder derivatives play a key role in forcing certain types of dynamics. However, the application of these techniques requires an analytic restriction on the Lagrangian that it satisfy a twist property. In this dissertation we approach this problem from the point of view of curve shortening in an effort to remove the twist condition. In classical curve shortening a family of curves evolves with a velocity which is normal to the curve and proportional to its curvature. The evolution of curves with decreasing action is more general, and in the first part of this dissertation we develop some results for curve shortening flows which shorten lengths with respect to a Finsler metric rather than a Riemannian metric. The second part of this dissertation focuses on analytic methods to accommodate the fact that the Finsler metric for secondorder Lagrangian system has singularities. We prove the existence of simple periodic solutions for a general class of systems without requiring the twist condition. Further; our results provide a frame work in which to try to further extend the topological forcing theorems to systems without the twist condition.
Show less  Date Issued
 2014
 PURL
 http://purl.flvc.org/fau/fd/FA00004175, http://purl.flvc.org/fau/fd/FA00004175
 Subject Headings
 Critical point theory (Mathematical analysis), Differentiable dynamical systems, Geometry,Differential, Lagrange equations, Lagrangian functions, Mathematical optimization, Surfaces of constant curvature
 Format
 Document (PDF)
 Title
 Dynamics of twoactor cooperation–competition conflict models.
 Creator
 Liebovitch, Larry S., Naudot, Vincent, Vallacher, Robin R., Nowak, Andrzej, BuiWrzosinska, Lan, Coleman, Peter T.
 Date Issued
 20081101
 PURL
 http://purl.flvc.org/fau/165475
 Subject Headings
 Nonlinear theories, Social systemsMathematical models, Conflict management, Cooperativeness, Differential equations, Competition, DynamicsMathematical models
 Format
 Document (PDF)
 Title
 Finite Element Modeling of Dislocation Multiplication in Silicon Carbide Crystals Grown by Physical Vapor Transport Method.
 Creator
 Chen, Qingde, Tsai, ChiTay, Florida Atlantic University, College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

Silicon carbide as a representative wide bandgap semiconductor has recently received wide attention due to its excellent physical, thermal and especially electrical properties. It becomes a promising material for electronic and optoelectronic device under hightemperature, highpower and highfrequency and intense radiation conditions. During the Silicon Carbide crystal grown by the physical vapor transport process, the temperature gradients induce thermal stresses which is a major cause of...
Show moreSilicon carbide as a representative wide bandgap semiconductor has recently received wide attention due to its excellent physical, thermal and especially electrical properties. It becomes a promising material for electronic and optoelectronic device under hightemperature, highpower and highfrequency and intense radiation conditions. During the Silicon Carbide crystal grown by the physical vapor transport process, the temperature gradients induce thermal stresses which is a major cause of the dislocations multiplication. Although large dimension crystal with low dislocation density is required for satisfying the fast development of electronic and optoelectronic device, high dislocation densities always appear in large dimension crystal. Therefore, reducing dislocation density is one of the primary tasks of process optimization. This dissertation aims at developing a transient finite element model based on the AlexanderHaasen model for computing the dislocation densities in a crystal during its growing process. Different key growth parameters such as temperature gradient, crystal size will be used to investigate their influence on dislocation multiplications. The acceptable and optimal crystal diameter and temperature gradient to produce the lowest dislocation density in SiC crystal can be obtained through a thorough numerical investigation using this developed finite element model. The results reveal that the dislocation density multiplication in SiC crystal are easily affected by the crystal diameter and the temperature gradient. Generally, during the iterative calculation for SiC growth, the dislocation density multiples very rapidly in the early growth phase and then turns to a relatively slow multiplication or no multiplication at all. The results also show that larger size and higher temperature gradient causes the dislocation density enters rapid multiplication phase sooner and the final dislocation density in the crystal is higher.
Show less  Date Issued
 2015
 PURL
 http://purl.flvc.org/fau/fd/FA00004489, http://purl.flvc.org/fau/fd/FA00004489
 Subject Headings
 Computational grids, Crystals  Mathematical models, Differential equations  Data processing, Dislocations in crystals, Engineering mathematics, Finite element method
 Format
 Document (PDF)
 Title
 Stochastic optimal impulse control of jump diffusions with application to exchange rate.
 Creator
 Perera, Sandun C., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

We generalize the theory of stochastic impulse control of jump diffusions introduced by Oksendal and Sulem (2004) with milder assumptions. In particular, we assume that the original process is affected by the interventions. We also generalize the optimal central bank intervention problem including market reaction introduced by Moreno (2007), allowing the exchange rate dynamic to follow a jump diffusion process. We furthermore generalize the approximation theory of stochastic impulse control...
Show moreWe generalize the theory of stochastic impulse control of jump diffusions introduced by Oksendal and Sulem (2004) with milder assumptions. In particular, we assume that the original process is affected by the interventions. We also generalize the optimal central bank intervention problem including market reaction introduced by Moreno (2007), allowing the exchange rate dynamic to follow a jump diffusion process. We furthermore generalize the approximation theory of stochastic impulse control problems by a sequence of iterated optimal stopping problems which is also introduced in Oksendal and Sulem (2004). We develop new results which allow us to reduce a given impulse control problem to a sequence of iterated optimal stopping problems even though the original process is affected by interventions.
Show less  Date Issued
 2009
 PURL
 http://purl.flvc.org/FAU/3174308
 Subject Headings
 Management, Mathematical models, Control theory, Stochastic differential equations, Distribution (Probability theory), Optimal stopping (Mathematical statistics), Economics, Mathematical
 Format
 Document (PDF)
 Title
 Dynamical response of multipatch, fluxbased models to the input of infected people: epidemic response to initiated events.
 Creator
 Liebovitch, Larry S., Schwartz, Ira B., Rho, YoungAh
 Date Issued
 20080721
 PURL
 http://purl.flvc.org/FAU/165229
 Subject Headings
 Communicable diseasesEpidemiologyMathematical models, Epidemiologic Methods, Differential equations, DynamicsMathematical models, Spatial systemsMathematical models, Population dynamics, Emerging infectious diseases
 Format
 Document (PDF)