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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 memory-e 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 join-irreducible attractor-repeller pairs in computing a Lyapunov function, rather than having to use all possible attractor-repeller 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 gradient-like 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 non-invertible 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., 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].
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 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.
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 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)
- 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 time-dependent 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 Rendleman-Bartter model, the Vasicek model, and the Cox-Ingersoll-Ross (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 Rendleman-Bartter model, the Vasicek model, and the Cox-Ingersoll-Ross (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 Hull-White one-factor model, which does provide this fit. After calibrating, simulating, and comparing these four models, we find that the Hull-White 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 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.
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 Navier-Stokes 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 pressure-based computer program for a general Navier-Stokes equations has been developed. Body-fitted coordinate system is employed to handle flows with complex geometry. Non-staggered 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 finite-difference equations are solved by TDMA, while the whole solution is obtained through an under-relaxed iterative process. The...
Show moreA pressure-based computer program for a general Navier-Stokes equations has been developed. Body-fitted coordinate system is employed to handle flows with complex geometry. Non-staggered 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 finite-difference equations are solved by TDMA, while the whole solution is obtained through an under-relaxed 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 k-epsilon 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 three-dimensional 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
- Navier-Stokes equations--Numerical solutions--Data processing, Algorithms, Flows (Differential dynamical systems)
- Format
- Document (PDF)
- Title
- Karhunen-Loeve decomposition for non stationary propulsor flow noise.
- Creator
- Kersulec, Jean-Luc., 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 Karhunen-Loeve 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 Karhunen-Loeve 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 second-order lagrangian.
- Creator
- Adams, Ronald Edward, Kalies, William D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
A second-order 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 lower-order derivatives play a key role in forcing certain types of dynamics. However, the application of...
Show moreA second-order 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 lower-order 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 second-order 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 two-actor cooperation–competition conflict models.
- Creator
- Liebovitch, Larry S., Naudot, Vincent, Vallacher, Robin R., Nowak, Andrzej, Bui-Wrzosinska, Lan, Coleman, Peter T.
- Date Issued
- 2008-11-01
- PURL
- http://purl.flvc.org/fau/165475
- Subject Headings
- Nonlinear theories, Social systems--Mathematical models, Conflict management, Cooperativeness, Differential equations, Competition, Dynamics--Mathematical 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, Chi-Tay, Florida Atlantic University, College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
- Abstract/Description
-
Silicon carbide as a representative wide band-gap 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 high-temperature, high-power and high-frequency 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 band-gap 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 high-temperature, high-power and high-frequency 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 Alexander-Haasen 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 multi-patch, flux-based models to the input of infected people: epidemic response to initiated events.
- Creator
- Liebovitch, Larry S., Schwartz, Ira B., Rho, Young-Ah
- Date Issued
- 2008-07-21
- PURL
- http://purl.flvc.org/FAU/165229
- Subject Headings
- Communicable diseases--Epidemiology--Mathematical models, Epidemiologic Methods, Differential equations, Dynamics--Mathematical models, Spatial systems--Mathematical models, Population dynamics, Emerging infectious diseases
- Format
- Document (PDF)