Dynamical Systems Commons^™

Iterated Aluthge Transforms: A Brief Survey, Jorge Antezana, Enrique R. Pujals, Demetrio Stojanoff

Publications and Research

Given an r × r complex matrix T, if T = U|T| is the polar decomposition of T, then the Aluthge transform is defined by

∆(T) = |T|^1/2U|T|^1/2.

Let ∆ⁿ(T) denote the n-times iterated Aluthge transform of T, i.e. ∆⁰(T) = T and ∆ⁿ(T) = ∆(∆ⁿ⁻¹(T)), n ∈ N. In this paper we make a brief survey on the known properties and applications of …

Super Linear Algebra, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). This book expects the readers to be well-versed in linear algebra. Many theorems on super linear algebra and its properties are proved. Some theorems are left as exercises for the reader. These new class of super linear algebras which can be thought of as a set of linear algebras, following a stipulated condition, will find applications in several fields using computers. The authors feel that such a paradigm shift is essential in this computerized …

Hartman-Grobman Theorems Along Hyperbolic Stationary Trajectories, Edson A. Coayla-Teran, Salah-Eldin A. Mohammed, Paulo Régis C. Ruffino

Articles and Preprints

We extend the Hartman-Grobman theorems on discrete random dynamical systems (RDS), proved in [7], in two directions: For continuous RDS and for hyperbolic stationary trajectories. In this last case there exists a conjugacy between traveling neighbourhoods of trajectories and neighbourhoods of the origin in the corresponding tangent bundle. We present applications to deterministic dynamical systems.

Turing Patterns On Growing Spheres: The Exponential Case, Julijana Gjorgjieva, Jon T. Jacobsen

All HMC Faculty Publications and Research

We consider Turing patterns for reaction-diffusion systems on the surface of a growing sphere. In particular, we are interested in the effect of dynamic growth on the pattern formation. We consider exponential isotropic growth of the sphere and perform a linear stability analysis and compare the results with numerical simulations.

The Elementary Theory Of Normed Linear Spaces And Linear Functionals, Suresh Eswarthasan

Renée Crown University Honors Thesis Projects - All

Abstract not Included

A Remark On Conservative Diffeomorphisms, Jairo Bochi, Bassam R. Fayad, Enrique Pujals

Publications and Research

Abstract:

We show that a stably ergodic diffeomorphism can be C¹ approximated by a diffeomorphism having stably non-zero Lyapunov exponents.

Résumé:

On montre qu'un difféomorphisme stablement ergodique peut être C¹ approché par un difféomorphisme ayant des exposants de Lyapunov stablement non-nuls.

Geometric Field Stability And Normal Field Curvature Of Solution Sets Of Ordinary Differential Equations In Two Variables, Leslie L. Kerns

Theses, Dissertations and Capstones

The classical linearization approach to stability theory determines whether or not a system is stable in the vicinity of its equilibrium points. This classical approach partly depends on the validity of the linear approximation. The definition of stability developed in this article takes a different approach and uses a curvature function to assess the relative locations of solutions within a field of solutions (the underlying solution set of the ODE). The present approach involves calculations that directly yield stability information, without having to enter into the often lengthy eigenvalue-eigenvector method. The present results both complement and are compatible with the …

Solving Higher Order Dynamic Equations On Time Scales As First Order Systems, Elizabeth R. Duke

Theses, Dissertations and Capstones

Time scales calculus seeks to unite two disparate worlds: that of differential, Newtonian calculus and the difference calculus. As such, in place of differential and difference equations, time scales calculus uses dynamic equations. Many theoretical results have been developed concerning solutions of dynamic equations. However, little work has been done in the arena of developing numerical methods for approximating these solutions. This thesis work takes a first step in obtaining numerical solutions of dynamic equations|a protocol for writing higher-order dynamic equations as systems of first-order equations. This process proves necessary in obtaining numerical solutions of differential equations since the Runge-Kutta …

The Dynamics Of Newton's Method On Cubic Polynomials, Shannon N. Miller

Theses, Dissertations and Capstones

The field of dynamics is itself a huge part of many branches of science, including the motion of the planets and galaxies, changing weather patterns, and the growth and decline of populations. Consider a function f and pick x0 in the domain of f . If we iterate this function around the point x0, then we will have the sequence x0, f (x0), f (f (x0)), f (f (f (x0))), ..., which becomes our dynamical system. We are essentially interested in the end behavior of this system. Do …

A Mathematical Model For Treatment-Resistant Mutations Of Hiv, Helen Moore, Weiqing Gu

All HMC Faculty Publications and Research

In this paper, we propose and analyze a mathematical model, in the form of a system of ordinary differential equations, governing mutated strains of human immunodeficiency virus (HIV) and their interactions with the immune system and treatments. Our model incorporates two types of resistant mutations: strains that are not responsive to protease inhibitors, and strains that are not responsive to reverse transcriptase inhibitors. It also includes strains that do not have either of these two types of resistance (wild-type virus) and strains that have both types. We perform our analysis by changing the system of ordinary differential equations (ODEs) to …

Dynamic Equations On Changing Time Scales: Dynamics Of Given Logistic Problems, Parameterization, And Convergence Of Solutions, Kelli J. Hall

Theses, Dissertations and Capstones

In this thesis we use the theory of dynamic equations on time scales to understand the changes in dynamics between difference and differen- tial equations by parameterizing the underlying domains. To illustrate where and how these changes occur, we then construct a bifurcation diagram for a simple family of dynamic equations. However, these results are only true if we can move continuously through our domains, i.e, the time scales. In the last part of this thesis, we define what it means to have a convergent sequence of time scales. Then we use this definition to prove that the limit …

Discrete-Time Approximations Of Stochastic Delay Equations: The Milstein Scheme, Yaozhong Hu, Salah-Eldin A. Mohammed, Feng Yan

Articles and Preprints

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDE's). The scheme has convergence order 1. In order to establish the scheme, we prove an infinite-dimensional Itô formula for "tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the non-anticipating nature of the SDDE, the use of anticipating calculus methods appears to be novel.

Brownian Dynamics Simulation Studies Of Ion Throughput During Cellular Electroporation, Juan A. Gonzalez-Cuevas

Electrical & Computer Engineering Theses & Dissertations

The purpose of the present research work is to analyze the flow of ions through pores formed in the plasma membrane of biological cells during electroporation. In this regard, simulations have been conducted to track the movement of ions in 3-D space based on a Brownian dynamics model, which is mostly deterministic but has some stochastic properties to account for ionic scattering. From the Brownian dynamics simulations, it has been possible to estimate the pore's conductance as well as transport parameters such as the diffusion coefficient and mobility within the pore region, of which there is no experimental data currently …

Robustly Transitive Sets And Heterodimensional Cycles, Christian Bonatti, Lorenzo J. Díaz, Enrique R. Pujals, Jorge Rocha

Publications and Research

It is known that all non-hyperbolic robustly transitive sets Λ_φ have a dominated splitting and, generically, contain periodic points of different indices. We show that, for a C¹-dense open subset of diffeomorphisms φ, the indices of periodic points in a robust transitive set Λ_φ form an interval in ℕ. We also prove that the homoclinic classes of two periodic points in Λ_φ are robustly equal. Finally, we describe what sort of homoclinic tangencies may appear in Λ_φ by studying its dominated splittings.

Separable Preference Orders, Jonathan K. Hodge

Dissertations

Whenever a decision-maker must express simultaneously his or her preferences on several possibly related issues, the existence of interdependence among these preferences can lead to collective decisions that are unsatisfactory or even paradoxical. Intuitively, an individual’s preferences are said to be separable on a subset of issues if they do not depend on the choice of alternatives for issues outside the subset. Here we explore from a mathematical standpoint the properties of separable and nonseparable preference orders. We begin by formulating a general model of multidimensional preferences and we formally introduce the notions of separability and noninfluentiality. We study the …

Geometrical Models For Grain Dynamics, Giovani L. Vasconcelos, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

We study models for the gravity-driven, dissipative motion of a single grain on an inclined rough surface. Imposing some conditions on the momentum loss due to the collisions between the particle and the surface, we arrive at a class of models in which the grain dynamics is described by one-dimensional maps. The dynamics of these maps is studied in detail. We prove the existence of various dynamical phases and show that the presence of these phases is independent of the restitution law (within the class considered).

Coupled Electrodynamic-Monte Carlo Simulations Of Nanoscale Gaas Terahertz Optical Mixers, Jiang Li

Electrical & Computer Engineering Theses & Dissertations

The concept of mixing or heterodyning has traditionally been used for microwaves and for radio frequency communications. However, the concept can easily be extended into the optical frequency regime. By doing so, the photomixing process can serve as a very versatile tool for both the generation of ultrahigh frequency (terahertz) and the detection of weak optical signals.

The aim of this thesis is to perform a theoretical study of the photomixing process inside GaAs devices as the non-linear elements. A coupled approach which combines the Monte Carlo simulation scheme for the carrier transport, with Maxwell's equation for the electrodynamics, has …

Gnarled Defined, Rudy Rucker

SWITCH

The article is a reflection of the author’s conception of the term ‘gnarly’, extending the term’s meaning from its origins in California surfer slang. 'Gnarly' is often used in a colloquial context, however, the author believes that the term is able to be used in an academic field as it pertains to outcomes and results of equations. Discussions towards the application of the term 'gnarly' showcase how it can be used in a scientific, mathematical, and artistic context through seemingly random patterns. In order to be gnarly, things must lie and exist between the realm of orderly and chaotic often …

Divergence Diagrams: More Than Cantor Dust Lies At The Edge Of Feigenbaum Diagrams, John H. Rickert, Aaron Klebanoff

Mathematical Sciences Technical Reports (MSTR)

The dynamical system analysis of the logistic map f(x)=ax(1-x) is studied for values of a greater than 4.

Virtual Celluoid, Switch Staffs

SWITCH

The article is an analysis of the author’s research pertaining to films relating to or containing the concept of virtual reality. The author lists several films such as Johnny Mnemnonic, Virtuosity, The Net, and Disclosure and provides a brief synopsis and review of each movie. Each film explains the concept of virtual reality through differing plots and methods such as cyberspace, progressive software, and artificial intelligence. The author also gives their own insight into and ratings of the films, explaining what they think is the most relatable in terms of overall storyline as well as how realisticly the movie portrays …

Vr Products, P.D. Quick

SWITCH

The article is an analysis of the author’s experience testing several virtual reality items at Siggraph, an annual convention displaying computer machinery and interactive technology products. The author explains each device and how they work, the company behind the invention, as well as how it can help with future technological advancements. Several products are explained in more depth; the i-Glasses by Virtual I/O, Red Planet by Virtual World Entertainment Inc., Venturer S-2 by Thomson Entertainment Systems, and several more. Each item explores virtual reality in differing ways such as interactive video games, more user-friendly 3D modeling, and virtual movie theater …

Population Genetics: Estimation Of Distributions Through Systems Of Non-Linear Differential Equations, Nacer E. Abrouk, Robert J. Lopez

Mathematical Sciences Technical Reports (MSTR)

In stochastic population genetics, the fundamental quantity used for describing the genetic composition of a Mendelian population is the gene frequency. The process of change in the gene frequency is generally modeled as a stochastic process satisfying a stochastic differential equation. The drift and diffusion coefficients in this equation reflect such mechanisms as mutation, selection, and migration that affect the population. Except in very simple cases, it is difficult to determine the probability law of the stochastic process of change in gene frequency. We present a method for obtaining approximations of this process, enabling us to study models more realistic …

Atamm Multicomputer System Design, William Robert Tymchyshyn

Electrical & Computer Engineering Theses & Dissertations

The Algorithm To Architecture Mapping Model, or ATAMM, is a graph theoretic design methodology that has been created to resolve design and performance issues involved with concurrent processing. Petri-net marked graphs are used to represent the computational environment.

This thesis describes the development of a multicomputer system which will operate within the bounds specified by the ATAMM model. The system is first designed and implemented using the framework of standard multicomputer design theory. A validation is then performed by the comparison of the system's computational performance to results predicted by ATAMM. This validation is shown to be successful for three …

Equations Of Variation For Ordinary Differential Equations On Manifolds, J. B. Bennett

Journal of the Arkansas Academy of Science

No abstract provided.

Linear Estimation: The Kalman-Bucy Filter, William Douglas Schindel

Graduate Theses - Mathematics

The problem of linear dynamic estimation, its solution as developed by Kalman and Bucy, and interpretations, properties and illustrations of that solution are discussed. The central problem considered is the estimation of the system state vector X, describing a linear dynamic system governed by

dx/dt = F(t)X(t) + G(t)U(t)

Y(t) = H(t)X(t) + V(t)

for observations of Y (system output), where V is a random observation-corrupting process, and U is a random system driving process.

An extension of the Kalman-Bucy filter to estimation in the absence of priori knowledge of the random process U and V is developed and illustrated.