Dynamical Systems Commons^™

Euler-Poincar´E Equations For G-Strands, Darryl Holm, Rossen Ivanov

Conference papers

The G-strand equations for a map R×R into a Lie group G are associated to a G-invariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The G-strand itself is the map g(t,s):R×R→G, where t and s are the independent variables of the G-strand equations. The Euler-Poincar'e reduction of the variational principle leads to a formulation where the dependent variables of the G-strand equations take values in the corresponding Lie algebra g and its co-algebra, g^∗ with respect to the pairing provided by the variational derivatives of the Lagrangian. We review examples of different G-strand …

Fractal Powers In Serrin's Swirling Vortex Solutions, Pavel Bělík, Douglas P. Dokken, Kurt Scholz, Mikhail M. Shvartsman

Faculty Authored Articles

We consider a modification of the fluid flow model for a tornado-like swirling vortex developed by Serrin [Phil. Trans. Roy. Soc. London, Series A, Math & Phys. Sci. 271(1214) (1972), 325–360], where velocity decreases as the reciprocal of the distance from the vortex axis. Recent studies, based on radar data of selected severe weather events [Mon. Wea. Rev. 133(9) (2005), 2535–2551; Mon. Wea. Rev. 128(7) (2000), 2135–2164; Mon. Wea. Rev. 133(1) (2005), 97–119], indicate that the angular momentum in a tornado may not be constant with the radius, and thus suggest a different scaling of the velocity/radial distance dependence. Motivated …

Mechanical Visualization Of A Second Order Dynamic Equation On Varying Time Scales, Molly Kathryn Peterson

Theses, Dissertations and Capstones

In this work, we give an introduction to Time Scales Calculus, the properties of the exponential function on an arbitrary time scale, and use it to solve linear dynamic equation of second order. Time Scales Calculus was introduced by Stefan Hilger in 1988. It brings together the theories of difference and differential equations into one unified theory. By using the properties of the delta derivative and the delta anti-derivative, we analyze the behavior of a second order linear homogeneous dynamic equation on various time scales. After the analytical discussion, we will graphically evaluate the second order dynamic equation in Marshall’s …

Integrability, Recursion Operators And Soliton Interactions, Boyka Aneva, Georgi Grahovski, Rossen Ivanov, Dimitar Mladenov

Book chapter/book

This volume contains selected papers based on the talks,presentedat the Conference Integrability, Recursion Operators and Soliton Interactions, held in Sofia, Bulgaria (29-31 August 2012) at the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences. Included are also invited papers presenting new research developments in the thematic area. The Conference was dedicated to the 65-th birthday of our esteemed colleague and friend Vladimir Gerdjikov. The event brought together more than 30 scientists, from 6 European countries to celebrate Vladimir's scientific achievements. All participants enjoyed a variety of excellent talks in a friendly and stimulating atmosphere. …

Relative Equilibria Of Isosceles Triatomic Molecules In Classical Approximation, Damaris Miriam Mckinley

Theses and Dissertations (Comprehensive)

In this thesis we study relative equilibria of di-atomic and isosceles tri-atomic molecules in classical approximations with repulsive-attractive interaction. For di-atomic systems we retrieve well-known results. The main contribution consists of the study of the existence and stability of relative equilibria in a three-atom system formed by two identical atoms of mass $m$ and a third of mass $m_3$, constrained in an isosceles configuration at all times.

Given the shape of the binary potential only, we discuss the existence of equilibria and relative equilibria. We represent the results in the form of energy-momentum diagrams. We find that fixing the masses …

Viscosity Dependence Of Faraday Wave Formation Thresholds, Lisa Michelle Slaughter

Physics

This experiment uses an electromagnetic shaker to produce standing wave patterns on the surface of a vertically oscillating sample of silicon liquid. These surface waves, known as Faraday waves, form shapes such as squares, lines, and hexagons. They are known to be dependent upon the frequency and amplitude of the forcing as well as on the viscosity and depth of the liquid in the dish. At a depth of 4mm and for various silicon liquids having kinematic viscosities of 10, 20, and 38 cSt, we determined the acceleration at which patterns form for frequencies between 10 and 60 Hz. For …

Identifiability Of Additive, Time-Varying Actuator And Sensor Faults By State Augmentation, Jason M. Upchurch

Electrical & Computer Engineering Theses & Dissertations

Faults in dynamical systems can have serious safety and reliability implications. For example, actuator and sensor faults have been factors in past incidents and mishaps in many aerospace systems. A large body of research is devoted to developing methods to detect and identify actuator and sensor faults in such systems.

One fault detection and identification menthol employs state augmentation, whereby a set of time-varying faults of interest are modeled as outputs of exogenous linear, time-invariant systems and augmented to the state of the nominal system model. The resulting model represents the system dynamics due to a particular actuator-sensor fault configuration. …

Long-Wave Model For Strongly Anisotropic Growth Of A Crystal Step, Mikhail Khenner

Mathematics Faculty Publications

A continuum model for the dynamics of a single step with the strongly anisotropic line energy is formulated and analyzed. The step grows by attachment of adatoms from the lower terrace, onto which atoms adsorb from a vapor phase or from a molecular beam, and the desorption is nonnegligible (the “one-sided” model). Via a multiscale expansion, we derived a long-wave, strongly nonlinear, and strongly anisotropic evolution PDE for the step profile. Written in terms of the step slope, the PDE can be represented in a form similar to a convective Cahn-Hilliard equation. We performed the linear stability analysis and computed …

Dynamics Of The Fitzhugh-Nagumo Neuron Model, Zechariah Thurman

Physics

In this paper, the dynamical behavior of the Fitzhugh-Nagumo model is examined. The relationship between neuron input current and the firing frequency of the neuron is characterized. Various coupling schemes are also examined, and their effects on the dynamics of the system is discussed. The phenomenon of stochastic resonance is studied for a single uncoupled Fitzhugh-Nagumo neuron.

Boundary Value Problems For Discrete Fractional Equations, Pushp R. Awasthi

Department of Mathematics: Dissertations, Theses, and Student Research

In this dissertation we develop certain aspects of the theory of discrete fractional calculus. The author begins with an introduction to the discrete delta calculus together with the fractional delta calculus which is used throughout this dissertation. The Cauchy function, the Green's function and some of their important properties for a fractional boundary value problem for are developed. This dissertation is comprised of four chapters. In the first chapter we introduce the delta fractional calculus. In the second chapter we give some preliminary definitions, properties and theorems for the fractional delta calculus and derive the appropriate Green's function and give …

Solutions Of Dynamic Equations On Time Scales With Jumps, Kayode Daniel Olumoyin

Theses, Dissertations and Capstones

To obtain the solution of first order dynamic equations on time scales with jumps, a good question to ask is, how many initial conditions will be needed? We shall show that you only need the initial condition that gives you either the initial position or the initial velocity. The solution at each left scattered point in the time scale can be obtained analytically. With this approach we shall write the general form of the solution of a first order dynamic equations on time scales with jumps. To do this we shall use the Hilger derivative, anti-derivatives, the Hilger Complex plane, …

Rational Map Of Cp^2 With No Invariant Foliation, Scott R. Kaschner, Rodrigo A. Perez, Roland K.W. Roeder

Scholarship and Professional Work - LAS

Conference Poster presented at: Midwest Dynamical Systems Conference, Champaign/Urbana, IL November 1-3, 2013.

Control, Stability, And Qualitative Theory Of Dynamical Systems, Nazim Idrisoglu Mahmudov, Mark A. Mckibben, Sakthivel Rathinasamy, Yong Ren

Mathematics Faculty Publications

No abstract provided.

G-Strands, Darryl Holm, Rossen Ivanov, James Percival

Articles

A G-strand is a map g(t,s): RxR --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. The SO(3)-strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3)_K-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincar'e system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3) …

The Octonions And The Exceptional Lie Algebra G2, Ian M. Anderson

Research Vignettes

The octonions O are an 8-dimensional non-commutative, non-associative normed real algebra. The set of all derivations of O form a real Lie algebra. It is remarkable fact, first proved by E. Cartan in 1908, that the the derivation algebra of O is the compact form of the exceptional Lie algebra G2. In this worksheet we shall verify this result of Cartan and also show that the derivation algebra of the split octonions is the split real form of G2.

PDF and Maple worksheets can be downloaded from the links below.

Special Dual Like Numbers And Lattices, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book the authors introduce a new type of dual numbers called special dual like numbers. These numbers are constructed using idempotents in the place of nilpotents of order two as new element. That is x = a + bg is a special dual like number where a and b are reals and g is a new element such that g2 =g. The collection of special dual like numbers forms a ring. Further lattices are the rich structures which contributes to special dual like numbers. These special dual like numbers x = a + bg; when a and b …

Euler Equations On A Semi-Direct Product Of The Diffeomorphisms Group By Itself, Joachim Escher, Rossen Ivanov, Boris Kolev

Articles

The geodesic equations of a class of right invariant metrics on the semi-direct product of two Diff(S) groups are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra are found.

A Mathematician Weighs In On The Evolution Debate, Kris H. Green

Journal of Humanistic Mathematics

There are a variety of reasons underlying the lack of public acceptance for the theory of evolution in the United States. An overlooked cause is related to problems with the mathematics curriculum in the K-12 setting. In this essay, we examine this relationship and propose changes to the mathematics curriculum that could improve mathematical thinking while also providing a basis for understanding theories, like evolution, that are poorly understood.

A Group-Permutation Algorithm To Solve The Generalized Sudoku, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Sudoku can be generalized to squares whose dimensions are n^2 × n^2 , where n ≥ 2, using various symbols (numbers, letters, mathematical symbols, etc.), written just one time on each row and on each column; and the large square is divided into n 2 small squares with the side n × n and each will contain all n 2 symbols written only once. In this paper we present an elementary solution for the generalized sudoku based on a group-permutation algorithm.

Deterministic And Stochastic Bellman's Optimality Principles On Isolated Time Domains And Their Applications In Finance, Nezihe Turhan

Masters Theses & Specialist Projects

The concept of dynamic programming was originally used in late 1949, mostly during the 1950s, by Richard Bellman to describe decision making problems. By 1952, he refined this to the modern meaning, referring specifically to nesting smaller decision problems inside larger decisions. Also, the Bellman equation, one of the basic concepts in dynamic programming, is named after him. Dynamic programming has become an important argument which was used in various fields; such as, economics, finance, bioinformatics, aerospace, information theory, etc. Since Richard Bellman's invention of dynamic programming, economists and mathematicians have formulated and solved a huge variety of sequential decision …