The Phase Flow Method
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Chapter 5. Flow Maps and Dynamical Systems
Chapter 5 Flow Maps and Dynamical Systems Main concepts: In this chapter we introduce the concepts of continuous and discrete dynamical systems on phase space. Keywords are: classical mechanics, phase space, vector field, linear systems, flow maps, dynamical systems Figure 5.1: A solar system is well modelled by classical mechanics. (source: Wikimedia Commons) In Chapter 1 we encountered a variety of ODEs in one or several variables. Only in a few cases could we write out the solution in a simple closed form. In most cases, the solutions will only be obtainable in some approximate sense, either from an asymptotic expansion or a numerical computation. Yet, as discussed in Chapter 3, many smooth systems of differential equations are known to have solutions, even globally defined ones, and so in principle we can suppose the existence of a trajectory through each point of phase space (or through “almost all” initial points). For such systems we will use the existence of such a solution to define a map called the flow map that takes points forward t units in time from their initial points. The concept of the flow map is very helpful in exploring the qualitative behavior of numerical methods, since it is often possible to think of numerical methods as approximations of the flow map and to evaluate methods by comparing the properties of the map associated to the numerical scheme to those of the flow map. 25 26 CHAPTER 5. FLOW MAPS AND DYNAMICAL SYSTEMS In this chapter we will address autonomous ODEs (recall 1.13) only: 0 d y = f(y), y, f ∈ R (5.1) 5.1 Classical mechanics In Chapter 1 we introduced models from population dynamics. -
Perturbation Theory and Exact Solutions
PERTURBATION THEORY AND EXACT SOLUTIONS by J J. LODDER R|nhtdnn Report 76~96 DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS J J. LODOER ASSOCIATIE EURATOM-FOM Jun»»76 FOM-INST1TUUT VOOR PLASMAFYSICA RUNHUIZEN - JUTPHAAS - NEDERLAND DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS by JJ LODDER R^nhuizen Report 76-95 Thisworkwat performed at part of th«r«Mvchprogmmncof thcHMCiattofiafrccmentof EnratoniOTd th« Stichting voor FundtmenteelOiutereoek der Matctk" (FOM) wtihnnmcWMppoft from the Nederhmdie Organiutic voor Zuiver Wetemchap- pcigk Onderzoek (ZWO) and Evntom It it abo pabHtfMd w a the* of Ac Univenrty of Utrecht CONTENTS page SUMMARY iii I. INTRODUCTION 1 II. GENERALIZED FUNCTIONS DEFINED ON DISCONTINUOUS TEST FUNC TIONS AND THEIR FOURIER, LAPLACE, AND HILBERT TRANSFORMS 1. Introduction 4 2. Discontinuous test functions 5 3. Differentiation 7 4. Powers of x. The partie finie 10 5. Fourier transforms 16 6. Laplace transforms 20 7. Hubert transforms 20 8. Dispersion relations 21 III. PERTURBATION THEORY 1. Introduction 24 2. Arbitrary potential, momentum coupling 24 3. Dissipative equation of motion 31 4. Expectation values 32 5. Matrix elements, transition probabilities 33 6. Harmonic oscillator 36 7. Classical mechanics and quantum corrections 36 8. Discussion of the Pu strength function 38 IV. EXACTLY SOLVABLE MODELS FOR DISSIPATIVE MOTION 1. Introduction 40 2. General quadratic Kami1tonians 41 3. Differential equations 46 4. Classical mechanics and quantum corrections 49 5. Equation of motion for observables 51 V. SPECIAL QUADRATIC HAMILTONIANS 1. Introduction 53 2. Hamiltcnians with coordinate coupling 53 3. Double coordinate coupled Hamiltonians 62 4. Symmetric Hamiltonians 63 i page VI. DISCUSSION 1. Introduction 66 ?. -
Chapter 8 Stability Theory
Chapter 8 Stability theory We discuss properties of solutions of a first order two dimensional system, and stability theory for a special class of linear systems. We denote the independent variable by ‘t’ in place of ‘x’, and x,y denote dependent variables. Let I ⊆ R be an interval, and Ω ⊆ R2 be a domain. Let us consider the system dx = F (t, x, y), dt (8.1) dy = G(t, x, y), dt where the functions are defined on I × Ω, and are locally Lipschitz w.r.t. variable (x, y) ∈ Ω. Definition 8.1 (Autonomous system) A system of ODE having the form (8.1) is called an autonomous system if the functions F (t, x, y) and G(t, x, y) are constant w.r.t. variable t. That is, dx = F (x, y), dt (8.2) dy = G(x, y), dt Definition 8.2 A point (x0, y0) ∈ Ω is said to be a critical point of the autonomous system (8.2) if F (x0, y0) = G(x0, y0) = 0. (8.3) A critical point is also called an equilibrium point, a rest point. Definition 8.3 Let (x(t), y(t)) be a solution of a two-dimensional (planar) autonomous system (8.2). The trace of (x(t), y(t)) as t varies is a curve in the plane. This curve is called trajectory. Remark 8.4 (On solutions of autonomous systems) (i) Two different solutions may represent the same trajectory. For, (1) If (x1(t), y1(t)) defined on an interval J is a solution of the autonomous system (8.2), then the pair of functions (x2(t), y2(t)) defined by (x2(t), y2(t)) := (x1(t − s), y1(t − s)), for t ∈ s + J (8.4) is a solution on interval s + J, for every arbitrary but fixed s ∈ R. -
ATTRACTORS: STRANGE and OTHERWISE Attractor - in Mathematics, an Attractor Is a Region of Phase Space That "Attracts" All Nearby Points As Time Passes
ATTRACTORS: STRANGE AND OTHERWISE Attractor - In mathematics, an attractor is a region of phase space that "attracts" all nearby points as time passes. That is, the changing values have a trajectory which moves across the phase space toward the attractor, like a ball rolling down a hilly landscape toward the valley it is attracted to. PHASE SPACE - imagine a standard graph with an x-y axis; it is a phase space. We plot the position of an x-y variable on the graph as a point. That single point summarizes all the information about x and y. If the values of x and/or y change systematically a series of points will plot as a curve or trajectory moving across the phase space. Phase space turns numbers into pictures. There are as many phase space dimensions as there are variables. The strange attractor below has 3 dimensions. LIMIT CYCLE (OR PERIODIC) ATTRACTOR STRANGE (OR COMPLEX) ATTRACTOR A system which repeats itself exactly, continuously, like A strange (or chaotic) attractor is one in which the - + a clock pendulum (left) Position (right) trajectory of the points circle around a region of phase space, but never exactly repeat their path. That is, they do have a predictable overall form, but the form is made up of unpredictable details. Velocity More important, the trajectory of nearby points diverge 0 rapidly reflecting sensitive dependence. Many different strange attractors exist, including the Lorenz, Julian, and Henon, each generated by a Velocity Y different equation. 0 Z The attractors + (right) exhibit fractal X geometry. Velocity 0 ATTRACTORS IN GENERAL We can generalize an attractor as any state toward which a Velocity system naturally evolves. -
Polarization Fields and Phase Space Densities in Storage Rings: Stroboscopic Averaging and the Ergodic Theorem
Physica D 234 (2007) 131–149 www.elsevier.com/locate/physd Polarization fields and phase space densities in storage rings: Stroboscopic averaging and the ergodic theorem✩ James A. Ellison∗, Klaus Heinemann Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, United States Received 1 May 2007; received in revised form 6 July 2007; accepted 9 July 2007 Available online 14 July 2007 Communicated by C.K.R.T. Jones Abstract A class of orbital motions with volume preserving flows and with vector fields periodic in the “time” parameter θ is defined. Spin motion coupled to the orbital dynamics is then defined, resulting in a class of spin–orbit motions which are important for storage rings. Phase space densities and polarization fields are introduced. It is important, in the context of storage rings, to understand the behavior of periodic polarization fields and phase space densities. Due to the 2π time periodicity of the spin–orbit equations of motion the polarization field, taken at a sequence of increasing time values θ,θ 2π,θ 4π,... , gives a sequence of polarization fields, called the stroboscopic sequence. We show, by using the + + Birkhoff ergodic theorem, that under very general conditions the Cesaro` averages of that sequence converge almost everywhere on phase space to a polarization field which is 2π-periodic in time. This fulfills the main aim of this paper in that it demonstrates that the tracking algorithm for stroboscopic averaging, encoded in the program SPRINT and used in the study of spin motion in storage rings, is mathematically well-founded. -
Phase Plane Methods
Chapter 10 Phase Plane Methods Contents 10.1 Planar Autonomous Systems . 680 10.2 Planar Constant Linear Systems . 694 10.3 Planar Almost Linear Systems . 705 10.4 Biological Models . 715 10.5 Mechanical Models . 730 Studied here are planar autonomous systems of differential equations. The topics: Planar Autonomous Systems: Phase Portraits, Stability. Planar Constant Linear Systems: Classification of isolated equilib- ria, Phase portraits. Planar Almost Linear Systems: Phase portraits, Nonlinear classi- fications of equilibria. Biological Models: Predator-prey models, Competition models, Survival of one species, Co-existence, Alligators, doomsday and extinction. Mechanical Models: Nonlinear spring-mass system, Soft and hard springs, Energy conservation, Phase plane and scenes. 680 Phase Plane Methods 10.1 Planar Autonomous Systems A set of two scalar differential equations of the form x0(t) = f(x(t); y(t)); (1) y0(t) = g(x(t); y(t)): is called a planar autonomous system. The term autonomous means self-governing, justified by the absence of the time variable t in the functions f(x; y), g(x; y). ! ! x(t) f(x; y) To obtain the vector form, let ~u(t) = , F~ (x; y) = y(t) g(x; y) and write (1) as the first order vector-matrix system d (2) ~u(t) = F~ (~u(t)): dt It is assumed that f, g are continuously differentiable in some region D in the xy-plane. This assumption makes F~ continuously differentiable in D and guarantees that Picard's existence-uniqueness theorem for initial d ~ value problems applies to the initial value problem dt ~u(t) = F (~u(t)), ~u(0) = ~u0. -
Calculus and Differential Equations II
Calculus and Differential Equations II MATH 250 B Linear systems of differential equations Linear systems of differential equations Calculus and Differential Equations II Second order autonomous linear systems We are mostly interested with2 × 2 first order autonomous systems of the form x0 = a x + b y y 0 = c x + d y where x and y are functions of t and a, b, c, and d are real constants. Such a system may be re-written in matrix form as d x x a b = M ; M = : dt y y c d The purpose of this section is to classify the dynamics of the solutions of the above system, in terms of the properties of the matrix M. Linear systems of differential equations Calculus and Differential Equations II Existence and uniqueness (general statement) Consider a linear system of the form dY = M(t)Y + F (t); dt where Y and F (t) are n × 1 column vectors, and M(t) is an n × n matrix whose entries may depend on t. Existence and uniqueness theorem: If the entries of the matrix M(t) and of the vector F (t) are continuous on some open interval I containing t0, then the initial value problem dY = M(t)Y + F (t); Y (t ) = Y dt 0 0 has a unique solution on I . In particular, this means that trajectories in the phase space do not cross. Linear systems of differential equations Calculus and Differential Equations II General solution The general solution to Y 0 = M(t)Y + F (t) reads Y (t) = C1 Y1(t) + C2 Y2(t) + ··· + Cn Yn(t) + Yp(t); = U(t) C + Yp(t); where 0 Yp(t) is a particular solution to Y = M(t)Y + F (t). -
Visualizing Quantum Mechanics in Phase Space
Visualizing quantum mechanics in phase space Heiko Baukea) and Noya Ruth Itzhak Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany (Dated: 17 January 2011) We examine the visualization of quantum mechanics in phase space by means of the Wigner function and the Wigner function flow as a complementary approach to illustrating quantum mechanics in configuration space by wave func- tions. The Wigner function formalism resembles the mathematical language of classical mechanics of non-interacting particles. Thus, it allows a more direct comparison between classical and quantum dynamical features. PACS numbers: 03.65.-w, 03.65.Ca Keywords: 1. Introduction 2. Visualizing quantum dynamics in position space or momentum Quantum mechanics is a corner stone of modern physics and technology. Virtually all processes that take place on an space atomar scale require a quantum mechanical description. For example, it is not possible to understand chemical reactionsor A (one-dimensional) quantum mechanical position space the characteristics of solid states without a sound knowledge wave function maps each point x on the position axis to a of quantum mechanics. Quantum mechanical effects are uti- time dependent complex number Ψ(x, t). Equivalently, one lized in many technical devices ranging from transistors and may consider the wave function Ψ˜ (p, t) in momentum space, Flash memory to tunneling microscopes and quantum cryp- which is given by a Fourier transform of Ψ(x, t), viz. tography, just to mention a few applications. Quantum effects, however, are not directly accessible to hu- 1 ixp/~ Ψ˜ (p, t) = Ψ(x, t)e− dx . (1) man senses, the mathematical formulations of quantum me- (2π~)1/2 chanics are abstract and its implications are often unintuitive in terms of classical physics. -
Phase Space Formulation of Quantum Mechanics
PHASE SPACE FORMULATION OF QUANTUM MECHANICS. INSIGHT INTO THE MEASUREMENT PROBLEM D. Dragoman* – Univ. Bucharest, Physics Dept., P.O. Box MG-11, 76900 Bucharest, Romania Abstract: A phase space mathematical formulation of quantum mechanical processes accompanied by and ontological interpretation is presented in an axiomatic form. The problem of quantum measurement, including that of quantum state filtering, is treated in detail. Unlike standard quantum theory both quantum and classical measuring device can be accommodated by the present approach to solve the quantum measurement problem. * Correspondence address: Prof. D. Dragoman, P.O. Box 1-480, 70700 Bucharest, Romania, email: [email protected] 1. Introduction At more than a century after the discovery of the quantum and despite the indubitable success of quantum theory in calculating the energy levels, transition probabilities and other parameters of quantum systems, the interpretation of quantum mechanics is still under debate. Unlike relativistic physics, which has been founded on a new physical principle, i.e. the constancy of light speed in any reference frame, quantum mechanics is rather a successful mathematical algorithm. Quantum mechanics is not founded on a fundamental principle whose validity may be questioned or may be subjected to experimental testing; in quantum mechanics what is questionable is the meaning of the concepts involved. The quantum theory offers a recipe of how to quantize the dynamics of a physical system starting from the classical Hamiltonian and establishes rules that determine the relation between elements of the mathematical formalism and measurable quantities. This set of instructions works remarkably well, but on the other hand, the significance of even its landmark parameter, the Planck’s constant, is not clearly stated. -
Why Must We Work in the Phase Space?
IPPT Reports on Fundamental Technological Research 1/2016 Jan J. Sławianowski, Frank E. Schroeck Jr., Agnieszka Martens WHY MUST WE WORK IN THE PHASE SPACE? Institute of Fundamental Technological Research Polish Academy of Sciences Warsaw 2016 http://rcin.org.pl IPPT Reports on Fundamental Technological Research ISSN 2299-3657 ISBN 978-83-89687-98-2 Editorial Board/Kolegium Redakcyjne: Wojciech Nasalski (Editor-in-Chief/Redaktor Naczelny), Paweł Dłużewski, Zbigniew Kotulski, Wiera Oliferuk, Jerzy Rojek, Zygmunt Szymański, Yuriy Tasinkevych Reviewer/Recenzent: prof. dr Paolo Maria Mariano Received on 21st January 2016 Copyright °c 2016 by IPPT PAN Instytut Podstawowych Problemów Techniki Polskiej Akademii Nauk (IPPT PAN) (Institute of Fundamental Technological Research, Polish Academy of Sciences) Pawińskiego 5B, PL 02-106 Warsaw, Poland Printed by/Druk: Drukarnia Braci Grodzickich, Piaseczno, ul. Geodetów 47A http://rcin.org.pl Why must we work in the phase space? Jan J. Sławianowski1, Frank E. Schroeck Jr.2, Agnieszka Martens1 1Institute of Fundamental Technological Research, Polish Academy of Sciences 2Department of Mathematics, University of Denver Abstract We are going to prove that the phase-space description is fundamental both in the classical and quantum physics. It is shown that many problems in statis- tical mechanics, quantum mechanics, quasi-classical theory and in the theory of integrable systems may be well-formulated only in the phase-space language. There are some misunderstandings and confusions concerning the concept of induced probability and entropy on the submanifolds of the phase space. First of all, they are restricted only to hypersurfaces in the phase space, i.e., to the manifolds of the defect of dimension equal to one. -
The Phase Space Elementary Cell in Classical and Generalized Statistics
Entropy 2013, 15, 4319-4333; doi:10.3390/e15104319 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article The Phase Space Elementary Cell in Classical and Generalized Statistics Piero Quarati 1;2;* and Marcello Lissia 2 1 DISAT, Politecnico di Torino, C.so Duca degli Abruzzi 24, Torino I-10129, Italy 2 Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari, Monserrato I-09042, Italy; E-Mail: [email protected] * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +39-0706754899; Fax: +39-070510212. Received: 05 September 2013; in revised form: 25 September 2013/ Accepted: 27 September 2013 / Published: 15 October 2013 Abstract: In the past, the phase-space elementary cell of a non-quantized system was set equal to the third power of the Planck constant; in fact, it is not a necessary assumption. We discuss how the phase space volume, the number of states and the elementary-cell volume of a system of non-interacting N particles, changes when an interaction is switched on and the system becomes or evolves to a system of correlated non-Boltzmann particles and derives the appropriate expressions. Even if we assume that nowadays the volume of the elementary cell is equal to the cube of the Planck constant, h3, at least for quantum systems, we show that there is a correspondence between different values of h in the past, with important and, in principle, measurable cosmological and astrophysical consequences, and systems with an effective smaller (or even larger) phase-space volume described by non-extensive generalized statistics. -
Flows of Continuous-Time Dynamical Systems with No Periodic Orbit As an Equivalence Class Under Topological Conjugacy Relation
Journal of Mathematics and Statistics 7 (3): 207-215, 2011 ISSN 1549-3644 © 2011 Science Publications Flows of Continuous-Time Dynamical Systems with No Periodic Orbit as an Equivalence Class under Topological Conjugacy Relation Tahir Ahmad and Tan Lit Ken Department of Mathematics, Faculty of Science, Nanotechnology Research Alliance, Theoretical and Computational Modeling for Complex Systems, University Technology Malaysia 81310 Skudai, Johor, Malaysia Abstract: Problem statement: Flows of continuous-time dynamical systems with the same number of equilibrium points and trajectories, and which has no periodic orbit form an equivalence class under the topological conjugacy relation. Approach: Arbitrarily, two trajectories resulting from two distinct flows of this type of dynamical systems were written as a set of points (orbit). A homeomorphism which maps between these two sets is then built. Using the notion of topological conjugacy, they were shown to conjugate topologically. By the arbitrariness in selection of flows and their respective initial states, the results were extended to all the flows of dynamical system of that type. Results: Any two flows of such dynamical systems were shown to share the same dynamics temporally along with other properties such as order isomorphic and homeomorphic. Conclusion: Topological conjugacy serves as an equivalence relation in the set of flows of continuous-time dynamical systems which have same number of equilibrium points and trajectories, and has no periodic orbit. Key words: Dynamical system, equilibrium points, trajectories, periodic orbit, equivalence class, topological conjugacy, order isomorphic, Flat Electroencephalography (Flat EEG), dynamical systems INTRODUCTION models. Nandhakumar et al . (2009) for example, the dynamics of robot arm is studied.