4 SecondOrder Systems


 Aldous Reeves
 1 years ago
 Views:
Transcription
1 4 SecondOrder Systems Secondorder autonomous systems occupy an important place in the study of nonlinear systems because solution trajectories can be represented in the plane. This allows for easy visualization of the qualitative behavior of the system. A secondorder autonomous system is represented by two scalar differential equations (, ) (, ) = f x x = f x x x x ( ) ( ) = x = x The locus in the x x x t for all t is a curve that passes through the point x. The x x plane is usually called the state plane or the phase plane. f and f plane of the solution ( ) expresses the tangent vector ( t) to the curve. =, then at x = (,), we draw an arrow pointing from (, ) to (,) + (,) = ( 3, ). Repeating this at every point in a grid covering the plane, we obtain Example 4.: If f ( x) ( x, x) a vector field diagram. Example 4.: Pendulum without friction = x = sin x Δ Δ 7
2 4. Qualitative Behavior of Order Systems Near Equilibrium Points Consider the linear timeinvariant system = Ax where A is a real matrix. The solution of the equation for a given state x is given by ( ) = exp( ) x t M J t M x r where J r is the real Jordan form of A and M is a real nonsingular matrix such that M AM = J r. Depending on the eigenvalues of A, the real Jordan form may take one of three forms, k, and α β β α where k is either or. The first form corresponds to the case when the eigenvalues and are real and distinct, the second form corresponds to the case when the eigenvalues are real and equal, and the third form corresponds to the case of complex eigenvalues = α ± β., j 4.. Real Distinct Eigenvalues In this case and M = v, v, where v and v are the real eigenvectors associated with and. The change of coordinates z = M x transforms the system into two decoupled firstorder differential equations, z = z, z = z are different from zero and [ ] whose solution, for a given initial state (, ) t ( ) =, z () t = z e z t z e t z z, is given by Eliminating t between the two equations, we obtain z = cz / / c= z z. where ( ) Three cases: 8
3 Stable node: Both eigenvalues are negative Unstable node: Both eigenvalues are positive Saddle point: Eigenvalues have different sign 9
4 4.. Complex Eigenvalues The change of coordinates z M x = transforms the system into the form z = α z β z, z = β z+ α z The solution of these equations is oscillatory and can be expressed more conveniently in polar coordinates. z r = z + z, θ = tan z where we have two uncoupled firstorder differential equation: r = αr and θ = β The solution for a given initial state (, ) r( t) = re αt and ( t) θ = θ + βt r θ is given by When α <, the spiral converges to the origin; when α >, it diverges away from the origin. When α =, the trajectory is a circle of radius r. Three cases Stable focus: α <
5 Unstable focus: α > Circle: α = 4..3 Nonzero Multiple Eigenvalues The change of coordinates z M x = transforms the system into the form z = z+ kz, z = z whose solution, for a given initial state (, ) t ( ) = ( + ), ( ) z t e z kz t z t = e z z z, is given by t Eliminating t, we obtain the trajectory equation
6 z k z z = z + ln z z Two cases: k = ( <, > ) : k = ( <, > ) : 4..4 One or more Eigenvalues are zero When one or both eigenvalues of A are zero, the phase portrait is in some sense degenerate. Here, the matrix A has a nontrivial null space. Any vector in the null space of A is an equilibrium point for the system; that is, the system has an equilibrium subspace, rather than an equilibrium point. The dimension of the null space could be one or two; if it is two, the matrix A will be the zero matrix. When the dimension of the null space is one, the shape of the Jordan form of A will depend on the multiplicity of the zero eigenvalue. When = and, the matrix M is given by M = [ v, v] where v and v are the associated eigenvectors.
7 Two cases: = and (, ) < > : = = : 3
Nonlinear Control Lecture 2:Phase Plane Analysis
Nonlinear Control Lecture 2:Phase Plane Analysis Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 r. Farzaneh Abdollahi Nonlinear Control Lecture 2 1/53
More information8.1 Bifurcations of Equilibria
1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integrodifferential equations
More informationDef. (a, b) is a critical point of the autonomous system. 1 Proper node (stable or unstable) 2 Improper node (stable or unstable)
Types of critical points Def. (a, b) is a critical point of the autonomous system Math 216 Differential Equations Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan November
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More informationMath 216 First Midterm 19 October, 2017
Math 6 First Midterm 9 October, 7 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationMath 312 Lecture Notes Linearization
Math 3 Lecture Notes Linearization Warren Weckesser Department of Mathematics Colgate University 3 March 005 These notes discuss linearization, in which a linear system is used to approximate the behavior
More informationMath 312 Lecture Notes Linear Twodimensional Systems of Differential Equations
Math 2 Lecture Notes Linear Twodimensional Systems of Differential Equations Warren Weckesser Department of Mathematics Colgate University February 2005 In these notes, we consider the linear system of
More information1. < 0: the eigenvalues are real and have opposite signs; the fixed point is a saddle point
Solving a Linear System τ = trace(a) = a + d = λ 1 + λ 2 λ 1,2 = τ± = det(a) = ad bc = λ 1 λ 2 Classification of Fixed Points τ 2 4 1. < 0: the eigenvalues are real and have opposite signs; the fixed point
More informationPhase Plane Analysis
Phase Plane Analysis Phase plane analysis is one of the most important techniques for studying the behavior of nonlinear systems, since there is usually no analytical solution for a nonlinear system. Background
More informationNonlinear FEM. Critical Points. NFEM Ch 5 Slide 1
5 Critical Points NFEM Ch 5 Slide Assumptions for this Chapter System is conservative: total residual is the gradient of a total potential energy function r(u,λ) = (u,λ) u Consequence: the tangent stiffness
More informationPhysics: springmass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics
Applications of nonlinear ODE systems: Physics: springmass system, planet motion, pendulum Chemistry: mixing problems, chemical reactions Biology: ecology problem, neural conduction, epidemics Economy:
More informationKinematics of fluid motion
Chapter 4 Kinematics of fluid motion 4.1 Elementary flow patterns Recall the discussion of flow patterns in Chapter 1. The equations for particle paths in a threedimensional, steady fluid flow are dx
More informationDesigning Information Devices and Systems II Fall 2015 Note 22
EE 16B Designing Information Devices and Systems II Fall 2015 Note 22 Notes taken by John Noonan (11/12) Graphing of the State Solutions Open loop x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) Closed loop x(k
More informationChapter 9 Global Nonlinear Techniques
Chapter 9 Global Nonlinear Techniques Consider nonlinear dynamical system 0 Nullcline X 0 = F (X) = B @ f 1 (X) f 2 (X). f n (X) x j nullcline = fx : f j (X) = 0g equilibrium solutions = intersection of
More information2 Lyapunov Stability. x(0) x 0 < δ x(t) x 0 < ɛ
1 2 Lyapunov Stability Whereas I/O stability is concerned with the effect of inputs on outputs, Lyapunov stability deals with unforced systems: ẋ = f(x, t) (1) where x R n, t R +, and f : R n R + R n.
More informationANSWERS Final Exam Math 250b, Section 2 (Professor J. M. Cushing), 15 May 2008 PART 1
ANSWERS Final Exam Math 50b, Section (Professor J. M. Cushing), 5 May 008 PART. (0 points) A bacterial population x grows exponentially according to the equation x 0 = rx, where r>0is the per unit rate
More informationAPPPHYS217 Tuesday 25 May 2010
APPPHYS7 Tuesday 5 May Our aim today is to take a brief tour of some topics in nonlinear dynamics. Some good references include: [Perko] Lawrence Perko Differential Equations and Dynamical Systems (SpringerVerlag
More informationNotation. 0,1,2,, 1 with addition and multiplication modulo
Notation Q,, The set of all natural numbers 1,2,3, The set of all integers The set of all rational numbers The set of all real numbers The group of permutations of distinct symbols 0,1,2,,1 with addition
More informationCHAPTER 5 KINEMATICS OF FLUID MOTION
CHAPTER 5 KINEMATICS OF FLUID MOTION 5. ELEMENTARY FLOW PATTERNS Recall the discussion of flow patterns in Chapter. The equations for particle paths in a threedimensional, steady fluid flow are dx 
More informationAn Undergraduate s Guide to the HartmanGrobman and PoincaréBendixon Theorems
An Undergraduate s Guide to the HartmanGrobman and PoincaréBendixon Theorems Scott Zimmerman MATH181HM: Dynamical Systems Spring 2008 1 Introduction The HartmanGrobman and PoincaréBendixon Theorems
More informationPart II. Dynamical Systems. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 34 Paper 1, Section II 30A Consider the dynamical system where β > 1 is a constant. ẋ = x + x 3 + βxy 2, ẏ = y + βx 2
More informationELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky
ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky Equilibrium points and linearization Eigenvalue decomposition and modal form State transition matrix and matrix exponential Stability ELEC 3035 (Part
More informationCh 10.1: Two Point Boundary Value Problems
Ch 10.1: Two Point Boundary Value Problems In many important physical problems there are two or more independent variables, so the corresponding mathematical models involve partial differential equations.
More informationDo not write below here. Question Score Question Score Question Score
MATH2240 Friday, May 4, 2012, FINAL EXAMINATION 8:00AM12:00NOON Your Instructor: Your Name: 1. Do not open this exam until you are told to do so. 2. This exam has 30 problems and 18 pages including this
More informationLocal Phase Portrait of Nonlinear Systems Near Equilibria
Local Phase Portrait of Nonlinear Sstems Near Equilibria [1] Consider 1 = 6 1 1 3 1, = 3 1. ( ) (a) Find all equilibrium solutions of the sstem ( ). (b) For each equilibrium point, give the linear approimating
More informationCharacterization of the stability boundary of nonlinear autonomous dynamical systems in the presence of a saddlenode equilibrium point of type 0
Anais do CNMAC v.2 ISSN 198482X Characterization of the stability boundary of nonlinear autonomous dynamical systems in the presence of a saddlenode equilibrium point of type Fabíolo M. Amaral Departamento
More informationOne Dimensional Dynamical Systems
16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with onedimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar
More informationTravelling waves. Chapter 8. 1 Introduction
Chapter 8 Travelling waves 1 Introduction One of the cornerstones in the study of both linear and nonlinear PDEs is the wave propagation. A wave is a recognizable signal which is transferred from one part
More informationAutonomous system = system without inputs
Autonomous system = system without inputs State space representation B(A,C) = {y there is x, such that σx = Ax, y = Cx } x is the state, n := dim(x) is the state dimension, y is the output Polynomial representation
More information20D  Homework Assignment 5
Brian Bowers TA for Hui Sun MATH D Homework Assignment 5 November 8, 3 D  Homework Assignment 5 First, I present the list of all matrix row operations. We use combinations of these steps to row reduce
More informationRepresent this system in terms of a block diagram consisting only of. g From Newton s law: 2 : θ sin θ 9 θ ` T
Exercise (Block diagram decomposition). Consider a system P that maps each input to the solutions of 9 4 ` 3 9 Represent this system in terms of a block diagram consisting only of integrator systems, represented
More informationHW  Chapter 10  Parametric Equations and Polar Coordinates
Berkeley City College Due: HW  Chapter 0  Parametric Equations and Polar Coordinates Name Parametric equations and a parameter interval for the motion of a particle in the xyplane are given. Identify
More informationTangent spaces, normals and extrema
Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3space, with a point a S where S looks smooth, i.e., without any fold or cusp or selfcrossing, we can intuitively define the tangent
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,
More informationLinear Algebra. Paul Yiu. 6D: 2planes in R 4. Department of Mathematics Florida Atlantic University. Fall 2011
Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 6D: 2planes in R 4 The angle between a vector and a plane The angle between a vector v R n and a subspace V is the
More informationChapter 8 Equilibria in Nonlinear Systems
Chapter 8 Equilibria in Nonlinear Sstems Recall linearization for Nonlinear dnamical sstems in R n : X 0 = F (X) : if X 0 is an equilibrium, i.e., F (X 0 ) = 0; then its linearization is U 0 = AU; A =
More informationSolutions Chapter 9. u. (c) u(t) = 1 e t + c 2 e 3 t! c 1 e t 3c 2 e 3 t. (v) (a) u(t) = c 1 e t cos 3t + c 2 e t sin 3t. (b) du
Solutions hapter 9 dode 9 asic Solution Techniques 9 hoose one or more of the following differential equations, and then: (a) Solve the equation directly (b) Write down its phase plane equivalent, and
More informationAsymptotic Stability by Linearization
Dynamical Systems Prof. J. Rauch Asymptotic Stability by Linearization Summary. Sufficient and nearly sharp sufficient conditions for asymptotic stability of equiiibria of differential equations, fixed
More informationMatrices and Linear transformations
Matrices and Linear transformations We have been thinking of matrices in connection with solutions to linear systems of equations like Ax = b. It is time to broaden our horizons a bit and start thinking
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More informationMath 273 (51)  Final
Name: Id #: Math 273 (5)  Final Autumn Quarter 26 Thursday, December 8, 266: to 8: Instructions: Prob. Points Score possible 25 2 25 3 25 TOTAL 75 Read each problem carefully. Write legibly. Show all
More informationLinear vector spaces and subspaces.
Math 2051 W2008 Margo Kondratieva Week 1 Linear vector spaces and subspaces. Section 1.1 The notion of a linear vector space. For the purpose of these notes we regard (m 1)matrices as mdimensional vectors,
More informationTwo Dimensional Linear Systems of ODEs
34 CHAPTER 3 Two Dimensional Linear Sstems of ODEs A firstder, autonomous, homogeneous linear sstem of two ODEs has the fm x t ax + b, t cx + d where a, b, c, d are real constants The matrix fm is 31
More informationDefinition (T invariant subspace) Example. Example
Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin
More informationStability lectures. Stability of Linear Systems. Stability of Linear Systems. Stability of Continuous Systems. EECE 571M/491M, Spring 2008 Lecture 5
EECE 571M/491M, Spring 2008 Lecture 5 Stability of Continuous Systems http://courses.ece.ubc.ca/491m moishi@ece.ubc.ca Dr. Meeko Oishi Electrical and Computer Engineering University of British Columbia,
More informationEigenvalues, Eigenvectors, and Diagonalization
Math 240 TA: Shuyi Weng Winter 207 February 23, 207 Eigenvalues, Eigenvectors, and Diagonalization The concepts of eigenvalues, eigenvectors, and diagonalization are best studied with examples. We will
More informationThe Jordan Canonical Form
The Jordan Canonical Form The Jordan canonical form describes the structure of an arbitrary linear transformation on a finitedimensional vector space over an algebraically closed field. Here we develop
More information2DVolterraLotka Modeling For 2 Species
Majalat AlUlum AlInsaniya wat  Tatbiqiya 2DVolterraLotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose
More information2. (i) Find the equation of the circle which passes through ( 7, 1) and has centre ( 4, 3).
Circle 1. (i) Find the equation of the circle with centre ( 7, 3) and of radius 10. (ii) Find the centre of the circle 2x 2 + 2y 2 + 6x + 8y 1 = 0 (iii) What is the radius of the circle 3x 2 + 3y 2 + 5x
More informationChapter 2 Hopf Bifurcation and Normal Form Computation
Chapter 2 Hopf Bifurcation and Normal Form Computation In this chapter, we discuss the computation of normal forms. First we present a general approach which combines center manifold theory with computation
More informationSample Solutions of Assignment 10 for MAT3270B
Sample Solutions of Assignment 1 for MAT327B 1. For the following ODEs, (a) determine all critical points; (b) find the corresponding linear system near each critical point; (c) find the eigenvalues of
More information3. Identify and find the general solution of each of the following first order differential equations.
Final Exam MATH 33, Sample Questions. Fall 6. y = Cx 3 3 is the general solution of a differential equation. Find the equation. Answer: y = 3y + 9 xy. y = C x + C is the general solution of a differential
More informationStability Analysis for ODEs
Stability Analysis for ODEs Marc R Roussel September 13, 2005 1 Linear stability analysis Equilibria are not always stable Since stable and unstable equilibria play quite different roles in the dynamics
More informationStress, Strain, Mohr s Circle
Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected
More informationPhase Portraits of 1D Autonomous Equations
Phase Portraits of 1D Autonomous Equations In each of the following problems [1][5]: (a) find all equilibrium solutions; (b) determine whether each of the equilibrium solutions is stable, asmptoticall
More informationPh.D. Katarína Bellová Page 1 Mathematics 2 (10PHYBIPMA2) EXAM  Solutions, 20 July 2017, 10:00 12:00 All answers to be justified.
PhD Katarína Bellová Page 1 Mathematics 2 (10PHYBIPMA2 EXAM  Solutions, 20 July 2017, 10:00 12:00 All answers to be justified Problem 1 [ points]: For which parameters λ R does the following system
More informationEdexcel past paper questions. Core Mathematics 4. Parametric Equations
Edexcel past paper questions Core Mathematics 4 Parametric Equations Edited by: K V Kumaran Email: kvkumaran@gmail.com C4 Maths Parametric equations Page 1 Coordinate Geometry A parametric equation of
More informationUsing Lyapunov Theory I
Lecture 33 Stability Analysis of Nonlinear Systems Using Lyapunov heory I Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science  Bangalore Outline Motivation Definitions
More informationStable Manifolds of Saddle Equilibria for Pendulum Dynamics on S 2 and SO(3)
2011 50th IEEE Conference on Decision and Control and European Control Conference (CDCECC) Orlando, FL, USA, December 1215, 2011 Stable Manifolds of Saddle Equilibria for Pendulum Dynamics on S 2 and
More informationReview Problems for Exam 2
Review Problems for Exam 2 This is a list of problems to help you review the material which will be covered in the final. Go over the problem carefully. Keep in mind that I am going to put some problems
More informationPHASE PLANE DIAGRAMS OF DIFFERENCE EQUATIONS. 1. Introduction
PHASE PLANE DIAGRAMS OF DIFFERENCE EQUATIONS TANYA DEWLAND, JEROME WESTON, AND RACHEL WEYRENS Abstract. We will be determining qalitatie featres of a discrete dynamical system of homogeneos difference
More informationTWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 SYMMETRY. P. Yu 1,2 and M. Han 1
COMMUNICATIONS ON Website: http://aimsciences.org PURE AND APPLIED ANALYSIS Volume 3, Number 3, September 2004 pp. 515 526 TWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 SYMMETRY P. Yu 1,2
More information28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)
28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.
More information( ) a (graphical) transformation of y = f ( x )? x 0,2π. f ( 1 b) = a if and only if f ( a ) = b. f 1 1 f
WarmUp: Solve sinx = 2 for x 0,2π 5 (a) graphically (approximate to three decimal places) y (b) algebraically BY HAND EXACTLY (do NOT approximate except to verify your solutions) x x 0,2π, xscl = π 6,y,,
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationCONTROL SYSTEMS, ROBOTICS AND AUTOMATION  Vol. XII  Lyapunov Stability  Hassan K. Khalil
LYAPUNO STABILITY Hassan K. Khalil Department of Electrical and Computer Enigneering, Michigan State University, USA. Keywords: Asymptotic stability, Autonomous systems, Exponential stability, Global asymptotic
More informationCalculus III. Math 233 Spring Final exam May 3rd. Suggested solutions
alculus III Math 33 pring 7 Final exam May 3rd. uggested solutions This exam contains twenty problems numbered 1 through. All problems are multiple choice problems, and each counts 5% of your total score.
More information27. Topological classification of complex linear foliations
27. Topological classification of complex linear foliations 545 H. Find the expression of the corresponding element [Γ ε ] H 1 (L ε, Z) through [Γ 1 ε], [Γ 2 ε], [δ ε ]. Problem 26.24. Prove that for any
More informationSystems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University
Systems of Linear Equations By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Standard of Competency: Understanding the properties of systems of linear equations, matrices,
More informationAn angle in the Cartesian plane is in standard position if its vertex lies at the origin and its initial arm lies on the positive xaxis.
Learning Goals 1. To understand what standard position represents. 2. To understand what a principal and related acute angle are. 3. To understand that positive angles are measured by a counterclockwise
More informationMany Coupled Oscillators
Many Coupled Oscillators A VIBRATING STRING Say we have n particles with the same mass m equally spaced on a string having tension τ. Let y k denote the vertical displacement if the k th mass. Assume the
More informationEigenvalues. Matrices: Geometric Interpretation. Calculating Eigenvalues
Eigenvalues Matrices: Geometric Interpretation Start with a vector of length 2, for example, x =(1, 2). This among other things give the coordinates for a point on a plane. Take a 2 2 matrix, for example,
More informationSolutions to Final Exam Sample Problems, Math 246, Spring 2011
Solutions to Final Exam Sample Problems, Math 246, Spring 2 () Consider the differential equation dy dt = (9 y2 )y 2 (a) Identify its equilibrium (stationary) points and classify their stability (b) Sketch
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationExam Basics. midterm. 1 There will be 9 questions. 2 The first 3 are on premidterm material. 3 The next 1 is a mix of old and new material.
Exam Basics 1 There will be 9 questions. 2 The first 3 are on premidterm material. 3 The next 1 is a mix of old and new material. 4 The last 5 questions will be on new material since the midterm. 5 60
More informationTopic 14 Notes Jeremy Orloff
Topic 4 Notes Jeremy Orloff 4 Row reduction and subspaces 4. Goals. Be able to put a matrix into row reduced echelon form (RREF) using elementary row operations.. Know the definitions of null and column
More information1. Select the unique answer (choice) for each problem. Write only the answer.
MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +
More informationVectors. In kinematics, the simplest concept is position, so let s begin with a position vector shown below:
Vectors Extending the concepts of kinematics into two and three dimensions, the idea of a vector becomes very useful. By definition, a vector is a quantity with both a magnitude and a spatial direction.
More information19 Jacobian Linearizations, equilibrium points
169 19 Jacobian Linearizations, equilibrium points In modeling systems, we see that nearly all systems are nonlinear, in that the differential equations governing the evolution of the system s variables
More informationReport EProject Henriette Laabsch Toni Luhdo Steffen Mitzscherling Jens Paasche Thomas Pache
Potsdam, August 006 Report EProject Henriette Laabsch 7685 Toni Luhdo 7589 Steffen Mitzscherling 7540 Jens Paasche 7575 Thomas Pache 754 Introduction From 7 th February till 3 rd March, we had our laboratory
More informationA Mathematical Trivium
A Mathematical Trivium V.I. Arnold 1991 1. Sketch the graph of the derivative and the graph of the integral of a function given by a freehand graph. 2. Find the limit lim x 0 sin tan x tan sin x arcsin
More informationGRE Math Subject Test #5 Solutions.
GRE Math Subject Test #5 Solutions. 1. E (Calculus) Apply L Hôpital s Rule two times: cos(3x) 1 3 sin(3x) 9 cos(3x) lim x 0 = lim x 2 x 0 = lim 2x x 0 = 9. 2 2 2. C (Geometry) Note that a line segment
More informationEQUATIONS OF MOTION: CYLINDRICAL COORDINATES (Section 13.6)
EQUATIONS OF MOTION: CYLINDRICAL COORDINATES (Section 13.6) Today s Objectives: Students will be able to analyze the kinetics of a particle using cylindrical coordinates. APPLICATIONS The forces acting
More informationDynamical Systems. Bernard Deconinck Department of Applied Mathematics University of Washington Campus Box Seattle, WA, 98195, USA
Dynamical Systems Bernard Deconinck Department of Applied Mathematics University of Washington Campus Box 352420 Seattle, WA, 98195, USA June 4, 2009 i Prolegomenon These are the lecture notes for Amath
More informationLyapunov functions and stability problems
Lyapunov functions and stability problems Gunnar Söderbacka, Workshop Ghana, 29.510.5, 2013 1 Introduction In these notes we explain the power of Lyapunov functions in determining stability of equilibria
More informationFundamentals of Matrices
Maschinelles Lernen II Fundamentals of Matrices Christoph Sawade/Niels Landwehr/Blaine Nelson Tobias Scheffer Matrix Examples Recap: Data Linear Model: f i x = w i T x Let X = x x n be the data matrix
More informationDIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS
WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Series Editor: Leon O. Chua Series A Vol. 66 DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS JeanMarc Ginoux Université du Sud, France World Scientific
More informationA Tridiagonal Matrix
A Tridiagonal Matrix We investigate the simple n n real tridiagonal matrix: α β 0 0... 0 0 0 1 0 0... 0 0 β α β 0... 0 0 1 0 1 0... 0 0 0 β α β... 0 0 0 1 0 1... 0 0 M =....... = αi + β....... = αi + βt,
More informationName: Lab Partner: Section: In this experiment vector addition, resolution of vectors into components, force, and equilibrium will be explored.
Chapter 3 Vectors Name: Lab Partner: Section: 3.1 Purpose In this experiment vector addition, resolution of vectors into components, force, and equilibrium will be explored. 3.2 Introduction A vector is
More informationMAT1035 Analytic Geometry
MAT1035 Analytic Geometry Lecture Notes R.A. Sabri Kaan Gürbüzer Dokuz Eylül University 2016 2 Contents 1 Review of Trigonometry 5 2 Polar Coordinates 7 3 Vectors in R n 9 3.1 Located Vectors..............................................
More informationMath 2331 Linear Algebra
5. Eigenvectors & Eigenvalues Math 233 Linear Algebra 5. Eigenvectors & Eigenvalues ShangHuan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ ShangHuan Chiu,
More informationLinearization problem. The simplest example
Linear Systems Lecture 3 1 problem Consider a nonlinear timeinvariant system of the form ( ẋ(t f x(t u(t y(t g ( x(t u(t (1 such that x R n u R m y R p and Slide 1 A: f(xu f(xu g(xu and g(xu exist and
More informationEigenvalues and Eigenvectors, More Direction Fields and Systems of ODEs
Eigenvalues and Eigenvectors, More Direction Fields and Systems of ODEs First let us speak a bit about eigenvalues. Defn. An eigenvalue λ of an nxn matrix A means a scalar (perhaps a complex number) such
More informationMatrices and Vectors. Definition of Matrix. An MxN matrix A is a twodimensional array of numbers A =
30 MATHEMATICS REVIEW G A.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a twodimensional array of numbers A = a 11 a 12... a 1N a 21 a 22... a 2N...... a M1 a M2... a MN A matrix can
More informationControl Systems. Internal Stability  LTI systems. L. Lanari
Control Systems Internal Stability  LTI systems L. Lanari definitions (AS)  A system S is said to be asymptotically stable if its state zeroinput response converges to the origin for any initial condition
More information+ 2gx + 2fy + c = 0 if S
CIRCLE DEFINITIONS A circle is the locus of a point which moves in such a way that its distance from a fixed point, called the centre, is always a constant. The distance r from the centre is called the
More informationFor a rigid body that is constrained to rotate about a fixed axis, the gravitational torque about the axis is
Experiment 14 The Physical Pendulum The period of oscillation of a physical pendulum is found to a high degree of accuracy by two methods: theory and experiment. The values are then compared. Theory For
More informationHomework 2. Solutions T =
Homework. s Let {e x, e y, e z } be an orthonormal basis in E. Consider the following ordered triples: a) {e x, e x + e y, 5e z }, b) {e y, e x, 5e z }, c) {e y, e x, e z }, d) {e y, e x, 5e z }, e) {
More information2 Scalars and Vectors
2 Scalars and Vectors Scalars : have magnitude only : Length, time, mass, speed and volume is example of scalar. v Vectors : have magnitude and direction. v The magnitude of is written v v Position, displacement,
More informationEQUATIONS OF MOTION: NORMAL AND TANGENTIAL COORDINATES (Section 13.5)
EQUATIONS OF MOTION: NORMAL AND TANGENTIAL COORDINATES (Section 13.5) Today s Objectives: Students will be able to apply the equation of motion using normal and tangential coordinates. APPLICATIONS Race
More information