Introduction to the Phase Plane

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Introduction to the Phase Plane June, 06 The Phase Line A single first order differential equation of the form = f(y) () makes no mention of t in the function f. Such a differential equation is called autonomous, that is time independent variable t does not appear explicitly. When the independent variable is time, then the system is also known as time invariant. This is the content of the next exercise. Exercise. If y(t)) is a solution of (), and t 0 is a real number, then so is y(t 0 + t). We analyze the qualitative properties of (??) using the phase line through an example. Example. Consider the autonomous differential equation = A(y y )(y y )(y y 3 ). with y < y < y 3. We can give the qualitative behavior using the phase line. This is a line with arrow in the plane pointing to the right at y if f(y) is positive and to the left if f(y) is negative. Thus Figure : Plot of A(y y y )(y y )(y y 3 )., A =, y =, y =, y 3 = 3

2 Solutions y(t) move the direction of the arrows as time increases. A point y 0 are called an equilibrium of critical point if f(y 0 ) = 0. This give the trivial solution y(t) = y 0 for all t. Example 3. Describe the equilibrium points of For the equilibrium points = sin t. If both arrows point toward the critical point, it is called stable or a sink. Nearby solutions will converge to the equilibrium point. The solution is stable under small perturbations, meaning that if the solution is disturbed, it will return. y is a stable point. If both arrows point away from the critical point, it is called unstable or source. Nearby solutions will diverge from the equilibrium point, The solution is unstable under small perturbations, meaning that if the solution is disturbed, it will not return. y is an unstable point. If one arrow points towards the equilibrium point, and one points away, it is semi-stable or a node: It is stable in one direction (where the arrow points towards the point), and unstable in the other direction (where the arrow points away from the point). y is a node. More generally, if f has a derivative. f (y) < 0 at stable equilibria, f (y) > 0 at unstable equilibria. f (y) = 0 at nodes. Representation as a System of First Order Equations For the second order linear ordinary differential equation. y + by + cy = f. We can write this as system of two first order equations y = v and v = (bv + cy)/a. In the case of mass-spring oscillator, v is the velocity. In the case of a circuit in series, we typically write the equation in terms of the charge q, aq + bq + cq = E, we can write this as the system In this case I is the current. q = I and I = (bi + cq)/a.

3 3 The Phase Plane Equation Now we will look at a system of two first order linear ordinary differential equations. = f(x, y) and = g(x, y) () This system is also called autonomous and time invariant. As above, we have the property. Exercise 4. If (x(t), y(t)) is a solution of (), and t 0 is a real number, then so is (x(t 0 + t), y(t 0 + t)). or By the chain rule = = / /. For time invariant systems in (), this allows us to consider the phase plane equation, g(x, y) = f(x, y) (3) Let us review what the phase plane equation tells us and what it does not tell us. A plot of (x(t), y(t)) is a solution of () is a curve in the plane, known as a trajectory. If (x(t), y(t)) is a solution of (), then the derivatives of the solution are vectors (/, /) = (f(x, y), g(x, y)) with slope / = g(x, y)/f(x, y) and magnitude (or speed) (/) + (/) = f(x, y) + g(x, y) The trajectory itself does not tell us when the solution arrives at a position (x, y) in the plane, not does it tell us the speed. Indeed, if we were t0 multliply both f and g by a constant α 0, it would not change the trajectory. It would change the speed and it would change the direction of the solution curve is α < 0. It does tell us the slope / which is the tangent line to the trajectory at a position (x, y). The solutions to (3) give us these trajectories. However, for (), t is the independent variable. x and y are dependent variables. The direction fields give us solutions to differential equations with x the independent variable and y dependent. So, even though these plots look very similar, they reveal different information. 3

4 Example 5. An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, having inductance L, and a capacitor, having capacitance C, the governing equations for the chage q are L d q + C q = 0. If we have the inttial value problem, q(0) = 0, q (0) = I 0, then the solution is q(t) = I 0 ω 0 sin ω 0 t where ω 0 = /LC. The current, Thus the trajectory I(t) = dq (t) = I 0 cos ω 0 t. (q(t), I(t)) = I 0 ( ω 0 sin ω 0 t, cos ω 0 t). is an ellipse. ω 0q + I = I 0. In addition, (t) = d q (t) = I 0ω 0 sin ω 0 t. In the qiplane the solution (q(t), I(t)) has slope Thus, dq = / dq/ = I 0ω 0 sin ω 0 t = ω 0 tan ω 0 t. I 0 cos ω 0 t. If now, we look at this same equation at a first order system, we have that dq = I, and = LC q dq = / dq/ = q/lc I From the solution to the differential equation, we have as before. = ω 0q I I ω 0 q = I 0 sin ω 0 t ω 0 I 0 cos ω 0 t = ω 0 cot ω 0 t 4

5 If we focus on a solution to dq I dq I = ω 0q I = ω0q dq dq = ω0qdq I = ω 0q + c ω0q + I = c If the trajectory has the point (0, I 0 ), then c = I 0, the ellipse we saw before. The difference is that we do not have the time course for the trace of the differential equation along the ellipse. 4 Equilibrium Points As in the of one equation, a point x 0, y 0 satisfying f(x 0, y 0 ) = 0 and g(x 0, y 0 ) = 0 called a critical point, or equilibrium point, of the autonomous system () The corresponding constant solution (x(t), y(t)) = (x 0, y 0 ) is called an equilibrium solution. The set of all critical points is called the critical point set. Example 6. For = x, and = y The direction field = y x Gives vectors with positive slope in quadrants I and III and negative slope in quadrants I and IV. So the vector etiher aim toward the equilibrium point (0, 0) or away. Since both x and y are decreasing with time in quadrant I, we can add arrows that point to the origin and the origin is stable. To solve the phase plane equation y = x y = x y = x ln y = ln x + c y = cx 5

6 For = x, and = y We have the same equilibrium point and the same phase plane equation, but the origins is unstable and the arrows point autward Exercise 7. Give the equilibrium points and use software to describe them for the following pairs of differential equations (See aeb09/pplane.html.).. 3. We can now develop a general procedure: If (x 0, y 0 ) is an equilibrium point. = y x and = sin x = x + y and = x = y and = x x3 Create a linearize equation about this point, setting ( x, ỹ) = (x x 0, y y 0 ) d x = x f x (x 0, y 0 ) + ỹ f y (x 0, y 0 ) and = x g x (x 0, y 0 ) + ỹ g y (x 0, y 0 ) Create a single second order linear constant coefficient ordinary differential equation. Find the roots r and r + of the associated auxiliary equation. The type of equilibrium point is determined by the following table r and r + real and both negative real and both positive real, one positive and one negative purely imaginary complex, negative real part complex, positive real part type of equilibrium point stable unstable saddle node oscillatory stable spiral unstable spiral 6

Def. (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

4 Second-Order Systems

4 Second-Order Systems Second-order 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

Math 312 Lecture Notes Linear Two-dimensional Systems of Differential Equations

Math 2 Lecture Notes Linear Two-dimensional Systems of Differential Equations Warren Weckesser Department of Mathematics Colgate University February 2005 In these notes, we consider the linear system of

Section 9.3 Phase Plane Portraits (for Planar Systems)

Section 9.3 Phase Plane Portraits (for Planar Systems) Key Terms: Equilibrium point of planer system yꞌ = Ay o Equilibrium solution Exponential solutions o Half-line solutions Unstable solution Stable

2.2. PLANAR LINEAR SYSTEMS 3 2.2. Planar Linear Systems We consider the linear system of two first order differential equations or equivalently, = ax + by (2.7) dy = cx + dy [ d x x = A x, where x =, and

STABILITY. Phase portraits and local stability

MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),

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

154 Chapter 9 Hints, Answers, and Solutions The particular trajectories are highlighted in the phase portraits below.

54 Chapter 9 Hints, Answers, and Solutions 9. The Phase Plane 9.. 4. The particular trajectories are highlighted in the phase portraits below... 3. 4. 9..5. Shown below is one possibility with x(t) and

Department of Mathematics IIT Guwahati

Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati A system of first order differential equations is called autonomous if the system can be written in the form dx 1 dt = g 1(x 1,

+ i. cos(t) + 2 sin(t) + c 2.

MATH HOMEWORK #7 PART A SOLUTIONS Problem 7.6.. Consider the system x = 5 x. a Express the general solution of the given system of equations in terms of realvalued functions. b Draw a direction field,

. For each initial condition y(0) = y 0, there exists a. unique solution. In fact, given any point (x, y), there is a unique curve through this point,

1.2. Direction Fields: Graphical Representation of the ODE and its Solution Section Objective(s): Constructing Direction Fields. Interpreting Direction Fields. Definition 1.2.1. A first order ODE of the

A plane autonomous system is a pair of simultaneous first-order differential equations,

Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, ẋ = f(x, y), ẏ = g(x, y). This system has an equilibrium

PES 1120 Spring 2014, Spendier Lecture 35/Page 1

PES 0 Spring 04, Spendier Lecture 35/Page Today: chapter 3 - LC circuits We have explored the basic physics of electric and magnetic fields and how energy can be stored in capacitors and inductors. We

Problem set 7 Math 207A, Fall 2011 Solutions

Problem set 7 Math 207A, Fall 2011 s 1. Classify the equilibrium (x, y) = (0, 0) of the system x t = x, y t = y + x 2. Is the equilibrium hyperbolic? Find an equation for the trajectories in (x, y)- phase

AC Circuits Homework Set

Problem 1. In an oscillating LC circuit in which C=4.0 μf, the maximum potential difference across the capacitor during the oscillations is 1.50 V and the maximum current through the inductor is 50.0 ma.

2.10 Saddles, Nodes, Foci and Centers

2.10 Saddles, Nodes, Foci and Centers In Section 1.5, a linear system (1 where x R 2 was said to have a saddle, node, focus or center at the origin if its phase portrait was linearly equivalent to one

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 2009 Farzaneh Abdollahi Nonlinear Control Lecture 2 1/68

Math 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

ENGI Duffing s Equation Page 4.65

ENGI 940 4. - Duffing s Equation Page 4.65 4. Duffing s Equation Among the simplest models of damped non-linear forced oscillations of a mechanical or electrical system with a cubic stiffness term is Duffing

Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits

Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the timevarying

Nonlinear Autonomous Systems of Differential

Chapter 4 Nonlinear Autonomous Systems of Differential Equations 4.0 The Phase Plane: Linear Systems 4.0.1 Introduction Consider a system of the form x = A(x), (4.0.1) where A is independent of t. Such

ANSWERS 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

Chapter 6 Nonlinear Systems and Phenomena. Friday, November 2, 12

Chapter 6 Nonlinear Systems and Phenomena 6.1 Stability and the Phase Plane We now move to nonlinear systems Begin with the first-order system for x(t) d dt x = f(x,t), x(0) = x 0 In particular, consider

Stability of Dynamical systems

Stability of Dynamical systems Stability Isolated equilibria Classification of Isolated Equilibria Attractor and Repeller Almost linear systems Jacobian Matrix Stability Consider an autonomous system u

Hello everyone, Best, Josh

Hello everyone, As promised, the chart mentioned in class about what kind of critical points you get with different types of eigenvalues are included on the following pages (The pages are an ecerpt from

L = 1 2 a(q) q2 V (q).

Physics 3550, Fall 2011 Motion near equilibrium - Small Oscillations Relevant Sections in Text: 5.1 5.6 Motion near equilibrium 1 degree of freedom One of the most important situations in physics is motion

Math 266: Phase Plane Portrait

Math 266: Phase Plane Portrait Long Jin Purdue, Spring 2018 Review: Phase line for an autonomous equation For a single autonomous equation y = f (y) we used a phase line to illustrate the equilibrium solutions

Definition of differential equations and their classification. Methods of solution of first-order differential equations

Introduction to differential equations: overview Definition of differential equations and their classification Solutions of differential equations Initial value problems Existence and uniqueness Mathematical

ENGI 9420 Lecture Notes 4 - Stability Analysis Page Stability Analysis for Non-linear Ordinary Differential Equations

ENGI 940 Lecture Notes 4 - Stability Analysis Page 4.01 4. Stability Analysis for Non-linear Ordinary Differential Equations A pair of simultaneous first order homogeneous linear ordinary differential

ẋ = f(x, y), ẏ = g(x, y), (x, y) D, can only have periodic solutions if (f,g) changes sign in D or if (f,g)=0in D.

4 Periodic Solutions We have shown that in the case of an autonomous equation the periodic solutions correspond with closed orbits in phase-space. Autonomous two-dimensional systems with phase-space R

Differential Equations Spring 2007 Assignments

Differential Equations Spring 2007 Assignments Homework 1, due 1/10/7 Read the first two chapters of the book up to the end of section 2.4. Prepare for the first quiz on Friday 10th January (material up

Old Math 330 Exams. David M. McClendon. Department of Mathematics Ferris State University

Old Math 330 Exams David M. McClendon Department of Mathematics Ferris State University Last updated to include exams from Fall 07 Contents Contents General information about these exams 3 Exams from Fall

8. Qualitative analysis of autonomous equations on the line/population dynamics models, phase line, and stability of equilibrium points (corresponds

c Dr Igor Zelenko, Spring 2017 1 8. Qualitative analysis of autonomous equations on the line/population dynamics models, phase line, and stability of equilibrium points (corresponds to section 2.5) 1.

Final Exam Review. Review of Systems of ODE. Differential Equations Lia Vas. 1. Find all the equilibrium points of the following systems.

Differential Equations Lia Vas Review of Systems of ODE Final Exam Review 1. Find all the equilibrium points of the following systems. (a) dx = x x xy (b) dx = x x xy = 0.5y y 0.5xy = 0.5y 0.5y 0.5xy.

Introductory Differential Equations

Introductory Differential Equations Lecture Notes June 3, 208 Contents Introduction Terminology and Examples 2 Classification of Differential Equations 4 2 First Order ODEs 5 2 Separable ODEs 5 22 First

Solutions to the Review Questions

Solutions to the Review Questions Short Answer/True or False. True or False, and explain: (a) If y = y + 2t, then 0 = y + 2t is an equilibrium solution. False: (a) Equilibrium solutions are only defined

Classification of Phase Portraits at Equilibria for u (t) = f( u(t))

Classification of Phase Portraits at Equilibria for u t = f ut Transfer of Local Linearized Phase Portrait Transfer of Local Linearized Stability How to Classify Linear Equilibria Justification of the

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv

Math 1270 Honors ODE I Fall, 2008 Class notes # 1 We have learned how to study nonlinear systems x 0 = F (x; y) y 0 = G (x; y) (1) by linearizing around equilibrium points. If (x 0 ; y 0 ) is an equilibrium

Appendix: A Computer-Generated Portrait Gallery

Appendi: A Computer-Generated Portrait Galler There are a number of public-domain computer programs which produce phase portraits for 2 2 autonomous sstems. One has the option of displaing the trajectories

Solutions to the Review Questions

Solutions to the Review Questions Short Answer/True or False. True or False, and explain: (a) If y = y + 2t, then 0 = y + 2t is an equilibrium solution. False: This is an isocline associated with a slope

8.1 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 integro-differential equations

Nonlinear dynamics & chaos BECS

Nonlinear dynamics & chaos BECS-114.7151 Phase portraits Focus: nonlinear systems in two dimensions General form of a vector field on the phase plane: Vector notation: Phase portraits Solution x(t) describes

L = 1 2 a(q) q2 V (q).

Physics 3550 Motion near equilibrium - Small Oscillations Relevant Sections in Text: 5.1 5.6, 11.1 11.3 Motion near equilibrium 1 degree of freedom One of the most important situations in physics is motion

The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d

Solving ODEs using Fourier Transforms The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d 2 dx + q f (x) R(x)

Practice Problems for Final Exam

Math 1280 Spring 2016 Practice Problems for Final Exam Part 2 (Sections 6.6, 6.7, 6.8, and chapter 7) S o l u t i o n s 1. Show that the given system has a nonlinear center at the origin. ẋ = 9y 5y 5,

MATH 215/255 Solutions to Additional Practice Problems April dy dt

. For the nonlinear system MATH 5/55 Solutions to Additional Practice Problems April 08 dx dt = x( x y, dy dt = y(.5 y x, x 0, y 0, (a Show that if x(0 > 0 and y(0 = 0, then the solution (x(t, y(t of the

Chapter 4. Systems of ODEs. Phase Plane. Qualitative Methods

Chapter 4 Systems of ODEs. Phase Plane. Qualitative Methods Contents 4.0 Basics of Matrices and Vectors 4.1 Systems of ODEs as Models 4.2 Basic Theory of Systems of ODEs 4.3 Constant-Coefficient Systems.

EE222 - Spring 16 - Lecture 2 Notes 1

EE222 - Spring 16 - Lecture 2 Notes 1 Murat Arcak January 21 2016 1 Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Essentially Nonlinear Phenomena Continued

Module 25: Outline Resonance & Resonance Driven & LRC Circuits Circuits 2

Module 25: Driven RLC Circuits 1 Module 25: Outline Resonance & Driven LRC Circuits 2 Driven Oscillations: Resonance 3 Mass on a Spring: Simple Harmonic Motion A Second Look 4 Mass on a Spring (1) (2)

dx n a 1(x) dy

HIGHER ORDER DIFFERENTIAL EQUATIONS Theory of linear equations Initial-value and boundary-value problem nth-order initial value problem is Solve: a n (x) dn y dx n + a n 1(x) dn 1 y dx n 1 +... + a 1(x)

MATH 415, WEEKS 7 & 8: Conservative and Hamiltonian Systems, Non-linear Pendulum

MATH 415, WEEKS 7 & 8: Conservative and Hamiltonian Systems, Non-linear Pendulum Reconsider the following example from last week: dx dt = x y dy dt = x2 y. We were able to determine many qualitative features

Physics for Scientists & Engineers 2

Electromagnetic Oscillations Physics for Scientists & Engineers Spring Semester 005 Lecture 8! We have been working with circuits that have a constant current a current that increases to a constant current

Autonomous Equations and Stability Sections

A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Autonomous Equations and Stability Sections 2.4-2.5 Dr. John Ehrke Department of Mathematics Fall 2012 Autonomous Differential

Entrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.

Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the

4.2 Homogeneous Linear Equations

4.2 Homogeneous Linear Equations Homogeneous Linear Equations with Constant Coefficients Consider the first-order linear differential equation with constant coefficients a 0 and b. If f(t) = 0 then this

Applications of Second-Order Differential Equations

Applications of Second-Order Differential Equations ymy/013 Building Intuition Even though there are an infinite number of differential equations, they all share common characteristics that allow intuition

Response of Second-Order Systems

Unit 3 Response of SecondOrder Systems In this unit, we consider the natural and step responses of simple series and parallel circuits containing inductors, capacitors and resistors. The equations which

Computer Problems for Methods of Solving Ordinary Differential Equations

Computer Problems for Methods of Solving Ordinary Differential Equations 1. Have a computer make a phase portrait for the system dx/dt = x + y, dy/dt = 2y. Clearly indicate critical points and separatrices.

Inductance, RL and RLC Circuits

Inductance, RL and RLC Circuits Inductance Temporarily storage of energy by the magnetic field When the switch is closed, the current does not immediately reach its maximum value. Faraday s law of electromagnetic

Solutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10)

Solutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10) Mason A. Porter 15/05/2010 1 Question 1 i. (6 points) Define a saddle-node bifurcation and show that the first order system dx dt = r x e x

Section 5. Graphing Systems

Section 5. Graphing Systems 5A. The Phase Plane 5A-1. Find the critical points of each of the following non-linear autonomous systems. x = x 2 y 2 x = 1 x + y a) b) y = x xy y = y + 2x 2 5A-2. Write each

Do not write below here. Question Score Question Score Question Score

MATH-2240 Friday, May 4, 2012, FINAL EXAMINATION 8:00AM-12:00NOON 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

INDUCTANCE Self Inductance

NDUTANE 3. Self nductance onsider the circuit shown in the Figure. When the switch is closed the current, and so the magnetic field, through the circuit increases from zero to a specific value. The increasing

Coordinate Curves for Trajectories

43 The material on linearizations and Jacobian matrices developed in the last chapter certainly expanded our ability to deal with nonlinear systems of differential equations Unfortunately, those tools

11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes

Parametric Differentiation 11.6 Introduction Often, the equation of a curve may not be given in Cartesian form y f(x) but in parametric form: x h(t), y g(t). In this section we see how to calculate the

Solutionbank Edexcel AS and A Level Modular Mathematics

Page of Exercise A, Question The curve C, with equation y = x ln x, x > 0, has a stationary point P. Find, in terms of e, the coordinates of P. (7) y = x ln x, x > 0 Differentiate as a product: = x + x

Sample 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

11.6. Parametric Differentiation. Introduction. Prerequisites. Learning Outcomes

Parametric Differentiation 11.6 Introduction Sometimes the equation of a curve is not be given in Cartesian form y f(x) but in parametric form: x h(t), y g(t). In this Section we see how to calculate the

2D-Volterra-Lotka Modeling For 2 Species

Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya 2D-Volterra-Lotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose

2. Higher-order Linear ODE s

2. Higher-order Linear ODE s 2A. Second-order Linear ODE s: General Properties 2A-1. On the right below is an abbreviated form of the ODE on the left: (*) y + p()y + q()y = r() Ly = r() ; where L is the

MATH 251 Final Examination December 19, 2012 FORM A. Name: Student Number: Section:

MATH 251 Final Examination December 19, 2012 FORM A Name: Student Number: Section: This exam has 17 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all

Nonlinear differential equations - phase plane analysis

Nonlinear differential equations - phase plane analysis We consider the general first order differential equation for y(x Revision Q(x, y f(x, y dx P (x, y. ( Curves in the (x, y-plane which satisfy this

Differential Equations and Modeling

Differential Equations and Modeling Preliminary Lecture Notes Adolfo J. Rumbos c Draft date: March 22, 2018 March 22, 2018 2 Contents 1 Preface 5 2 Introduction to Modeling 7 2.1 Constructing Models.........................

Chapter 31: RLC Circuits. PHY2049: Chapter 31 1

hapter 31: RL ircuits PHY049: hapter 31 1 L Oscillations onservation of energy Topics Damped oscillations in RL circuits Energy loss A current RMS quantities Forced oscillations Resistance, reactance,

B5.6 Nonlinear Systems

B5.6 Nonlinear Systems 4. Bifurcations Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Local bifurcations for vector fields 1.1 The problem 1.2 The extended centre

ODE. Philippe Rukimbira. Department of Mathematics Florida International University PR (FIU) MAP / 92

ODE Philippe Rukimbira Department of Mathematics Florida International University PR (FIU) MAP 2302 1 / 92 4.4 The method of Variation of parameters 1. Second order differential equations (Normalized,

APPPHYS 217 Tuesday 6 April 2010

APPPHYS 7 Tuesday 6 April Stability and input-output performance: second-order systems Here we present a detailed example to draw connections between today s topics and our prior review of linear algebra

Math 392 Exam 1 Solutions Fall (10 pts) Find the general solution to the differential equation dy dt = 1

Math 392 Exam 1 Solutions Fall 20104 1. (10 pts) Find the general solution to the differential equation = 1 y 2 t + 4ty = 1 t(y 2 + 4y). Hence (y 2 + 4y) = t y3 3 + 2y2 = ln t + c. 2. (8 pts) Perform Euler

Physics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics

Applications of nonlinear ODE systems: Physics: spring-mass system, planet motion, pendulum Chemistry: mixing problems, chemical reactions Biology: ecology problem, neural conduction, epidemics Economy:

Complex Dynamic Systems: Qualitative vs Quantitative analysis

Complex Dynamic Systems: Qualitative vs Quantitative analysis Complex Dynamic Systems Chiara Mocenni Department of Information Engineering and Mathematics University of Siena (mocenni@diism.unisi.it) Dynamic

Handout 10: Inductance. Self-Inductance and inductors

1 Handout 10: Inductance Self-Inductance and inductors In Fig. 1, electric current is present in an isolate circuit, setting up magnetic field that causes a magnetic flux through the circuit itself. This

The Harmonic Oscillator

The Harmonic Oscillator Math 4: Ordinary Differential Equations Chris Meyer May 3, 008 Introduction The harmonic oscillator is a common model used in physics because of the wide range of problems it can

Math 310 Introduction to Ordinary Differential Equations Final Examination August 9, Instructor: John Stockie

Make sure this exam has 15 pages. Math 310 Introduction to Ordinary Differential Equations inal Examination August 9, 2006 Instructor: John Stockie Name: (Please Print) Student Number: Special Instructions

1. < 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

( ) f (k) = FT (R(x)) = R(k)

Solving ODEs using Fourier Transforms The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d 2 dx + q f (x) = R(x)

Chapter #4 EEE8086-EEE8115. Robust and Adaptive Control Systems

Chapter #4 Robust and Adaptive Control Systems Nonlinear Dynamics.... Linear Combination.... Equilibrium points... 3 3. Linearisation... 5 4. Limit cycles... 3 5. Bifurcations... 4 6. Stability... 6 7.

Vectors, matrices, eigenvalues and eigenvectors

Vectors, matrices, eigenvalues and eigenvectors 1 ( ) ( ) ( ) Scaling a vector: 0.5V 2 0.5 2 1 = 0.5 = = 1 0.5 1 0.5 ( ) ( ) ( ) ( ) Adding two vectors: V + W 2 1 2 + 1 3 = + = = 1 3 1 + 3 4 ( ) ( ) a

How to measure complex impedance at high frequencies where phase measurement is unreliable.

Objectives In this course you will learn the following Various applications of transmission lines. How to measure complex impedance at high frequencies where phase measurement is unreliable. How and why

(Refer Slide Time: 00:32)

Nonlinear Dynamical Systems Prof. Madhu. N. Belur and Prof. Harish. K. Pillai Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 12 Scilab simulation of Lotka Volterra

B5.6 Nonlinear Systems

B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A

CDS 101/110a: Lecture 2.1 Dynamic Behavior

CDS 11/11a: Lecture.1 Dynamic Behavior Richard M. Murray 6 October 8 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium

Lecture 38. Almost Linear Systems

Math 245 - Mathematics of Physics and Engineering I Lecture 38. Almost Linear Systems April 20, 2012 Konstantin Zuev (USC) Math 245, Lecture 38 April 20, 2012 1 / 11 Agenda Stability Properties of Linear

MCE693/793: Analysis and Control of Nonlinear Systems

MCE693/793: Analysis and Control of Nonlinear Systems Systems of Differential Equations Phase Plane Analysis Hanz Richter Mechanical Engineering Department Cleveland State University Systems of Nonlinear

Stability 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,

Theoretical physics. Deterministic chaos in classical physics. Martin Scholtz

Theoretical physics Deterministic chaos in classical physics Martin Scholtz scholtzzz@gmail.com Fundamental physical theories and role of classical mechanics. Intuitive characteristics of chaos. Newton

Integral Curve (generic name for a member of the collection of known as the general solution)

Section 1.2 Solutions of Some Differential Equations Key Terms/Ideas: Phase Line (This topic is not in this section of the book.) Classification of Equilibrium Solutions: Source, Sink, Node SPECIAL CASE:

2. Determine whether the following pair of functions are linearly dependent, or linearly independent:

Topics to be covered on the exam include: Recognizing, and verifying solutions to homogeneous second-order linear differential equations, and their corresponding Initial Value Problems Recognizing and

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self-paced Course

SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self-paced Course MODULE 26 APPLICATIONS TO ELECTRICAL CIRCUITS Module Topics 1. Complex numbers and alternating currents 2. Complex impedance 3.