# The Phase Plane. Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations

Save this PDF as:

Size: px
Start display at page:

Download "The Phase Plane. Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations"

## Transcription

1 The Phase Plane Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations Phase Portraits of Linear Systems Consider a systems of linear differential equations x = Ax. Its phase portrait is a representative set of its solutions, plotted as parametric curves (with t as the parameter) on the Cartesian plane tracing the path of each particular solution (x, y) = (x 1 (t), x 2 (t)), < t <. Similar to a direction field, a phase portrait is a graphical tool to visualize how the solutions of a given system of differential equations would behave in the long run. In this context, the Cartesian plane where the phase portrait resides is called the phase plane. The parametric curves traced by the solutions are sometimes also called their trajectories. Remark: It is quite labor-intensive, but it is possible to sketch the phase portrait by hand without first having to solve the system of equations that it represents. Just like a direction field, a phase portrait can be a tool to predict the behaviors of a system s solutions. To do so, we draw a grid on the phase plane. Then, at each grid point x = (α, β), we can calculate the solution trajectory s instantaneous direction of motion at that point by using the given system of equations to compute the tangent / velocity vector, x. Namely plug in x = (α, β) to compute x = Ax. In the first section we will examine the phase portrait of linear system of differential equations. We will classify the type and stability the equilibrium solution of a given linear system by the shape and behavior of its phase portrait Zachary S Tseng D-2-1

2 Equilibrium Solution (a.k.a. Critical Point, or Stationary Point) An equilibrium solution of the system x = Ax is a point (x 1, x 2 ) where x = 0, that is, where x 1 = 0 = x 2. An equilibrium solution is a constant solution of the system, and is usually called a critical point. For a linear system x = Ax, an equilibrium solution occurs at each solution of the system (of homogeneous algebraic equations) Ax = 0. As we have seen, such a system has exactly one solution, located at the origin, if det(a) 0. If det(a) = 0, then there are infinitely many solutions. For our purpose, and unless otherwise noted, we will only consider systems of linear differential equations whose coefficient matrix A has nonzero determinant. That is, we will only consider systems where the origin is the only critical point. Note: A matrix could only have zero as one of its eigenvalues if and only if its determinant is also zero. Therefore, since we limit ourselves to consider only those systems where det(a) 0, we will not encounter in this section any matrix with zero as an eigenvalue Zachary S Tseng D-2-2

3 Classification of Critical Points Similar to the earlier discussion on the equilibrium solutions of a single first order differential equation using the direction field, we will presently classify the critical points of various systems of first order linear differential equations by their stability. In addition, due to the truly two-dimensional nature of the parametric curves, we will also classify the type of those critical points by their shapes (or, rather, by the shape formed by the trajectories about each critical point). Comment: The accurate tracing of the parametric curves of the solutions is not an easy task without electronic aids. However, we can obtain very reasonable approximation of a trajectory by using the very same idea behind the direction field, namely the tangent line approximation. At each point x = (x 1, x 2 ) on the ty-plane, the direction of motion of the solution curve the passes through the point is determined by the direction vector (i.e. the tangent vector) x, the derivative of the solution vector x, evaluated at the given point. The tangent vector at each given point can be calculated directly from the given matrix-vector equation x = Ax, using the position vector x = (x 1, x 2 ). Like working with a direction field, there is no need to find the solution first before performing this approximation Zachary S Tseng D-2-3

4 Given x = Ax, where there is only one critical point, at (0,0): Case I. Distinct real eigenvalues The general solution is x 2 r1 t r t = C1 k1 e + C2 k2 e. 1. When r 1 and r 2 are both positive, or are both negative The phase portrait shows trajectories either moving away from the critical point to infinite-distant away (when r > 0), or moving directly toward, and converge to the critical point (when r < 0). The trajectories that are the eigenvectors move in straight lines. The rest of the trajectories move, initially when near the critical point, roughly in the same direction as the eigenvector of the eigenvalue with the smaller absolute value. Then, farther away, they would bend toward the direction of the eigenvector of the eigenvalue with the larger absolute value The trajectories either move away from the critical point to infinite-distant away (when r are both positive), or move toward from infinite-distant out and eventually converge to the critical point (when r are both negative). This type of critical point is called a node. It is asymptotically stable if r are both negative, unstable if r are both positive Zachary S Tseng D-2-4

5 Two distinct real eigenvalues, both of the same sign Type: Node Stability: It is unstable if both eigenvalues are positive; asymptotically stable if they are both negative Zachary S Tseng D-2-5

6 2. When r 1 and r 2 have opposite signs (say r 1 > 0 and r 2 < 0) In this type of phase portrait, the trajectories given by the eigenvectors of the negative eigenvalue initially start at infinite-distant away, move toward and eventually converge at the critical point. The trajectories that represent the eigenvectors of the positive eigenvalue move in exactly the opposite way: start at the critical point then diverge to infinite-distant out. Every other trajectory starts at infinite-distant away, moves toward but never converges to the critical point, before changing direction and moves back to infinite-distant away. All the while it would roughly follow the 2 sets of eigenvectors. This type of critical point is called a saddle point. It is always unstable Zachary S Tseng D-2-6

7 Two distinct real eigenvalues, opposite signs Type: Stability: Saddle Point It is always unstable Zachary S Tseng D-2-7

8 Case II. Repeated real eigenvalue 3. When there are two linearly independent eigenvectors k 1 and k 2. The general solution is x = C 1 k 1 e rt + C 2 k 2 e rt = e rt (C 1 k 1 + C 2 k 2 ). Every nonzero solution traces a straight-line trajectory, in the direction given by the vector C 1 k 1 + C 2 k 2. The phase portrait thus has a distinct star-burst shape. The trajectories either move directly away from the critical point to infinite-distant away (when r > 0), or move directly toward, and converge to the critical point (when r < 0). This type of critical point is called a proper node (or a starl point). It is asymptotically stable if r < 0, unstable if r > 0. Note: For 2 2 systems of linear differential equations, this will occur if, and only if, when the coefficient matrix A is a constant multiple of the identity matrix: 1 A = α α = 0 0 α, α = any nonzero constant *. * 1 0 C1 In the case of α = 0, the solution is x = C1 + C2 =. Every solution is an equilibrium 0 1 C2 solution. Therefore, every trajectory on its phase portrait consists of a single point, and every point on the phase plane is a trajectory Zachary S Tseng D-2-8

9 A repeated real eigenvalue, two linearly independent eigenvectors Type: Proper Node (or Star Point) Stability: It is unstable if the eigenvalue is positive; asymptotically stable if the eigenvalue is negative Zachary S Tseng D-2-9

10 4. When there is only one linearly independent eigenvector k. Then the general solution is x = C 1 k e rt + C 2 (k t e rt + η e rt ). The phase portrait shares characteristics with that of a node. With only one eigenvector, it is a degenerated-looking node that is a cross between a node and a spiral point (see case 6 below). The trajectories either all diverge away from the critical point to infinite-distant away (when r > 0), or all converge to the critical point (when r < 0). This type of critical point is called an improper node. It is asymptotically stable if r < 0, unstable if r > Zachary S Tseng D-2-10

11 A repeated real eigenvalue, one linearly independent eigenvector Type: Improper Node Stability: It is unstable if the eigenvalue is positive; asymptotically stable if the eigenvalue is negative Zachary S Tseng D-2-11

12 Case III. Complex conjugate eigenvalues The general solution is λ t ( a µ t) b sin( µ t) ) + C e ( a sin( µ t) + b cos( µ )) λ t x= C1e cos( 2 t 5. When the real part λ is zero. In this case the trajectories neither converge to the critical point nor move to infinite-distant away. Rather, they stay in constant, elliptical (or, rarely, circular) orbits. This type of critical point is called a center. It has a unique stability classification shared by no other: stable (or neutrally stable). It is NOT asymptotically stable and one should not confuse them. 6. When the real part λ is nonzero. The trajectories still retain the elliptical traces as in the previous case. However, with each revolution, their distances from the critical point grow/decay exponentially according to the term e λt. Therefore, the phase portrait shows trajectories that spiral away from the critical point to infinite-distant away (when λ > 0). Or trajectories that spiral toward, and converge to the critical point (when λ < 0). This type of critical point is called a spiral point. It is asymptotically stable if λ < 0, it is unstable if λ > Zachary S Tseng D-2-12

13 Complex eigenvalues, with real part zero (purely imaginary numbers) Type: Center Stability: Stable (but not asymptotically stable); sometimes it is referred to as neutrally stable Zachary S Tseng D-2-13

14 Complex eigenvalues, with nonzero real part Type: Spiral Point Stability: It is unstable if the eigenvalues have positive real part; asymptotically stable if the eigenvalues have negative real part Zachary S Tseng D-2-14

15 Summary of Stability Classification Asymptotically stable All trajectories of its solutions converge to the critical point as t. A critical point is asymptotically stable if all of A s eigenvalues are negative, or have negative real part for complex eigenvalues. Unstable All trajectories (or all but a few, in the case of a saddle point) start out at the critical point at t, then move away to infinitely distant out as t. A critical point is unstable if at least one of A s eigenvalues is positive, or has positive real part for complex eigenvalues. Stable (or neutrally stable) Each trajectory move about the critical point within a finite range of distance. It never moves out to infinitely distant, nor (unlike in the case of asymptotically stable) does it ever go to the critical point. A critical point is stable if A s eigenvalues are purely imaginary. In short, as t increases, if all (or almost all) trajectories 1. converge to the critical point asymptotically stable, 2. move away from the critical point to infinitely far away unstable, 3. stay in a fixed orbit within a finite (i.e., bounded) range of distance away from the critical point stable (or neutrally stable) Zachary S Tseng D-2-15

16 Nonhomogeneous Linear Systems with Constant Coefficients Now let us consider the nonhomogeneous system x = Ax + b. Where b is a constant vector. The system above is explicitly: x 1 = a x 1 + b x 2 + g 1 x 2 = c x 1 + d x 2 + g 2 As before, we can find the critical point by setting x 1 = x 2 = 0 and solve the resulting nonhomgeneous system of algebraic equations. The origin will no longer be a critical point, since the zero vector is never a solution of a nonhomogeneous linear system. Instead, the unique critical point (as long as A has nonzero determinant, there remains exactly one critical point) will be located at the solution of the system of algebraic equations: 0 = a x 1 + b x 2 + g 1 0 = c x 1 + d x 2 + g 2 Once we have found the critical point, say it is the point (x 1, x 2 ) = (α, β), it then could be moved to (0, 0) via the translations χ 1 = x 1 α and χ 2 = x 2 β. The result after the translation would be the homogeneous linear system χ = Aχ. The two systems (before and after the translations) have the same coefficient matrix.* Their respective critical points will also have identical type and stability classification. Therefore, to determine the type and stability of the critical point of the given nonhomogeneous system, all we need to do is to disregard b, then take its coefficient matrix A and use its eigenvalues for the determination, in exactly the same way as we would do with the corresponding homogeneous system of equations Zachary S Tseng D-2-16

17 * (Optional topic) Note: Here is the formal justification of our ability to discard the vector b when determining the type and stability of the critical point of the nonhomogenous system x = Ax + b. Suppose (x 1, x 2 ) = (α, β) is the critical point of the system. That is 0 = a α + b β + g 1 0 = c α + d β + g 2 Now apply the translations χ 1 = x 1 α and χ 2 = x 2 β. We see that x 1 = χ 1 + α, x 2 = χ 2 + β, χ 1 = x 1, and χ 2 = x 2. Substitute them into the system x = Ax + b: x 1 = χ 1 = a x 1 + b x 2 + g 1 = a(χ 1 + α) + b(χ 2 + β) + g 1 = a χ 1 + b χ 2 + (a α + b β + g 1 ) = a χ 1 + b χ 2 x 2 = χ 2 = c x 1 + d x 2 + g 2 = c(χ 1 + α) + d(χ 2 + β) + g 2 = c χ 1 + d χ 2 + (c α + d β + g 2 ) = c χ 1 + d χ 2 That is, with the new variables χ 1 and χ 2 the given system has become χ 1 = a χ 1 + b χ 2 χ 2 = c χ 1 + d χ 2 Notice it has the form χ = Aχ. That is, it is a homogeneous system (with the critical point at the origin) whose coefficient matrix is exactly A, the same as the original system s. Hence, b could be disregarded, and a determination of the type and stability of the critical point of the system (with or without the translations) could be based solely on the coefficient matrix A Zachary S Tseng D-2-17

18 Example: x 1 = x 1 2 x 2 1 x 2 = 2 x 1 3x 2 3 The critical point is at (3, 1). The matrix A has characteristic equation r 2 + 2r + 1 = 0. It has a repeated eigenvalue r = 1 that has only one linearly independent eigenvector. Therefore, the critical point at (3, 1) is an asymptotically stable improper node. The phase portrait is shown below Zachary S Tseng D-2-18

19 Example: x 1 = 2 x 1 6 x x 2 = 8 x 1 + 4x 2 12 The critical point is at (1, 1). The matrix A has characteristic equation r 2 2r + 40 = 0. It has complex conjugate eigenvalues with positive real part, λ = 1. Therefore, the critical point at (1, 1) is an unstable spiral point. The phase portrait is shown below Zachary S Tseng D-2-19

20 Exercises D-2.1: 1 8 Determine the type and stability or the critical point at (0,0) of each system below x = x. 2. x = 5 10 x x = x. 4. x = 1 4 x x = x. 6. x = 1 1 x x = x. 8. x = 2 3 x (i) For what value(s) of b will the system below have an improper node at (0,0)? (ii) For what value(s) of b will the system below have a spiral point at (0,0)? 5 b x = x Find the critical point of each nonhomogeneous linear system given. Then determine the type and stability of the critical point x = x x = 24 x x = x x = 7 x x = x Zachary S Tseng D-2-20

21 Answers D-2.1: 1. Node, asymptotically stable 2. Center, stable 3. Improper node, unstable 4. Node, unstable 5. Spiral point, asymptotically stable 6. Proper node (or start point), asymptotically stable 7. Saddle point, unstable 8. Improper node, unstable 9. (i) b = 9/2, (ii) b < 9/2 10. ( 2, 4) is a saddle point, unstable. 11. ( 4, 1) is a proper node (star point), unstable. 12. (1, 0.5) is a center, stable. 13. (0, 2) is an improper node, asymptotically stable 14. ( 1, 3) is a spiral point, unstable Zachary S Tseng D-2-21

22 Nonlinear Systems Consider a nonlinear system of differential equations: x = F(x,y) y = G(x,y) Where F and G are functions of two variables: x = x(t) and y = y(t); and such that F and G are not both linear functions of x and y. Unlike a linear system, a nonlinear system could have none, one, two, three, or any number of critical points. Like a linear system, however, the critical points are found by setting x = y = 0, and solve the resulting system 0 = F(x,y) 0 = G(x,y) Any and every solution of this system of algebraic equations is a critical point of the given system of differential equations. Since there might be multiple critical points present on the phase portrait, each trajectory could be influenced by more than one critical point. This results in a much more chaotic appearance of the phase portrait. Consequently, the type and stability of each critical point need to be determined locally (in a small neighborhood on the phase plane around the critical point in question) on a case-by-case basis. Without detailed calculation, we could estimate (meaning, the result is not necessarily 100% accurate) the type and stability by a little bit of multi-variable calculus. We will approximate the behavior of the nearby trajectories using the linearizations (i.e. the tangent approximations) of F and G about each critical point. This converts the nonlinear system into a linear system whose phase portrait approximates the local behavior of the original nonlinear system near the critical point. To wit, start with the lineariztions of F and G (recall that such a linearization is just the 3 lowest order terms in the Taylor series expansion of each function) about the critical point (x, y) = (α, β) Zachary S Tseng D-2-22

23 x = F(x,y) F(α, β) + F x (α, β)(x α) + F y (α, β)(y β) y = G(x,y) G(α, β) + G x (α, β)(x α) + G y (α, β)(y β) But since (α, β) is a critical point, so F(α, β) = 0 = G(α, β), the above linearizations become x F x (α, β)(x α) + F y (α, β)(y β) y G x (α, β)(x α) + G y (α, β)(y β) As before, the critical point could be translated to (0, 0) and still retains its type and stability, using the substitutions χ = x α and γ = y β. After the translation, the approximated system becomes x = F x (α, β) x + F y (α, β) y y = G x (α, β) x + G y (α, β) y It is now a homogeneous linear system with a coefficient matrix A = F G x x ( α, β ) ( α, β ) F G y y ( α, β ) ( α, β ). That is, it is a matrix calculate by plugging in x = α and y = β into the matrix of first partial derivatives J = F G x x F G y y. This matrix of first partial derivatives, J, is often called the Jacobian matrix. It just needs to be calculated once for each nonlinear system. For each critical point of the system, all we need to do is to compute the coefficient matrix of the linearized system about the given critical point (x, y) = (α, β), and then use its eigenvalues to determine the (approximated) type and stability Zachary S Tseng D-2-23

24 Example: x = x y y = x 2 + y 2 2 The critical points are at (1, 1) and ( 1, 1). The Jacobian matrix is J = 1 1 2x 2y. At (1, 1), the linearized system has coefficient matrix: A = The eigenvalues are spiral point. r 3± 7i =. The critical point is an unstable 2 At ( 1, 1), the linearized system has coefficient matrix: A = The eigenvalues are saddle point. r = 1± The critical point is an unstable The phase portrait is shown on the next page Zachary S Tseng D-2-24

25 (1, 1) is an unstable spiral point. ( 1, 1) is an unstable saddle point Zachary S Tseng D-2-25

26 Example: x = x xy y = y + 2 xy The critical points are at (0, 0) and ( 1/2, 1). The Jacobian matrix is J = 1 y 2y x 1+ 2x At (0, 0), the linearized system has coefficient matrix: A = There is a repeated eigenvalue r = 1. A linear system would normally have an unstable proper node (star point) here. But as a nonlinear system it actually has an unstable node. (Didn t I say that this approximation using linearization is not always 100% accurate?) At ( 1/2, 1), the linearized system has coefficient matrix: A = 0 2 1/ 2 0. The eigenvalues are r = 1 and 1. Thus, the critical point is an unstable saddle point. The phase portrait is shown on the next page Zachary S Tseng D-2-26

27 (0, 0) is an unstable node. ( 1/2, 1) is an unstable saddle point Zachary S Tseng D-2-27

28 Exercises D-2.2: Find all the critical point(s) of each nonlinear system given. Then determine the type and stability of each critical point. 1. x = xy + 3y y = xy 3x 2. x = x 2 3xy + 2x y = x + y 1 3. x = x 2 + y 2 13 y = xy 2x 2y x = 2 x 2 y 2 y = x 2 y 2 5. x = x 2 y + 3xy 10y y = xy 4x Answer D-2.2: 1. Critical points are (0, 0) and ( 3, 3). (0, 0) is a stable center, and ( 3, 3) is an unstable saddle point. 2. Critical points are (0, 1) and (1/4, 3/4). (0, 1) is an unstable saddle point, and (1/4, 3/4) is an unstable spiral point. 3. Critical points are (2, 3), (2, 3), (3, 2), and ( 3, 2). (2, 3) is an unstable saddle point, (2, 3) is an unstable saddle point, (3, 2) is an unstable node, and ( 3, 2) is an asymptotically stable node. 4. Critical points are (1, 1), (1, 1), ( 1, 1), and ( 1, 1). (1, 1) is an asymptotically stable spiral point, (1, 1) and ( 1, 1) both are unstable saddle points, and ( 1, 1) is an unstable spiral point. 5. Critical points are (0, 0), (2, 4), and ( 5, 4). (0, 0) is an unstable saddle point, (2, 4) is an unstable node, and ( 5, 4) is an asymptotically stable node Zachary S Tseng D-2-28

### Math 410(510) Notes (4) Qualitative behavior of Linear systems a complete list. Junping Shi

A. Two different positive eigenvalues Math 4(5 Notes (4 Qualitative behavior of Linear systems a complete list Junping Shi Eigenvalues: λ = and λ = 4 Eigenvectors: V = (,, V = (, ( ( General solution:

### Linear systems of ordinary differential equations

Linear systems of ordinary differential equations (This is a draft and preliminary version of the lectures given by Prof. Colin Atkinson FRS on 2st, 22nd and 25th April 2008 at Tecnun Introduction. This

### 2.5 Complex Eigenvalues

1 25 Complex Eigenvalues Real Canonical Form A semisimple matrix with complex conjugate eigenvalues can be diagonalized using the procedure previously described However, the eigenvectors corresponding

### Nonlinear Systems of Ordinary Differential Equations

Differential Equations Massoud Malek Nonlinear Systems of Ordinary Differential Equations Dynamical System. A dynamical system has a state determined by a collection of real numbers, or more generally

### Chapter 8. Nonlinear systems. 8.1 Linearization, critical points, and equilibria

Chapter 8 Nonlinear systems 8. Linearization, critical points, and equilibria Note: lecture, 6. 6. in [EP], 9. 9.3 in [BD] Except for a few brief detours in chapter, we considered mostly linear equations.

### Orbits of the Lennard-Jones Potential

Orbits of the Lennard-Jones Potential Prashanth S. Venkataram July 28, 2012 1 Introduction The Lennard-Jones potential describes weak interactions between neutral atoms and molecules. Unlike the potentials

### The Predator-Prey Equations. x = a x α xy y = c y + γ xy

The Predator-Prey Equations An application of the nonlinear system of differential equations in mathematical biology / ecology: to model the predator-prey relationship of a simple eco-system. Suppose in

### Example 1: Competing Species

Local Linear Analysis of Nonlinear Autonomous DEs Local linear analysis is the process by which we analyze a nonlinear system of differential equations about its equilibrium solutions (also known as critical

### Autonomous Equations / Stability of Equilibrium Solutions. y = f (y).

Autonomous Equations / Stability of Equilibrium Solutions First order autonomous equations, Equilibrium solutions, Stability, Longterm behavior of solutions, direction fields, Population dynamics and logistic

### Dynamics. Figure 1: Dynamics used to generate an exemplar of the letter A. To generate

Dynamics Any physical system, such as neurons or muscles, will not respond instantaneously in time but will have a time-varying response termed the dynamics. The dynamics of neurons are an inevitable constraint

### 3.2 Sources, Sinks, Saddles, and Spirals

3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients

### Eigenvalues, Eigenvectors, and Differential Equations

Eigenvalues, Eigenvectors, and Differential Equations William Cherry April 009 (with a typo correction in November 05) The concepts of eigenvalue and eigenvector occur throughout advanced mathematics They

### 3. Reaction Diffusion Equations Consider the following ODE model for population growth

3. Reaction Diffusion Equations Consider the following ODE model for population growth u t a u t u t, u 0 u 0 where u t denotes the population size at time t, and a u plays the role of the population dependent

### Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.

Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard

### Maximum and Minimum Values

Jim Lambers MAT 280 Spring Semester 2009-10 Lecture 8 Notes These notes correspond to Section 11.7 in Stewart and Section 3.3 in Marsden and Tromba. Maximum and Minimum Values In single-variable calculus,

### 22 Matrix exponent. Equal eigenvalues

22 Matrix exponent. Equal eigenvalues 22. Matrix exponent Consider a first order differential equation of the form y = ay, a R, with the initial condition y) = y. Of course, we know that the solution to

### MVE041 Flervariabelanalys

MVE041 Flervariabelanalys 2015-16 This document contains the learning goals for this course. The goals are organized by subject, with reference to the course textbook Calculus: A Complete Course 8th ed.

### PURE MATHEMATICS AM 27

AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)

### Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

### PURE MATHEMATICS AM 27

AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and

### Second Order Linear Partial Differential Equations. Part I

Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables; - point boundary value problems; Eigenvalues and Eigenfunctions Introduction

### Solutions to Linear Algebra Practice Problems

Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the

### Lecture 12 Basic Lyapunov theory

EE363 Winter 2008-09 Lecture 12 Basic Lyapunov theory stability positive definite functions global Lyapunov stability theorems Lasalle s theorem converse Lyapunov theorems finding Lyapunov functions 12

### Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

### Dynamical Systems: Part 4. 2 Discrete and Continuous Dynamical Systems

Dynamical Systems: Part 4 2 Discrete and Continuous Dynamical Systems 2.1 Linear Discrete and Continuous Dynamical Systems There are two kinds of dynamical systems: discrete time and continuous time. For

### Math 497C Sep 9, Curves and Surfaces Fall 2004, PSU

Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say

### LS.6 Solution Matrices

LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

### 3.7 Non-autonomous linear systems of ODE. General theory

3.7 Non-autonomous linear systems of ODE. General theory Now I will study the ODE in the form ẋ = A(t)x + g(t), x(t) R k, A, g C(I), (3.1) where now the matrix A is time dependent and continuous on some

### 6 Systems of Linear Equations

6 Systems of Linear Equations A system of equations of the form or is called a linear system of equations. x + y = 7 x y = 5p 6q + r = p + q 5r = 7 6p q + r = Definition. An equation involving certain

### Two mass-three spring system. Math 216 Differential Equations. Forces on mass m 1. Forces on mass m 2. Kenneth Harris

Two mass-three spring system Math 6 Differential Equations Kenneth Harris kaharri@umich.edu m, m > 0, two masses k, k, k 3 > 0, spring elasticity t), t), displacement of m, m from equilibrium. Positive

### Matrices in Statics and Mechanics

Matrices in Statics and Mechanics Casey Pearson 3/19/2012 Abstract The goal of this project is to show how linear algebra can be used to solve complex, multi-variable statics problems as well as illustrate

### 3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### Module 3: Second-Order Partial Differential Equations

Module 3: Second-Order Partial Differential Equations In Module 3, we shall discuss some general concepts associated with second-order linear PDEs. These types of PDEs arise in connection with various

### Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

### LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12,

LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12, 2000 45 4 Iterative methods 4.1 What a two year old child can do Suppose we want to find a number x such that cos x = x (in radians). This is a nonlinear

### Solution: APPM 2360 Exam1 Spring 2012

APPM 60 Exam1 Spring 01 Problem 1: (0 points) Consider the following initial value problem where y is the radius of a raindrop and y = y / ; y 0 (1) y(0) = 1, () models the growth of the radius (as water

### C 1 x(t) = e ta C = e C n. 2! A2 + t3

Matrix Exponential Fundamental Matrix Solution Objective: Solve dt A x with an n n constant coefficient matrix A x (t) Here the unknown is the vector function x(t) x n (t) General Solution Formula in Matrix

### Applied Linear Algebra

Applied Linear Algebra OTTO BRETSCHER http://www.prenhall.com/bretscher Chapter 7 Eigenvalues and Eigenvectors Chia-Hui Chang Email: chia@csie.ncu.edu.tw National Central University, Taiwan 7.1 DYNAMICAL

### People s Physics book 3e Ch 25-1

The Big Idea: In most realistic situations forces and accelerations are not fixed quantities but vary with time or displacement. In these situations algebraic formulas cannot do better than approximate

### Section 5.4 (Systems of Linear Differential Equation); Eigenvalues and Eigenvectors

Section 5.4 (Systems of Linear Differential Equation); Eigenvalues and Eigenvectors July 1, 2009 Today s Session Today s Session A Summary of This Session: Today s Session A Summary of This Session: (1)

### Solutions to Linear Algebra Practice Problems 1. form (because the leading 1 in the third row is not to the right of the

Solutions to Linear Algebra Practice Problems. Determine which of the following augmented matrices are in row echelon from, row reduced echelon form or neither. Also determine which variables are free

### Newton s Method on a System of Nonlinear Equations

Newton s Method on a System of Nonlinear Equations Nicolle Eagan, University at Bu alo George Hauser, Brown University Research Advisor: Dr. Timothy Flaherty, Carnegie Mellon University Abstract Newton

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Mechanics Lecture Notes. 1 Notes for lectures 12 and 13: Motion in a circle

Mechanics Lecture Notes Notes for lectures 2 and 3: Motion in a circle. Introduction The important result in this lecture concerns the force required to keep a particle moving on a circular path: if the

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### LECTURE NOTES: MATHEMATICAL EPIDEMIOLOGY

LECTURE NOTES: MATHEMATICAL EPIDEMIOLOGY E M Lungu 1, M Kgosimore 2, and F Nyabadza 3 February 2007 1 Department of Mathematics, University of Botswana, P/Bag UB 00704, Gaborone, E- mail: lunguem@mopipiubbw

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### 3.4. Solving simultaneous linear equations. Introduction. Prerequisites. Learning Outcomes

Solving simultaneous linear equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability

S. Widnall 16.07 Dynamics Fall 2009 Version 1.0 Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability Vibration, Instability An important class of problems in dynamics concerns the free

### 4 Lyapunov Stability Theory

4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### Math 54. Selected Solutions for Week Is u in the plane in R 3 spanned by the columns

Math 5. Selected Solutions for Week 2 Section. (Page 2). Let u = and A = 5 2 6. Is u in the plane in R spanned by the columns of A? (See the figure omitted].) Why or why not? First of all, the plane in

### MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3

MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................

### Higher Order Linear Differential Equations with Constant Coefficients

Higher Order Linear Differential Equations with Constant Coefficients Part I. Homogeneous Equations: Characteristic Roots Objectives: Solve n-th order homogeneous linear equations where a n,, a 1, a 0

### Absolute Maxima and Minima

Absolute Maxima and Minima Definition. A function f is said to have an absolute maximum on an interval I at the point x 0 if it is the largest value of f on that interval; that is if f( x ) f() x for all

### Markov Chains, part I

Markov Chains, part I December 8, 2010 1 Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X

### On the general equation of the second degree

On the general equation of the second degree S Kesavan The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai - 600 113 e-mail:kesh@imscresin Abstract We give a unified treatment of the

### Differentiation of vectors

Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where

### Lecture 31: Second order homogeneous equations II

Lecture 31: Second order homogeneous equations II Nathan Pflueger 21 November 2011 1 Introduction This lecture gives a complete description of all the solutions to any differential equation of the form

### R is a function which is of class C 1. We have already thought about its level sets, sets of the form

Manifolds 1 Chapter 5 Manifolds We are now going to begin our study of calculus on curved spaces. Everything we have done up to this point has been concerned with what one might call the flat Euclidean

### (a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,

Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We

### Equilibria and stability analysis

Equilibria and stability analysis Michael Kopp 11 May 2011 Introduction Models in ecology and evolutionary biology often look at how variables such as population size or allele frequencies change over

### Main page. Given f ( x, y) = c we differentiate with respect to x so that

Further Calculus Implicit differentiation Parametric differentiation Related rates of change Small variations and linear approximations Stationary points Curve sketching - asymptotes Curve sketching the

### System of First Order Differential Equations

CHAPTER System of First Order Differential Equations In this chapter, we will discuss system of first order differential equations. There are many applications that involving find several unknown functions

### Dynamical Systems Analysis II: Evaluating Stability, Eigenvalues

Dynamical Systems Analysis II: Evaluating Stability, Eigenvalues By Peter Woolf pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls Open Textbook version 1.0 Creative

### 1 Error in Euler s Method

1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of

### Vector-Valued Functions The Calculus of Vector-Valued Functions Motion in Space Curvature Tangent and Normal Vect. Calculus. Vector-Valued Functions

Calculus Vector-Valued Functions Outline 1 Vector-Valued Functions 2 The Calculus of Vector-Valued Functions 3 Motion in Space 4 Curvature 5 Tangent and Normal Vectors 6 Parametric Surfaces Vector Valued

### Chapter 1. Vector valued functions

Chapter 1. Vector valued functions This material is covered in Thomas (chapters 12 & 13 in the 11th edition, or chapter 10 in the 10th edition). The approach here will be only slightly different and we

### Fibonacci Numbers Modulo p

Fibonacci Numbers Modulo p Brian Lawrence April 14, 2014 Abstract The Fibonacci numbers are ubiquitous in nature and a basic object of study in number theory. A fundamental question about the Fibonacci

### Understanding Poles and Zeros

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function

### 1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### Advanced Higher Mathematics Course Assessment Specification (C747 77)

Advanced Higher Mathematics Course Assessment Specification (C747 77) Valid from August 2015 This edition: April 2016, version 2.4 This specification may be reproduced in whole or in part for educational

### Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

### Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem

S. Widnall 16.07 Dynamics Fall 008 Version 1.0 Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem The Three-Body Problem In Lecture 15-17, we presented the solution to the two-body

### Lecture 32: Systems of equations

Lecture 32: Systems of equations Nathan Pflueger 23 November 2011 1 Introduction So far in the differential equations unit, we have considered individual differential equations which describe how a single

### MATH 56A SPRING 2008 STOCHASTIC PROCESSES 31

MATH 56A SPRING 2008 STOCHASTIC PROCESSES 3.3. Invariant probability distribution. Definition.4. A probability distribution is a function π : S [0, ] from the set of states S to the closed unit interval

### Chapter 4: Systems of Equations and Ineq. Lecture notes Math 1010

Section 4.1: Systems of Equations Systems of equations A system of equations consists of two or more equations involving two or more variables { ax + by = c dx + ey = f A solution of such a system is an

### Mechanical Vibrations

Mechanical Vibrations A mass m is suspended at the end of a spring, its weight stretches the spring by a length L to reach a static state (the equilibrium position of the system). Let u(t) denote the displacement,

### Numerical Solution of Differential Equations

Numerical Solution of Differential Equations Dr. Alvaro Islas Applications of Calculus I Spring 2008 We live in a world in constant change We live in a world in constant change We live in a world in constant

### One-Dimensional Kinematics

One-Dimensional Kinematics Copyright 2010 Pearson Education, Inc. Position, Distance, and Displacement Average Speed and Velocity Instantaneous Velocity Acceleration Motion with Constant Acceleration Applications

### AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.

AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods

### Eigenvalues and eigenvectors of a matrix

Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a non-zero column vector V such that AV = λv then λ is called an eigenvalue of A and V is

### The Graphical Method: An Example

The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

### Phase Portraits for u = A u

Phase Portraits for u = A u How to Construct a Phase Portrait Badly Threaded Solution Curves Solution Curve Tangent Matching Phase Portrait Illustration Phase plot by computer Revised Computer Phase plot

### Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve. cosh y = x + sin y + cos y

Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve at the point (0, 0) is cosh y = x + sin y + cos y Answer : y = x Justification: The equation of the

### Extra Problems for Midterm 2

Extra Problems for Midterm Sudesh Kalyanswamy Exercise (Surfaces). Find the equation of, and classify, the surface S consisting of all points equidistant from (0,, 0) and (,, ). Solution. Let P (x, y,

### Notes on Linear Recurrence Sequences

Notes on Linear Recurrence Sequences April 8, 005 As far as preparing for the final exam, I only hold you responsible for knowing sections,,, 6 and 7 Definitions and Basic Examples An example of a linear

### Eigenvalues and Markov Chains

Eigenvalues and Markov Chains Will Perkins April 15, 2013 The Metropolis Algorithm Say we want to sample from a different distribution, not necessarily uniform. Can we change the transition rates in such

### 1. (1 pt) Write the given second order equation as its equivalent system of first order equations. u =

1 (1 pt) Write the given second order equation as its equivalent system of first order equations u + u + 5u = 0 Use v to represent the velocity function, ie v = u (t) Use v and u for the two functions,

### Math Department Student Learning Objectives Updated April, 2014

Math Department Student Learning Objectives Updated April, 2014 Institutional Level Outcomes: Victor Valley College has adopted the following institutional outcomes to define the learning that all students

### Elements of Linear Algebra: Q&A

Elements of Linear Algebra: Q&A A matrix is a rectangular array of objects (elements that are numbers, functions, etc.) with its size indicated by the number of rows and columns, i.e., an m n matrix A

### ME128 Computer-Aided Mechanical Design Course Notes Introduction to Design Optimization

ME128 Computer-ided Mechanical Design Course Notes Introduction to Design Optimization 2. OPTIMIZTION Design optimization is rooted as a basic problem for design engineers. It is, of course, a rare situation

### Introduction to models for two interacting species

Lecture 4 Introduction to models for two interacting species Reading: Much of the material here comes from Section 3.1 pages 79 83 of James D. Murray (2002), Mathematical Biology I: An Introduction, 3rd

### A First Course in Elementary Differential Equations. Marcel B. Finan Arkansas Tech University c All Rights Reserved

A First Course in Elementary Differential Equations Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 1 Basic Terminology 4 2 Qualitative Analysis: Direction Field of y = f(t, y)

### Math 2280 Section 002 [SPRING 2013] 1

Math 2280 Section 002 [SPRING 2013] 1 Today well learn about a method for solving systems of differential equations, the method of elimination, that is very similar to the elimination methods we learned

### Bindel, Fall 2012 Matrix Computations (CS 6210) Week 8: Friday, Oct 12

Why eigenvalues? Week 8: Friday, Oct 12 I spend a lot of time thinking about eigenvalue problems. In part, this is because I look for problems that can be solved via eigenvalues. But I might have fewer

### Stability Of Structures: Basic Concepts

23 Stability Of Structures: Basic Concepts ASEN 3112 Lecture 23 Slide 1 Objective This Lecture (1) presents basic concepts & terminology on structural stability (2) describes conceptual procedures for

### r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + 1 = 1 1 θ(t) 1 9.4.4 Write the given system in matrix form x = Ax + f ( ) sin(t) x y 1 0 5 z = dy cos(t)

Solutions HW 9.4.2 Write the given system in matrix form x = Ax + f r (t) = 2r(t) + sin t θ (t) = r(t) θ(t) + We write this as ( ) r (t) θ (t) = ( ) ( ) 2 r(t) θ(t) + ( ) sin(t) 9.4.4 Write the given system