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

Save this PDF as:
Size: px
Start display at page:

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

## Transcription

1 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 of this system based on the vector field diagram and linear stability analysis about the fixed points (0, 0) and (1, 1). We determined that (0, 0) behaved locally like a saddle point and were able to get a sense of the long-term behavior of solutions by considering how the nullclines partitioned the state space. There were still, however, several important questions which we were not able to answer. In particular, linearization about the second fixed point (1, 1) yielded the strictly complex eigenvalues λ 1,2 = ±i. Since the real part of the eigenvalues was zero, we were not able to apply the Hartman- Grobman Theorem to guarantee that the local picture (a center) extended to the nonlinear system. In fact, it is still possible for the point to locally behaved like a spiral source, a spiral sink, or a center. We would like to determine which is the case. To address this, we take a slight detour. We will develop a more comprehensive theory as this week progresses, but for now let s just jump to the punchline and consider the function H(x, y) = x3 3 (1) + xy y2 2. (2) An important first observation to make about (2) is that, just like potentials for one-dimensional systems, the function H above assigns to each point (x, y) in the state space a numerical value. For one-dimensional systems, this was not a tremendous simplification of the problem, since our state space was one-dimensional to begin with. For two-dimensional (and higher!) systems, however, this will turn out to be a significant advantage. We perform the same analysis we did with one-dimensional potentials. That is, we ask the question of how the value of H changes along solutions 1

2 x(t) = (x(t), y(t)) of (1). To compute this, we carefully apply the chain rule to get dh(x(t)) dt = dh dx dx dt + dh dy dy dt = ( x 2 + y)(x y) + (x y)(x 2 y) = 0. This computation tells us that the value of H(x, y) does not change along solutions x(t) of (1). This is quite remarkable! For instance, supposing we start our trajectory at a state x(0) which has the value H(x(0)) = 5, we know that the trajectory may only ever pass through points also corresponding to the value of 5. We will never encounter a time where H(x(t)) = 4, or H(x(t)) = 6, or even H(x(t)) = To make the mathematical consequences of this more clear, we integrate with respect to t to obtain dh(x(t)) dt = (0) dt = H(x(t)) = C (3) dt where C is some constant. We should recognize (3) from multivariate calculus (e.g. Math 234) as it corresponds to the equation for a level curve of the multivariate function H. A collection of level curves for various values of C is called a contour plot. We have determined is that every solution x(t) of (2) remains on the level curve corresponding to the initial condition x(0) = x 0. For this example, those level curves are difficult to obtain analytically, but we can easily plot them with the aid of a computer (see Figure 1). We make the following notes: 1. We will say that any nontrivial function H(x, y) which has the property (3) along solutions of a system is a conservation function, and that the corresponding system as a conservative system. 2. We can determine that (1, 1) is a local maximum of H(x, y) by computing the Hessian at (1, 1): 2 H x 2 2 H x y 2 H x y 2 H y 2 x=1,y=1 = [ 2x ] x=1,y=1 = [ ]. 2

3 Figure 1: The level curves of the function H(x, y) = x 3 /3 + xy y 2 /2 (solid lines) trap the trajectories of the system (1). To test for a maximum or minimum we check [ 2 H 2 ( H 2 )] x 2 y 2 H = ( 2)( 1) (1) 2 = 1 > 0 x y x=1,y=1 and 2 H = 2 < 0. x2 It follows that(1, 1) is a local maximum of H(x, y) by the secondderivative test for multivariate functions. This is important since it guarantees that the system behaves like a center near (1, 1). 3. Although solutions of (1) are restricted to the level curves of H(x, y), we do not know which direction they go along these curves. For this information, we need to consider the original vector field. 4. If we are careful, we can divide the long-term behavior of solutions in the following way: (a) There is a closed loop which begins at (0, 0) (from the right), wraps around (1, 1), and then ends at (0, 0) (from the top). This type of unusual curve is called a homoclinic orbit. (b) Inside the homoclinic orbit, all solutions form closed curves. That is, every solution is periodic (in a nonlinear way). 3

4 (c) Outside the homoclinic orbit, almost all solutions drift off to infinity in the second quadrant. The lone exception is the stable manifold in quadrant three which divides those curves which divert to the left of (0, 0) and those which divert to the right. (This fragile curve actually converges to (0, 0).) This is a very interesting result, but we have only considered one example. We have no idea how robust/common the phenomenon of having a conserved quantity is or how we might systematically search for one. As we will see in the next few sections, this phenomenon is surprisingly robust, and there are several hints that a system can given us as to whether there is such a function and, if so, what it is. 1 State-Dependent Mechanical Systems We do not have to look very far in order to determine a classification of systems which necessarily conservative. Reconsider the model for a physical system obeying Newton s second law Force = mass acceleration. In general, the force function may depend upon x, x, and t, so that we have the system of equations mx = F (t, x, x ) (4) where m is the mass of the physical object being modeled. Systems of the form (4) do not always admit a conservation function; however, a very broad subclass of such systems does. Suppose we can write mx = F (x). (5) We will call such systems state-dependent mechanical systems, since the force function depends only upon the state x. A simple example of a state-dependent mechanical system which we have already considered is the pendulum/spring system with a restoring force kx but no friction. This gave us the second-order system mx = kx = mx + kx = 0. This system is linear and can be solved explicitly to get the periodic solution x(t) = C 1 sin( k/m t) + C 2 cos( k/m t). We will consider a more physically realistic pendulum model shortly. 4

5 Now consider the following argument. In analogy with potentials for one-dimensional systems, we rewrite (5) in the form mx = dv dx (6) where V (x) is an unknown function. That is, we take F (x) = V (x). We now apply a trick which will seem very unusual, but will produce the desired result almost directly. We rearrange (5) and multiply the whole expression by x. This gives mx x + x dv dx = 0. This trick should not feel intuitive, but the key feature is that we can actually integrate this expression. Using the chain rule in reverse, we realize that we have d [ m dt 2 (x ) 2 + V (x)] = 0. In other words, if we define E(x, x ) = m 2 (x ) 2 +V (x), we have that the value of E(x, x ) does not change along trajectories of (5). So E(x, x ) is exactly the conservation function we have been looking for! The function E(x, x ) should be familiar to those well-versed in physics. The term m 2 (x ) 2 corresponds to the kinetic energy of the system while the term V (x) corresponds to the potential energy of the system. This result tells us that, for state-dependent mechanical systems, any increase in the kinetic, or potential, energy must be offset by a corresponding decrease in the other. In other words, the total energy E is conserved. Example 1: Determine the total energy function E for the undamped pendulum/spring model mx = kx. (7) Solution: The argument above holds for all systems of the form (5), so we do not need to go through it all over again. It suffices to determine the potential energy function V (x). We have that V (x) = kx = V (x) = k x2 2. It follows that the total energy function is given by E(x, x ) = m 2 (x ) 2 + k 2 x2. 5

6 Figure 2: The level curves of the function E(x, x ) = m 2 (x ) 2 + k 2 x2 trap the trajectories of the system (7). The system above is pictured with m = k = 1. If we look at the level sets of this function in the (x, x )-plane (which is the same as making the substitutions x 1 = x, x 2 = x ) we can see that trajectories must move along ovals (see Figure 2). This should not come as a surprise. This system is linear, so we could transform it into a system and see that the corresponding coefficient matrix has strictly complex eigenvalues. This guarantees that the system is a center. If were are unconvinced by argument, we can check explicitly that d dt E(x, x ) = d dt [ m 2 (x ) 2 + k 2 x2 = mx x + kx x = x [mx + kx] = 0 by the original differential equation. It follows that, as expected, the total energy of the system is conserved. Example 2: Determine the total energy function E for the statedependent mechanical system ] x = x x 3. (8) Solution: As before, because the system is state-dependent only, it suffices to find the potential energy V (x). We have V (x) = F (x) dx = (x 3 x) dx = x4 4 x2 2. 6

7 It follows that the required energy function is E(x, x ) = (x ) x4 4 x2 2. The level curves of E, which correspond to solutions of the original system, are shown in the (x, x )-plane in Figure 3. Figure 3: The level curves of the function E(x, x ) = (x ) 2 x x4 4 trap the trajectories of the system (8). The fixed points at ( 1, 0) and (1, 0) are nonlinear centers while the fixed point at (0, 0) is a nonlinear saddle. Notice the homoclinic orbits which extend from (0, 0), wrap around (1, 0) or ( 1, 0), and then return to (0, 0). 2 Nonlinear Pendulum We noted at the beginning of the course that the model mx = kx (9) for an undamped pendulum was at best an approximation. It captures the idea that if the pendulum is positioned to the right of its center, the force will be to the left, and that if it is positioned to the left, the force will be to the right. It does not, however, capture the dynamical behavior far from the center. For instance, this model does not permit the pendulum to swing over the top as we would expect of a realistic physical model. In order to accommodate this more exotic behavior associated with pendulums, we need to consider the nonlinear model mθ = k sin(θ) (10) 7

8 where θ R is the angle the pendulum makes with the vertical axis (that is, θ = 0 corresponds to a pendulum hanging straight down). This relationship can be derived by considering the relevant forces. For our purposes, we will just be interested in the observation that, when θ 0, this system behaves like the linear system because sin(θ) θ for small θ. For large θ, however, this model has behavior not characterized by the linear model (9). The nonlinear system (10) is certainly more challenging to analyze than the linear system (9). The methods of Math 319 and 320 do not apply to (10) in fact, there is no way to determine the explicit solution. Instead we will rely on the qualitative methods we have been developing. First of all, we rewrite this in the first-order system form dx 1 dt = x 2 dx 2 dt = k m sin(x 1). (11) where x 1 (t) = θ(t) and x 2 (t) = θ (t). We now consider the vector field and linear stability analysis. We start by identifying the nullclines and fixed points of the system. We have x 1 = 0 = x 2 = 0 and x 2 = 0 = k m sin(x 1) = 0 = x 1 = nπ, n Z. It follows that we have the fixed points ( x 1, x 2 ) = (nπ, 0), n Z. Accounting for vector flow in the bounded regions, we arrive at the general picture contained in Figure 4. We now consider linear stability analysis. We have so that and [ Df(x 1, x 2 ) = Df(2nπ, 0) = 0 1 k m cos(x 1) 0 [ 0 1 k m 0 [ ] 0 1 Df((2n + 1)π, 0) = ] k m 0 where, for both cases, we take n Z. The eigenvalues of Df(2nπ, 0) are λ = ±i which does not yield any information as a result of linearization. The 8 ]

9 x 2 '=0 x 2 '=0 x 2 '=0 x 1 '=0 Figure 4: Phase portrait in the (x 1, x 2 )-plane (i.e. (θ, θ )-plane) of the system (11) with k/m = 1. The nullclines x 1 = 0 (red) and x 2 = 0 (blue) are overlain. eigenvalue/eigenvectors pairs for Df((2k+1)π, 0), however, are λ 1 = k/m, v 1 = (1, k/m), and λ = k/m, v 2 = (1, k/m). We therefore have saddles at each point ((2n+1)π, 0) with the unstable manifold corresponding to a positively sloped curve through the point, and a stable manifold corresponding to a negatively sloped curve. This is consistent with our picture in Figure 4. A number of things are now clear. Solutions with high values of x 2 (i.e. high velocity) travel to the right, while solutions with low values (i.e. high negative velocity) travel to the left. These cases correspond to pendulums which spin over the top in one direction or the other. For solutions with small x 2 (i.e. low velocity), trajectories swirl from the left to the right of a given fixed point. This corresponds to pendulums wobbling about their equilibrium position. There is still, however, something mysterious about the fixed point ( x 1, x 2 ) = ((2n + 1)π, 0). Because linear stability analysis failed, we have not determined whether these fixed points correspond to spiral sources, spiral sinks, or a nonlinear center. All three are consistent with the rough picture contained in Figure 4. To answer this question, we consider the previous discussion of energy functions and conserved quantities. While the extension to the nonlinear system (10) has changed many properties of the system, and destroyed our ability to solve it explicitly, it is still a state-dependent mechanical system since the force function depends only upon our state variable θ. All of our previous analysis applies! We need only find the potential function V (x), 9

10 which is given by V (x) = k sin(θ) dθ = k cos(θ). We therefore have that the value of the energy function E(θ, θ ) = m 2 (θ ) 2 k cos(θ) (12) is conserved along solutions of (10). We can check this explicitly, if we are unconvinced. We have de dt = mθ θ + k sin(θ)θ = θ [ mθ + k sin(θ) ] = 0. The level curves of (12) is once again not something we plot by hand, but our computers will happily do it for us (see Figure 5). We can now see that the previously unclassified fixed points are nonlinear centers. For low amplitudes, trajectories just oscillate around these fixed points, which corresponds to the pendulum swinging back and forth like a grandfather clock. Figure 5: Level curves of the energy function E(x, x ) = m 2 (x ) 2 cos(x) overlain on the vector field plot of (11) reverted back to the (x, x )-plane. 3 Hamiltonian Systems As nice as state-dependent mechanical systems are, they do not capture every system for which there is a conservation relation. For example, we determined earlier that the function H(x, y) = x xy y2 2 was conserved

11 along solutions of the system dx dt = x y dy dt = x2 y. This system is not a state-dependent mechanical system, so how was the conserved quantity determined? There is another very general class of systems which guarantees the existence of a conserved quantity (which can be thought of as an energy, or just as a general function, depending on your preferred point of view). They are called Hamiltonian systems. As with conservative systems, the basis for Hamiltonian systems comes from conserved quantities in physics and, in particular, Hamiltonian mechanics. We will not be interested in delving too deeply into those waters. We have the following. Definition 3.1. Consider a two-dimensional autonomous system of differential equations. We will say the system is Hamiltonian if there exists a function H(x, y) such that dx dt = H y dy dt = H x The function H(x, y) is called a Hamiltonian function. (13) Theorem 3.1. The Hamiltonian function is conserved along solutions x(t) of a Hamiltonian system. Proof. This can be computed directly by the chain rule. We have dh(x(t)) dt = H dx x = H x dt + H y ( H y dy dt ) + H y ( H ) = 0. x What is even better is that, for two-dimensional systems, it turns out to be very easy to check whether a system is Hamiltonian or not. We have the following result. 11

12 Theorem 3.2. A two-dimensional system is Hamiltonian if and only if f 1 x + f 2 y = 0. Proof. We will start by rewriting the general nonlinear system in the form (13). That is, we consider dx dt = f 1(x, y) = H y dy dt = f 2(x, y) = H x. This is a notational association, since we do not know that such a function H(x, y) exists. We now consider equality of mixed-order partial derivatives which must hold if a (sufficiently smooth) function H(x, y) exists: 2 H x y = 2 H y x. This can be rearranged to give ( ) H = ( ) H x y y x x f 1(x, y) = y f 2(x, y). After rearranging we have the desired condition, and we are done. We can now verify that our previous example is a Hamiltonian system, and also determine the Hamiltonian function H(x, y). We have f 1 (x, y) = x y and f 2 (x, y) = x 2 y so that f 1 x + f 2 y = x (x y) + y (x2 y) = 1 + ( 1) = 0. It follows by Theorem 3.2 that the system is Hamiltonian, but it remains to determine the Hamiltonian function H(x, y). To accomplish this, we need to solve the system H y = f 1(x, y) = x y H x = f 2(x, y) = x 2 y. There are a number of ways to accomplish this. The most straight-forward is generally to integrate one expression and then taking the corresponding 12

13 partial derivative required to match it with the second. The trick will be to be careful with the variable dependencies. We compute H(x, y) = H(x, y) dy = y (x y) dy = xy y2 2 + g(x) (14) where g(x) is a general unknown function which depends on x alone. Taking the partial derivative of (14) gives H x = y + g (x). Comparing this expression with the earlier expression gives g (x) = x 2 = g(x) = x3 3. Substituting this into the earlier form for H(x, y) gives H(x, y) = x3 3 + xy y2 2 as expected, and we are done. Example: Show that the following system is Hamiltonian and determine the Hamiltonian function: dx dt = ln( y ) dy dt = x2 1. (15) Solution: We apply the test of Theorem 3.2. We have df 1 dx + df 2 dy = d dx [ln( y )] + d dy [x2 1] = (0) + (0) = 0. Although this should seem trivial, it is still sufficient to guarantee that the system is Hamiltonian. To determine the Hamiltonian function H(x, y), we consider the system of equations H y = ln( y ) H x = x2 1. (16) 13

14 To illustrate the flexibility allowed at this step, we integrate the second equation to get H(x, y) = (1 x 2 ) dx = x x3 3 + g(y). To match this with the remaining equation in (16) we differentiate with respect to y. We have H y = g (y) = ln( y ) = g(y) = y ln( y ) y. It follows that the Hamiltonian function is H(x, y) = x x3 3 + y ln( y ) y. (17) As nice as having this conserved quantity is, it is really only the beginning of the story. We would like to use this information to say something about the dynamical behavior, but the function H(x, y) is too complicated to imagine graphing by hand. Instead, we go to our favorite computer graphic software package, and obtain the contour plot contained in Figure 6. We can now see very clearly that the fixed points (1, 1) and ( 1, 1) are saddles and the fixed points (1, 1) and ( 1, 1) are nonlinear centers. Each center has a clearly demarcated region of influence, outside of which solutions seem to slingshot around the fixed points. If you look closely, you can see one particularly interesting solution which is whipped around both centers before eventually being thrown off in the first quadrant (although we might wonder what really happens when it passes through y = 0 since this is not strictly allowed by (15). Figure 6: Level curves of the Hamiltonian function H(x, y) (17) overlain on the vector field plot of (15). 14

### MATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation

MATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation 1 Bifurcations in Multiple Dimensions When we were considering one-dimensional systems, we saw that subtle changes in parameter

### MATH 353 LECTURE NOTES: WEEK 1 FIRST ORDER ODES

MATH 353 LECTURE NOTES: WEEK 1 FIRST ORDER ODES J. WONG (FALL 2017) What did we cover this week? Basic definitions: DEs, linear operators, homogeneous (linear) ODEs. Solution techniques for some classes

### 1 The pendulum equation

Math 270 Honors ODE I Fall, 2008 Class notes # 5 A longer than usual homework assignment is at the end. The pendulum equation We now come to a particularly important example, the equation for an oscillating

### 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

### MATH 320, WEEK 2: Slope Fields, Uniqueness of Solutions, Initial Value Problems, Separable Equations

MATH 320, WEEK 2: Slope Fields, Uniqueness of Solutions, Initial Value Problems, Separable Equations 1 Slope Fields We have seen what a differential equation is (relationship with a function and its derivatives),

### 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),

### 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

### MATH 116, LECTURE 13, 14 & 15: Derivatives

MATH 116, LECTURE 13, 14 & 15: Derivatives 1 Formal Definition of the Derivative We have seen plenty of limits so far, but very few applications. In particular, we have seen very few functions for which

### Why are Discrete Maps Sufficient?

Why are Discrete Maps Sufficient? Why do dynamical systems specialists study maps of the form x n+ 1 = f ( xn), (time is discrete) when much of the world around us evolves continuously, and is thus well

### 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

### MATH 319, WEEK 2: Initial Value Problems, Existence/Uniqueness, First-Order Linear DEs

MATH 319, WEEK 2: Initial Value Problems, Existence/Uniqueness, First-Order Linear DEs 1 Initial-Value Problems We have seen that differential equations can, in general, given rise to multiple solutions.

### 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

### 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

### MATH 415, WEEKS 14 & 15: 1 Recurrence Relations / Difference Equations

MATH 415, WEEKS 14 & 15: Recurrence Relations / Difference Equations 1 Recurrence Relations / Difference Equations In many applications, the systems are updated in discrete jumps rather than continuous

### Math 232, Final Test, 20 March 2007

Math 232, Final Test, 20 March 2007 Name: Instructions. Do any five of the first six questions, and any five of the last six questions. Please do your best, and show all appropriate details in your solutions.

### 3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:

3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable

### 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

### 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

### LECTURE 8: DYNAMICAL SYSTEMS 7

15-382 COLLECTIVE INTELLIGENCE S18 LECTURE 8: DYNAMICAL SYSTEMS 7 INSTRUCTOR: GIANNI A. DI CARO GEOMETRIES IN THE PHASE SPACE Damped pendulum One cp in the region between two separatrix Separatrix Basin

### 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

### Nonlinear Oscillators: Free Response

20 Nonlinear Oscillators: Free Response Tools Used in Lab 20 Pendulums To the Instructor: This lab is just an introduction to the nonlinear phase portraits, but the connection between phase portraits and

### Lecture 6, September 1, 2017

Engineering Mathematics Fall 07 Lecture 6, September, 07 Escape Velocity Suppose we have a planet (or any large near to spherical heavenly body) of radius R and acceleration of gravity at the surface of

### 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

### September Math Course: First Order Derivative

September Math Course: First Order Derivative Arina Nikandrova Functions Function y = f (x), where x is either be a scalar or a vector of several variables (x,..., x n ), can be thought of as a rule which

### 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 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

### 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

### Predator - Prey Model Trajectories and the nonlinear conservation law

Predator - Prey Model Trajectories and the nonlinear conservation law James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 28, 2013 Outline

### The Liapunov Method for Determining Stability (DRAFT)

44 The Liapunov Method for Determining Stability (DRAFT) 44.1 The Liapunov Method, Naively Developed In the last chapter, we discussed describing trajectories of a 2 2 autonomous system x = F(x) as level

### 4.9 Anti-derivatives. Definition. An anti-derivative of a function f is a function F such that F (x) = f (x) for all x.

4.9 Anti-derivatives Anti-differentiation is exactly what it sounds like: the opposite of differentiation. That is, given a function f, can we find a function F whose derivative is f. Definition. An anti-derivative

### 8.3 Partial Fraction Decomposition

8.3 partial fraction decomposition 575 8.3 Partial Fraction Decomposition Rational functions (polynomials divided by polynomials) and their integrals play important roles in mathematics and applications,

### Figure 1: Doing work on a block by pushing it across the floor.

Work Let s imagine I have a block which I m pushing across the floor, shown in Figure 1. If I m moving the block at constant velocity, then I know that I have to apply a force to compensate the effects

### Mass on a Horizontal Spring

Course- B.Sc. Applied Physical Science (Computer Science) Year- IInd, Sem- IVth Subject Physics Paper- XIVth, Electromagnetic Theory Lecture No. 22, Simple Harmonic Motion Introduction Hello friends in

### AP Calculus (BC) Chapter 10 Test No Calculator Section. Name: Date: Period:

AP Calculus (BC) Chapter 10 Test No Calculator Section Name: Date: Period: Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The graph in the xy-plane represented

### Updated 2013 (Mathematica Version) M1.1. Lab M1: The Simple Pendulum

Updated 2013 (Mathematica Version) M1.1 Introduction. Lab M1: The Simple Pendulum The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are

### MATH 415, WEEK 12 & 13: Higher-Dimensional Systems, Lorenz Equations, Chaotic Behavior

MATH 415, WEEK 1 & 13: Higher-Dimensional Systems, Lorenz Equations, Chaotic Behavior 1 Higher-Dimensional Systems Consider the following system of differential equations: dx = x y dt dy dt = xy y dz dt

### Learning Objectives for Math 165

Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

### 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

### Daba Meshesha Gusu and O.Chandra Sekhara Reddy 1

International Journal of Basic and Applied Sciences Vol. 4. No. 1 2015. Pp.22-27 Copyright by CRDEEP. All Rights Reserved. Full Length Research Paper Solutions of Non Linear Ordinary Differential Equations

### Compound Damped Pendulum: An Example

Compound Damped Pendulum: An Example Temple H. Fay Department of Mathematical Technology 1 Tshwane University of Technology Pretoria, South Africa thf ay@hotmail:com Abstract: In this article, we use an

### Math 234 Exam 3 Review Sheet

Math 234 Exam 3 Review Sheet Jim Brunner LIST OF TOPIS TO KNOW Vector Fields lairaut s Theorem & onservative Vector Fields url Divergence Area & Volume Integrals Using oordinate Transforms hanging the

### Computers, Lies and the Fishing Season

1/47 Computers, Lies and the Fishing Season Liz Arnold May 21, 23 Introduction Computers, lies and the fishing season takes a look at computer software programs. As mathematicians, we depend on computers

### 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

### MATH 320, WEEK 4: Exact Differential Equations, Applications

MATH 320, WEEK 4: Exact Differential Equations, Applications 1 Exact Differential Equations We saw that the trick for first-order differential equations was to recognize the general property that the product

### Question: Total. Points:

MATH 308 May 23, 2011 Final Exam Name: ID: Question: 1 2 3 4 5 6 7 8 9 Total Points: 0 20 20 20 20 20 20 20 20 160 Score: There are 9 problems on 9 pages in this exam (not counting the cover sheet). Make

### Copyright (c) 2006 Warren Weckesser

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

### CDS 101 Precourse Phase Plane Analysis and Stability

CDS 101 Precourse Phase Plane Analysis and Stability Melvin Leok Control and Dynamical Systems California Institute of Technology Pasadena, CA, 26 September, 2002. mleok@cds.caltech.edu http://www.cds.caltech.edu/

### Lecture 10. (2) Functions of two variables. Partial derivatives. Dan Nichols February 27, 2018

Lecture 10 Partial derivatives Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 University of Massachusetts February 27, 2018 Last time: functions of two variables f(x, y) x and y are the independent

### MCE693/793: Analysis and Control of Nonlinear Systems

MCE693/793: Analysis and Control of Nonlinear Systems Lyapunov Stability - I Hanz Richter Mechanical Engineering Department Cleveland State University Definition of Stability - Lyapunov Sense Lyapunov

### 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,

### Half of Final Exam Name: Practice Problems October 28, 2014

Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half

### Parametric Equations, Function Composition and the Chain Rule: A Worksheet

Parametric Equations, Function Composition and the Chain Rule: A Worksheet Prof.Rebecca Goldin Oct. 8, 003 1 Parametric Equations We have seen that the graph of a function f(x) of one variable consists

### Midterm 1 Review. Distance = (x 1 x 0 ) 2 + (y 1 y 0 ) 2.

Midterm 1 Review Comments about the midterm The midterm will consist of five questions and will test on material from the first seven lectures the material given below. No calculus either single variable

### Physics 351, Spring 2017, Homework #4. Due at start of class, Friday, February 10, 2017

Physics 351, Spring 2017, Homework #4. Due at start of class, Friday, February 10, 2017 Course info is at positron.hep.upenn.edu/p351 When you finish this homework, remember to visit the feedback page

### Physics 141 Energy 1 Page 1. Energy 1

Physics 4 Energy Page Energy What I tell you three times is true. Lewis Carroll The interplay of mathematics and physics The mathematization of physics in ancient times is attributed to the Pythagoreans,

### 4 Partial Differentiation

4 Partial Differentiation Many equations in engineering, physics and mathematics tie together more than two variables. For example Ohm s Law (V = IR) and the equation for an ideal gas, PV = nrt, which

### Math 212-Lecture 8. The chain rule with one independent variable

Math 212-Lecture 8 137: The multivariable chain rule The chain rule with one independent variable w = f(x, y) If the particle is moving along a curve x = x(t), y = y(t), then the values that the particle

### 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.

### MATH 521, WEEK 2: Rational and Real Numbers, Ordered Sets, Countable Sets

MATH 521, WEEK 2: Rational and Real Numbers, Ordered Sets, Countable Sets 1 Rational and Real Numbers Recall that a number is rational if it can be written in the form a/b where a, b Z and b 0, and a number

### FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS III: Autonomous Planar Systems David Levermore Department of Mathematics University of Maryland

FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS III: Autonomous Planar Systems David Levermore Department of Mathematics University of Maryland 4 May 2012 Because the presentation of this material

### Slope Fields: Graphing Solutions Without the Solutions

8 Slope Fields: Graphing Solutions Without the Solutions Up to now, our efforts have been directed mainly towards finding formulas or equations describing solutions to given differential equations. Then,

### Systems of Linear ODEs

P a g e 1 Systems of Linear ODEs Systems of ordinary differential equations can be solved in much the same way as discrete dynamical systems if the differential equations are linear. We will focus here

### Autonomous Systems and Stability

LECTURE 8 Autonomous Systems and Stability An autonomous system is a system of ordinary differential equations of the form 1 1 ( 1 ) 2 2 ( 1 ). ( 1 ) or, in vector notation, x 0 F (x) That is to say, an

### 28. 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.

### Complex Differentials and the Stokes, Goursat and Cauchy Theorems

Complex Differentials and the Stokes, Goursat and Cauchy Theorems Benjamin McKay June 21, 2001 1 Stokes theorem Theorem 1 (Stokes) f(x, y) dx + g(x, y) dy = U ( g y f ) dx dy x where U is a region of the

### Chapter 4. Oscillatory Motion. 4.1 The Important Stuff Simple Harmonic Motion

Chapter 4 Oscillatory Motion 4.1 The Important Stuff 4.1.1 Simple Harmonic Motion In this chapter we consider systems which have a motion which repeats itself in time, that is, it is periodic. In particular

### Infinite series, improper integrals, and Taylor series

Chapter 2 Infinite series, improper integrals, and Taylor series 2. Introduction to series In studying calculus, we have explored a variety of functions. Among the most basic are polynomials, i.e. functions

### Calculus Review Session. Brian Prest Duke University Nicholas School of the Environment August 18, 2017

Calculus Review Session Brian Prest Duke University Nicholas School of the Environment August 18, 2017 Topics to be covered 1. Functions and Continuity 2. Solving Systems of Equations 3. Derivatives (one

### Limit. Chapter Introduction

Chapter 9 Limit Limit is the foundation of calculus that it is so useful to understand more complicating chapters of calculus. Besides, Mathematics has black hole scenarios (dividing by zero, going to

### Substitutions and by Parts, Area Between Curves. Goals: The Method of Substitution Areas Integration by Parts

Week #7: Substitutions and by Parts, Area Between Curves Goals: The Method of Substitution Areas Integration by Parts 1 Week 7 The Indefinite Integral The Fundamental Theorem of Calculus, b a f(x) dx =

### Topic 5 Notes Jeremy Orloff. 5 Homogeneous, linear, constant coefficient differential equations

Topic 5 Notes Jeremy Orloff 5 Homogeneous, linear, constant coefficient differential equations 5.1 Goals 1. Be able to solve homogeneous constant coefficient linear differential equations using the method

### Basic Theory of Dynamical Systems

1 Basic Theory of Dynamical Systems Page 1 1.1 Introduction and Basic Examples Dynamical systems is concerned with both quantitative and qualitative properties of evolution equations, which are often ordinary

### VANDERBILT UNIVERSITY. MATH 2300 MULTIVARIABLE CALCULUS Practice Test 1 Solutions

VANDERBILT UNIVERSITY MATH 2300 MULTIVARIABLE CALCULUS Practice Test 1 Solutions Directions. This practice test should be used as a study guide, illustrating the concepts that will be emphasized in the

### McGill University April Calculus 3. Tuesday April 29, 2014 Solutions

McGill University April 4 Faculty of Science Final Examination Calculus 3 Math Tuesday April 9, 4 Solutions Problem (6 points) Let r(t) = (t, cos t, sin t). i. Find the velocity r (t) and the acceleration

### FINAL EXAM MATH303 Theory of Ordinary Differential Equations. Spring dx dt = x + 3y dy dt = x y.

FINAL EXAM MATH0 Theory of Ordinary Differential Equations There are 5 problems on 2 pages. Spring 2009. 25 points Consider the linear plane autonomous system x + y x y. Find a fundamental matrix of the

### SYDE 112, LECTURE 7: Integration by Parts

SYDE 112, LECTURE 7: Integration by Parts 1 Integration By Parts Consider trying to take the integral of xe x dx. We could try to find a substitution but would quickly grow frustrated there is no substitution

### 2t t dt.. So the distance is (t2 +6) 3/2

Math 8, Solutions to Review for the Final Exam Question : The distance is 5 t t + dt To work that out, integrate by parts with u t +, so that t dt du The integral is t t + dt u du u 3/ (t +) 3/ So the

### Physics 132 3/31/17. March 31, 2017 Physics 132 Prof. E. F. Redish Theme Music: Benny Goodman. Swing, Swing, Swing. Cartoon: Bill Watterson

March 31, 2017 Physics 132 Prof. E. F. Redish Theme Music: Benny Goodman Swing, Swing, Swing Cartoon: Bill Watterson Calvin & Hobbes 1 Outline The makeup exam Recap: the math of the harmonic oscillator

### 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

### Differential Equations Handout A

Differential Equations Handout A Math 1b November 11, 2005 These are problems for the differential equations unit of the course, some topics of which are not covered in the main textbook. Some of these

### Calculus I Review Solutions

Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.

### AIMS Exercise Set # 1

AIMS Exercise Set #. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest

### 3. On the grid below, sketch and label graphs of the following functions: y = sin x, y = cos x, and y = sin(x π/2). π/2 π 3π/2 2π 5π/2

AP Physics C Calculus C.1 Name Trigonometric Functions 1. Consider the right triangle to the right. In terms of a, b, and c, write the expressions for the following: c a sin θ = cos θ = tan θ =. Using

### Stability of Nonlinear Systems An Introduction

Stability of Nonlinear Systems An Introduction Michael Baldea Department of Chemical Engineering The University of Texas at Austin April 3, 2012 The Concept of Stability Consider the generic nonlinear

### for changing independent variables. Most simply for a function f(x) the Legendre transformation f(x) B(s) takes the form B(s) = xs f(x) with s = df

Physics 106a, Caltech 1 November, 2018 Lecture 10: Hamiltonian Mechanics I The Hamiltonian In the Hamiltonian formulation of dynamics each second order ODE given by the Euler- Lagrange equation in terms

### Relevant sections from AMATH 351 Course Notes (Wainwright): 1.3 Relevant sections from AMATH 351 Course Notes (Poulin and Ingalls): 1.1.

Lecture 8 Qualitative Behaviour of Solutions to ODEs Relevant sections from AMATH 351 Course Notes (Wainwright): 1.3 Relevant sections from AMATH 351 Course Notes (Poulin and Ingalls): 1.1.1 The last few

### 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

### Integrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61

Integrals D. DeTurck University of Pennsylvania January 1, 2018 D. DeTurck Math 104 002 2018A: Integrals 1 / 61 Integrals Start with dx this means a little bit of x or a little change in x If we add up

### Differential Equations

Electricity and Magnetism I (P331) M. R. Shepherd October 14, 2008 Differential Equations The purpose of this note is to provide some supplementary background on differential equations. The problems discussed

### Math 2930 Worksheet Final Exam Review

Math 293 Worksheet Final Exam Review Week 14 November 3th, 217 Question 1. (* Solve the initial value problem y y = 2xe x, y( = 1 Question 2. (* Consider the differential equation: y = y y 3. (a Find the

### Sin, Cos and All That

Sin, Cos and All That James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 9, 2017 Outline 1 Sin, Cos and all that! 2 A New Power Rule 3

### MATH 320, WEEK 6: Linear Systems, Gaussian Elimination, Coefficient Matrices

MATH 320, WEEK 6: Linear Systems, Gaussian Elimination, Coefficient Matrices We will now switch gears and focus on a branch of mathematics known as linear algebra. There are a few notes worth making before

### Antiderivatives and Indefinite Integrals

Antiderivatives and Indefinite Integrals MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Fall 2018 Objectives After completing this lesson we will be able to use the definition

### Exam 2 Study Guide: MATH 2080: Summer I 2016

Exam Study Guide: MATH 080: Summer I 016 Dr. Peterson June 7 016 First Order Problems Solve the following IVP s by inspection (i.e. guessing). Sketch a careful graph of each solution. (a) u u; u(0) 0.

### Chapter 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

### Coordinate systems and vectors in three spatial dimensions

PHYS2796 Introduction to Modern Physics (Spring 2015) Notes on Mathematics Prerequisites Jim Napolitano, Department of Physics, Temple University January 7, 2015 This is a brief summary of material on

### Higher Mathematics Course Notes

Higher Mathematics Course Notes Equation of a Line (i) Collinearity: (ii) Gradient: If points are collinear then they lie on the same straight line. i.e. to show that A, B and C are collinear, show that

### Maps and differential equations

Maps and differential equations Marc R. Roussel November 8, 2005 Maps are algebraic rules for computing the next state of dynamical systems in discrete time. Differential equations and maps have a number