MAT267H1 Advanced Ordinary Differential Equations

Winter 2019

Preliminary Course Outline:

  1. Introduction
  2. First order ODE's
  3. Systems of first order ODE's
  4. Linear n-th order ODE's
  5. Nonlinear Differential Equations and Stability
  6. To be announced (as time allows)

Readings:

Ideally, you will have had a chance to do the readings (at least at a skim level) before class. After class, you would do the readings at a deeper level. Try all of the exercises. I expect that you're at a level of sophistication where you can recognize when some subset of problems are quite similar to one another. If you're able to do this mental "clumping" then work the problems in order until you feel that you've mastered whatever it is that you think the author is trying to get you to master. If you haven't exhausted the problems, now try the last problem in the clump. If you can do it, then you can check that clump off your list.

Errors in "Differential Equations, Dynamical Systems, and an Introduction to Chaos" by Hirsch, Smale, and Devaney.

Jan 8: What is a differential equation? What is a solution of a differential equation?
Tenenbaum & Pollard: Lesson 1 (optional), Lesson 2 (skim and slow down and read carefully if there's anything that you don't know at a deep level), Lesson 3 (read carefully & do exercises), Lesson 4 (read carefully if you're interested, otherwise skim).

Jan 10: Direction Fields, Separable Equations
Tenenbaum & Pollard: Lessons 5 and 6 (read carefully & do exercises).

Here're some quick notes on non-uniqueness of solutions.

Here're some notes on separable equations, including the brief in-class discussion on one forms.

Jan 15 & Jan 17: Homogenous ODEs, Exact ODEs, Integrating Factors, ODEs of the form y' + p(x) y = q(x)
Tenenbaum & Pollard: Lessons 7, 9, 10, and 11 (read carefully & do exercises).

If you're looking for some extra exercises see this file. I've starred exercises that I consider more interesting. Most of the other exercises should be straightforward if you've done enough of the Tenenbaum & Pollard exercises.

Here're the solutions to the Jan 18 tutorial quizzes.

Jan 22: Osgood's Uniqueness Theorem. Here're the lecture notes. There are so many uniqueness theorems!

Jan 24: The Picard-Lindelof Theorem, proven in various ways. Here're the lecture notes

Here're the
solutions to the Jan 25 tutorial quizzes.

Here're some theory exercises.

Jan 29 & 31: Systems of first-order, constant coefficient, homogeneous linear equations (X' = AX). This is chapters 2 and 3 of the Hirsch, Smale, and Devaney book --- make sure you can do the exercises! For help with plotting things, play around with this plotter, by Darryl Nester. If you select the "system" tab then you can play around with systems of 2 coupled ODEs. This includes both X'=AX as well as nonlinear systems. (For some fun, try the simple pendulum... x' = y, y' = - sin(x) - b y. b=0 corresponds to no friction, b>0 corresponds to friction. When b=0 you should get closed orbits if your initial data's not too large.) When using his plotter for tricky systems, you might want to play around with the four possible time-steppers (Euler, Heun, Midpoint, Fourth-order Runge-Kutta, and Runge-Kutta 3/8 Method).

Here're the solutions to the Feb 1 tutorial quizzes. I underestimated the time you'd need to write the solution up carefully; I'll ask the grading TA to grade gently. Note that the solution includes the "continuation" argument. Make sure you understand it. It's important and I always refer to it casually in class but never proved it for you.

Feb 5: midterm exam.

Feb 7: Phase plane plots for 2d linear systems that have complex eigenvalues. Introduction to exp(A) and exp(tA).
Note: I had a sign error in the matrix V from class. Here're the notes for the SVD decomposition of that 2x2 matrix. In terms of studying you should study chapters 4 and 5 of the Hirsch, Smale, and Devaney book --- make sure you can do the exercises! You don't need to read section 5.6 although it's good to know that generically matrices are diagonalizable.

Here're some notes on how to plot solutions in the phase plane when you have a 2x2 system with a degenerate eigenvalue (i.e. the only eigenvalue is lambda and you can't find a pair of linearly independent eigenvectors).

Here're some notes on how to plot solutions in the phase plane when you have a 2x2 system with a pair of complex eigenvalues. Here's a matlab script you can play with, if you want. And here's a nice write-up of the geometric meaning of the SVD for 2x2 matrices.

Here're the solutions to the Feb 8 tutorial quizzes. Note: the phase-plane plot on page two is not as accurate as one might like --- the trajectories should be asymptoting onto the unstable manifold as t goes to +infinity and they should be asymptoting onto the stable manifold as t goes to -infinity. Here, the "unstable manifold" is the span of the vector [2;3] and the "stable manifold" is the span of the vector [1;2]. (I don't know if we'll get to stable and unstable manifolds for nonlinear systems of ODEs but it's good for you to be exposed to the language.)

Feb 12: I cancelled class because of heavy snow.

Feb 14: Rigorous treatment of exponentiation of matrices: section 6.4 of Hirsch, Smale, and Devaney. The material on exp(A) and exp(tA) in our book is somewhat terse. If you'd like to have an additional source for exp(A) and exp(tA), check out these eight pages from the second edition of Hirsch & Smale's Differential equations, dynamical systems, and linear algebra. It's a great book! As mentioned in class, the norm used in our book doesn't satisfy ||AB|| <= ||A|| ||B||. In class I spoke about different possible norms one might use for matrices. The norm-induced matrix norms do satisfy this submultiplicativity property, making the Banach space of square matrices into a Banach Algebra. (You don't need to know any of this but if you're interested in functional analysis, spectral theory, quantum mechanics, etc you're going to see this stuff again in your fourth-year courses or in grad school.)

Feb 19 & 21 : Reading week!

Feb 26: Section 6.5 of Hirsch, Smale, and Devaney. We will be studying nonhomogenous systems X' - AX = B and X' - AX = B(t). Note: the variation of parameters method that we use for X' - AX = B(t) generalizes to nonhomogenous linear evolution equations as Duhamel's Principle.
Here are homework problems based on the Feb 26 lecture. If you're going to hand them in, please do so by 5pm on Saturday March 2.

Feb 26: Unforced linear oscillators. We will be studying oscillators, both forced and unforced, with and without damping. Section 6.2 of Hirsch, Smale, and Devaney has some material on linear oscillators but they don't really consider the effect of forcing. Chapter 6 of Tenenbaum & Pollard is all about oscillators but it's more than we have time for, sadly. So here's some material on unforced linear oscillators. Here's a video that shows you how a spring constant is measured. Three weights are chosen; we know the force on the spring due to the weight. The weight stretches the spring --- the stretching of the spring causes a force in the opposite direction. Once the mass is at rest, the two forces are equal to one another, and you can calculate the spring force. For small weights the force is approximately linear in the deviation: gravitational force = k times the displacement length. From the video you can compute that k = 49.

March 5: Forced linear oscillators. Here's some additional material on forced linear oscillators.
Here's a video which shows a mass hanging on a spring, being forced in a periodic manner. It's not exciting but you can see that when the forcing is below the natural frequency of the spring, the mass' motion is in phase with the forcing and when the forcing's above the natural frequency the motion is out of phase with the forcing.
Here are three resonance videos. And here's a nice one; it's cued up to a key point but the whole thing's neat, if you have the time.

Here's a nice audio demo of beats And here's some interesting stuff on acoustics and tuning and the like..

Do you know someone who'd like to learn some physics in a friendly, bite-sized manner? Send them to the Professor Julius Sumner Miller playlist.

Warning: the following is all physics/PDE, not ODE, so skip reading it if you're so inclined. In reality, a solid object has infinitely many natural frequencies, not just one. These frequencies depend on various things including the object's shape and material and how it's being held (boundary conditions). If you force the object at any one of these natural frequencies it will respond strongly. Here's an example of a square Chladni plate; there's a speaker under the plate playing a sound at one frequency. As the frequency is changed, the configuration of the sand changes. Where the sand accumulates is where plate is not moving up and down. As the frequency increases, the area of the regions that are moving up and down gets smaller. Here's a visualization of the first few modes of a circular drum head; you can write them down in terms of friendly functions and here's an experiment demonstrating them. You can write the modes down in terms of sines, cosines, and Bessel functions (yay! solving an ODE using a power series and naming the solution because it's so important and useful!). A very natural question is: given that each object has certain natural frequencies; can you reverse engineer this? If someone gave you a list of all the natural frequencies, could you predict its shape? Can one hear the shape of a drum? This was (and continues to be) an interesting area of study in PDEs. UofT math professor Victor Ivrii has done work in this area. Here's the classic video of the Tacoma Narrows Bridge being driven at one of its natural frequencies. Here's some more on that bridge collapse. If you got this far, here's a cute music video that plays around with natural frequencies in various physical systems.

Here is the second homework assignment. If you're going to hand it in, please do so by 5pm on Tuesday March 12. You should have gotten a link from crowdmark.

The midterm on March 12 will cover the material up to and including forced oscillators. It will focus on the January 29 - March 5 material: linear systems, phase plane plots, exp(tA), and oscillators. No SVDs will need to be computed. Here are some comments about things that you should be comfortable with for the exam. It's not exhaustive and I'll add to it. Note that existence and uniqueness stuff is still relevant --- you just need to understand how it works for systems in general and how it applies to specific systems. (Where "in specific" would mean specific systems like those mentioned in the study notes.) You should review the quizzes, the homework assignments, and the lecture material as well. There are extra problems to work in Hirsch, Smale, and Devaney, in Tennenbaum and Pollard, and in the scanned pages that I provided on oscillators.

March 7: For the rest of the course, we'll be relying nearly completely on Hirsch, Smale, and Devaney's book. We will start studying nonlinear systems, using the language and viewpoint of dynamical systems. We'll start with section 7.1 (introduction to language).

Here is the third homework assignment. If you're going to hand it in, please do so by 11:59pm on Tuesday March 19. You should have gotten a link from crowdmark.

March 12: midterm exam.

March 14: Existence and Uniqueness theorem: continuation of solutions forward in time and backward in time. Lyapunov functions as a method of ``trapping'' solutions in compact sets, allowing for continuation in time.

Here's a Mark Calculator. It's an excel spreadsheet. Don't have excel but do have a gmail account? If you download it and then go to google sheets and press the folder icon to the far right then you should be able to upload the sheet and use it.

March 19 and 21: Section 7.3 (continuous dependence of solutions on initial data) and section 7.4 (the variational equation). Here are some notes on the continuous dependence of solutions on initial data.

Here is the fourth homework assignment. If you're going to hand it in, please do so by 11:59pm on Tuesday March 26. You should have gotten a link from crowdmark.

March 26: Please work through section 8.1 before class and bring any questions you have to class. In class, we'll cover section 8.2.

Here is the fifth homework assignment. If you're going to hand it in, please do so by 11:59pm on Tuesday April 2. You should have gotten a link from crowdmark.

March 28: We'll start with a quick finishing up of 8.2 and will move on to section 8.3. Please review the example on pages 160-161 before class; it's a demonstration of the stable curve theorem.

April 2: We'll finish section 8.3. Please review the section before class. You're responsible for reading and understanding section 8.4 on your own. Please ask questions if you have any.

Here is the sixth homework assignment. If you're going to hand it in, please do so by 11:59pm on Tuesday April 9. You should have gotten a link from crowdmark.

April 4: Last day of class --- bifurcation theory! Section 8.5. Please review the section before class. The last example in class was a bit hurried. Here're notes on that example.

Preparing for the final exam: The final exam is on Tuesday April 16, 2-5pm in GB 405. I'll be holding my regular office hours until then. I'll have office hours on Monday April 8 (11-12, 1-2), Wednesday April 10 (1-2), and Monday April 15 (11-12, 1-2). NB: the other course I'm teaching has its final exam on the exact same day and time as this one so don't be surprised if there are Calculus students vying with you for my attention.

April 9: Your essay is due by 11:59pm on Tuesday April 9.