MAT267H1 Advanced Ordinary Differential Equations
Winter 2019
Preliminary Course Outline:
- Introduction
- Types of differential equations
- Direction fields for first order ODE's, isoclines
- First order ODE's
- Separable ODE's
- Homogeneous ODE's
- Linear first order ODE's: Integrating factor
- ODE's corresponding to exact differential 1-forms
- Some other types of explicitly solvable ODE's
- Theory for first-order ODEs: Peano's Existence Theorem, Osgoode's Uniqueness Theorem, Picard Iternations (the usual existence and uniqueness theorem).
- Systems of first order ODE's
- Existence and uniqueness
- Constant coefficient systems
- Complex eigenvalues, repeated eigenvalues
- Fundamental matrices
- Inhomogeneous systems
- Linear n-th order ODE's
- Reduction to first-order ODE's
- Existence and uniqueness theorem
- Fundamental systems of solutions, Wronskians
- Constant coefficients
- Complex roots
- Repeated roots, reduction of order
- Inhomogeneous equations: Undetermined coefficients,
variation of parameters
- Applications: Oscillations, vibrations
- Nonlinear Differential Equations and Stability
- Phase portraits
- Autonomous systems and stability
- Linearizations
- Applications
- 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.