Here I will try to keep an updated account of what has been
done so far in the course.
date |
what we did |
comments |
19/1 |
- introduction to the course
- the definition of dynamical systems, T:ch 6
- examples of discrete dynamical systems: iterated maps
- what are important questions about dynamical systems?
- solutions of ode:s as dynamical systems
- rewriting non autonomous, higher order, differential
equations as autonomous systems
- solving first order dynamical scalar ode:s: examples of
non uniqueness and exploding solutions
- the initial value problem rewritten as an integral
equations
- most of the material is from T: ch1.
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- About the literature: the notes by Teschl contains most of
the material covered in the course, but the part of dynamical
systems is a little different from the book by Arrowsmith and
Place. I recommend using both.
- The computer assignment will soon be available. For
those of you who are not very familiar with matlab, I will hand
out a set of notes, a "Crash course".
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22/1 |
- definition of a Banach space (as a complete normed
vector space)
- the contraction mapping principle
- application to the initial value problem of an ode. We
almost finished the proof of that.
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- About the exercises for the 26/1: you are not required
to participate for passing the course, but it counts when the
grade is decided.
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26/1 |
- finished the Picard-Lindelöv theorem
- Gronwall's lemma
- Continuous dependence on initial data
- in the exercises we did 1.11, 1.14, 1.19, 2.3, 2.5, 2.7
from Teschl
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- There is a misprint in the computer assignment, an error
in the definition of the function H . Please look at
the corrected version.
- I had used an older version of Teschl's lecture notes,
and the numbering of the exercises doesn't agree with the up to
date version. From now on I have the new version.
-
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29/1 |
- We proved extension of solutions, and global existence
of solutions to odes when the right-hand side does not grow
faster than linearly; we also discussed differentiability with
respect to initial data.
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2/2 |
- We did most of T 3.1 about the matrix exponential: the
matrix operator norm, convergence of the Taylor series for the
exponential.
- We discussed the Jordan decomposition of matrices, and
how this can be used for computing the matrix exponential in a
practical way.
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- We spent some extra time on problem T1.29, which gives
a good example on how one can find qualitative results on the
solutions to an equation without actually solving it.
- Ehsan Harati has found a nice
matlab routine, dfield7.m that can
be used for visualizing the solutions to ODEs. It is very
nice. However, for the computer exercise I prefer if you use the
more elementary methods that are built into Matlab.
- We saw a proof of a Gronwall inequality. As so often
happens, there are many paths to the same result, and I have put
a different version in a paper with solutions to some
exercises.
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5/2 |
- there is no lecture, it will be done later
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9/2 |
- Only two hours. We finished the part about Jordan
forms, invariant subspaces (from the extra set of notes), showed
the phase planes for linear systems in two dimensions.
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- N.B. Because of the Charm activities, we could not do
the second part of the lecture.
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12/2 |
- Theorem about stable and unstable subspaces: that if
x_0 belongs to the stable subspace, x(t) converges to zero, etc.
- Definition of stable and asymptotically stable systems
- Linear, non autonomous systems: the principal matrix
solution, fundamental matrix solutions, the Wronskian
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16/2 |
- In the lectures we completed the proof of the formula
for the Wronskian, and the formula for solving dx/dt =
A(t)x +h(x) .
- We then looked at differential equations as dynamical
systems, and in particular a phase portrait of a
two-dimensional system (see the figure ), and defined stationary points, domain of
attraction, integral curves, separatrices. In the figure, there
are three fixed points (one stable, and two unstable). For the
stable one, the domain of attraction is in red color.
- An important question from the break: we proved earlier
that the solutions to an ode depends continuously on the initial
data. How does this agree with the fact that a very small change
of initial data can move you from one side of a separatrix to
another. But trajectories starting sufficiently close to a
separatrix remain close to the separatrix for a finite time
interval, so there is no contradiction.
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- In the exercises we looked at 3.1, 3.9, 3.11, 3.13.
In problem 3.11 we were asked to determine whether some functions
could be the solutions to first-order autonomous homogeneous
systems, and were presented several solutions that neither I nor
the author of the book had expected: if we allow non-linear
first-order autonomous systems, there may be many different
solutions also when there is no linear system that works.
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17/2 |
- We proved Lemma 6.2 (how to straighten out vector
fields), and discussed some implications
- For theorem 6.1 we discussed that it really means: what
is the definition of W , how one should think of
it. The proof that W is a very useful exercise to try
to understand, but I did not have time to present it, and it
will not be examined.
- Definitions of orbits, completeness (in the sens that
the trajectory starting at a point x is defined for
all times; there are many other uses of the word complete in
Mathematics), periodic orbits, invariant sets. One example of an
invariant set is γ(x).
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- I will write a short note with some facts about
topology and C1-maps: something about open,
closed and compact sets, definitions of continuous functions,
the inverse function theorem etc. It will take a couple of days.
- I asked that everyone thing of which is the most
difficult exercise they have tried to do, and then to tell me. I
will then try to discuss some of these in class.
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19/2 |
- We worked our way through ch. 6.3 in Teschl
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- The small note on
topology that I have promised is now available
- In the computer exercise you are asked to find the
domain of attraction (at least approximately) for the
attractive fixed points of the given dynamical system. You
should look at the phase portrait that we discussed in
class. Here is a proper definition.
The domain of attraction to
a fixed point x is the set
{ y such that lim t → ∞
Φt(y) = x }. So to find the points y
in the domain of attraction, you should start at a large
number of points y in M , take y
as initial data to the differential equation, and solve it
for so long time that you are convinced whether the solution
converges to x or not. If it does, keep it in a
list, and then plot all those points in some suitable color.
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23/2 |
- Problem 6.9 on closedness and invariance of
non-wandering sets
- Liapunov functions and how they serve to prove
stability of fixed points.
- in the exercise part, we worked with problem 6.2,6.3,
6.5 and 6.8
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- One comment on topology: in my lectures I have always
assumed that a neighborhood of a point is an open set containing
the point. As was pointed out to me, the common definition is that
a neighborhood of a point x that contains an open set
that contains the point. The important fact is that a point is
in the interior of any of its neighborhoods. But in for example
the book by Simmons, a neighborhood is defined as an open set
that contains the point (the notion that I have used).
- My explanation of the proof of Lemma 6.9 was not very
clear, and I will try to repair that tomorrow.
- A comment on problem 6.2: the notion of flow is
important, and this exercise is a very good one for
understanding what flow means.
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24/2 |
- We repeated the proof of Lemma 6.9, and then started on
ch 7.1 with linear systems again. The definitions of stable and
unstable subspaces (manifolds)
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- We did some exercises: 2.7, 6.6, 3.28. The first two
focus on the definition of a flow.
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26/2-9/3 |
- We have finished the course: quite a lot about the
Hartman-Grobmann theorem, and also some about maps,
homeomorphisms and diffeomorphisms. In particular we looked at
maps on the circle, and the definition of lifts from
S2 to R1. Parts of my handwritten notes deal with
this. Otherwise it can be found in the book by Arrowsmith and Place.
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Last modified: Tue Mar 09 16:33:54 +0100 2010