TMA4165 Differential equations and dynamical systems 2009: Lectures

Week 3
Monday (2009-01-12): Opening lecture, to give some idea what it is all about. Then I start with Ch 1 from my notes, on well-posedness for ODEs. Without well-posedness almost nothing from the rest of the course would make any sense, but apart from that it will not play a very central role.
Thursday (2009-01-15): Various examples from Ch 1 of the book. Then the uniqueness part of Ch 1 of my notes.
Week 4
Monday (2009-01-19): Finished Ch 1 of my notes, with the local existence theorem, the maximal interval of existence, and continuous dependence on data.
Thursday (2009-01-22): From Ch 2 of the book: Linearization and the phase plane for linear 2×2 systems. I got as far as classifying and drawing phase planes for the cases with two distinct, nonzero real eigenvalues.
Week 5: More on linearization and phase plane analysis. Roughly, the rest of Ch 2 from the book plus parts of Ch 2 from my notes.
Monday (2009-01-26): Treated the case of non-real eigenvalues and explained the bifurcation diagram that is Figure 2.10 page 70 in the book.
Thursday (2009-01-29): On linearization at an equilibrium in general, and then onto the two-dimensional cases. Basically chapter 2 of my notes to the middle of page 12. I also gave the definitions of what it means for an equilibrium point to be stable or asymptotically stable (note that an asymptotically stable eqilibrium point is stable by definition, which means you have to check it). I also proved (in too much detail?) that these notions are invariant under linear coordinate changes.
Week 6: Finished chapter 2 of my notes, then move forward in ch 2 of the book.
Monday (2009-02-02): From chapter 2 of my notes: Saddles and foci.
Thursday (2009-02-05): Finished chapter 2 of the book, with Hamiltonian systems. Note that the book (only the 4th edition) has a bad misprint in the very definition of Hamiltonian system: There is a minus sign missing in the formula for Y (2.70) on page 75! (That minus sign makes a big difference.) Another misprint: the arrows in Figure 2.13 are reversed. I just barely got started on Chapter 3, with index theory.
Week 7: Mostly index theory.
Monday (2009-02-09): Index theory, from Ch 3 of the book. I based the theory on that of homotopy rather than just using Green's theorem as in the book, but tried to keep it intuitive and not overly rigorous. In particular, my version of the book's Theorem 3.1 says that two closed paths have the same index if they are homotopic via paths that never come across a zero of the vector field. I found the index of any node, focus or center (+1) as well as for centers (−1).
Thursday (2009-02-12): More index theory. I opened by outlining a more solid proof of the fact that the index of a zero of a vector field is +1 if the linearization has positive determinant and −1 if the linearization has negative determinant (based on the continuous deformation of the vector field into one of a few normal forms). I did Example 3.4 from the book, explaining how this could be done by counting the number of times the vector field (X,Y) crosses any given axis (taking account of direction) in the XY plane. Much preferable, I think, to Theorem 3.3. I showed by example that any integer can be the index of a point. In the end I mentioned, for the second or third time, I think, Theorem 3.2 stating that the index of a closed curve is the sum of the index of each point inside it.
Week 8: More index theory, limit cycles, etc.
Monday (2009-02-16): Chapter 5 of my notes, on Bendixson's index formula. The index and phase portrait at infinity. I relied on the stereographic projection to make sense of the vector field around infinity and work out its index. I showed a short video explaing the Möbius transform to help the intuition a little bit. I ended up noting that we cannot comb the hair of sphere, also known as the hairy ball theorem.
Thursday (2009-02-19): A bit on limit cycles and closed paths, and a bit from Section 13.1 on the Poincaré map. (We shall see how far I get.) Not yet sure where I'll go next. Into Ch 8, or possibly start a digression on fractals and chaos.
Week 9: The flow, the straightening theorem, Poincaré maps, Poincaré–Bendixsson theorem.
Monday (2009-02-23): The Poincaré map (13.1 in the book). But first I want to tell you about the flow of a dynamical system, and also the straightening theorem for vector fields.

The flow for a dynamical system is the function Φ(t,x) (more frequently written as Φt(x)) defined as the value at time t of the solution with initial data x at t=0. In fact, it is common to turn this inside out and define the flow to be the dynamical system.

The straightening theorem says that every nonequlibrium point has a neighbourhood on which you can choose new coordinates (u,v) so that the dynamical system becomes the trivial (or “straight”) u̇=1, v̇=0. (that should be u-dot and v-dot: Not all web browser display the dot.)

Thursday (2009-02-26): The Poincaré–Bendixsson theorem (for which I followed my notes).
Week 10: Chapter 11; Liénard equations, including the famous van der Pol equation.
Week 11: Fractals
Monday (2009-03-09): Iterated function systems (IFSs) as a means of generating fractal sets. The Hausdorff metric.
Thursday (2009-03-12): Fractal dimension.
Week 12: Chaos
Monday (2009-03-16): I spent the first half tidying up loose ends regarding fractal dimension. I found the fractal dimension of the harmonic sequence (not series) to be 1/2.
Most of the second half I spent talking about the Verhulst system and period doubling towards chaos. The Maple worksheet I used is available, also as an HTML page. The picture I showed, in such a ridiculously large resolution, is also available.
Thursday (2009-03-19): What is chaos? Really? We looked at two simple chaotic systems. I couldn't resist saying a few word about the Smale horseshoe, based on two short articles in the May 2005 issue of the AMS Notices.
In the time left over, I mentioned the definitions of Poincaré and Liapunov stability (from the start of Ch. 8 in the book) and illustraded with two examples (circular pendulum and standard saddle).
Week 13: We continue in Ch 8.
Monday (2009-03-23): Poincaré and Liapunov stability. Linear systems with variable coefficients. Sections 8.5–8.6 in the book make the easy too difficult, so I lectured from Chapter 4 in my notes instead.
Thursday (2009-03-26): More on linear systems: Section 8.7 and 8.8 of the book. A linear, homogeneous system with constant coefficients is stable if and only if all solutions are bounded, also if and only if all eigenvalues have real parts less than or equal to zero, and those with real parts equal to zero are associated only with trivial (i.e., 1×1) Jordan blocks. I barely got started on introducing Chapter 3 of my notes.
Week 14: Most of the students were in Japan (日本). Hence, no lectures this week.
Week 15 is Easter week. Obviously no lectures in week 15.
Week 16:
Thursday (2009-04-16): Sections 8.10–8.11 in the book. But I used proof techniques from chapter 3 of my notes (see Proposition 7, plus the new chapter 8 in the revised version).
Week 17:
Monday (2009-04-20): Finished chapter 3 of my notes, except the Hartman–Grobman theorem. That is, stability at equlibrium points in general.
Thursday (2009-04-23): The Hartman–Grobman theorem (end of ch 3 in my notes). Then I get started on Liapunov functions (ch 10 of the book).
Week 18: Finish Ch 10 of the book (Liapunov theory), and the first two sections of Ch 12 (a taste of bifurcation theory).
Monday (2009-04-27): There is less work remaining to do in Ch 10 than even I realized:
10.1 is just an introduction. 10.2 introduces topographical systems, which I will mostly ignore (just remember the pictures), and a wrong statement of the Poincaré–Bendixson theorem (see Ch 6 of my notes for a better one). 10.3–10.4 do the theory in two dimensions using topographical systems. I ignore that and use instead the discussion in 10.6. The instability result in 10.7 is a weaker version of Lemma 8 in Ch 3 of my notes. 10.8 and 10.10 are well covered by Ch 3 of my notes, and 1.9 by Ch 4.

What remains is 10.5 (asymptotic stability from a weak Liapunov function). On second thought I'll skip 10.11.

Thursday (2009-04-30): I had forgotten about sections in Ch 12. So I took a look at 12.1–12.4.
Week 19: Just the Monday lecture, making up for the second day of Easter.
Monday (2009-05-04): I went through the solution for the June 2008 exam. (In preparing this, I had found some flaws in the posted solution on the net. This is now updated.)

Harald Hanche-Olsen Last update: 2009–05–27 12:24