# TMA4165 Differential Equations and Dynamical Systems - Spring 2019

## Lectures

• The plan is filled in as we go
• “J&S” refers to the textbook (Jordan & Smith)
• “Notes” refers to my notes (see under Course information)
• The black diamond ◆ is intended to mark the upcoming lecture(s)
Week Day Topic(s) Sections Remarks
2 Mon Introduction J&S 1.1–1.3
Tue Uniqueness, local existence Notes Ch 1 to the middle of p 5
3 Mon Katrin Grunert lectured
Well-posedness
Notes Ch 1 p 5
Tue Continuous dependence, …, Flow Notes Ch 1 pp 6–9
Linear 2×2 systems J&S 2.4–2.6
4 Mon Linear 2×2 systems J&S 2.4–2.6, 10.9
Tue
5 Mon Equilibria for nonlinear 2×2 systems Notes Ch 2
J&S 10.8
But first, a look at Example 2.10 (p. 74 in J&S).
Tue Equilibira … cont'd, then Hamiltonian systems Notes Ch 2, J&S 2.8
6 Plane autonomous systems J&S 3.1
Bendixson's index formula Notes Ch 5
7 Mon Closed paths, Bendixson's negative criterion, limit cycles, heteroclinic & homoclinic orbits
Introduction to Poincaré–Bendixson
J&S 3.4
Tue The Poincaré–Bendixson theorem
Liénard equations
J&S Ch 11.1, Notes Ch 6
8 Mon Centres and limit cycles
Especially for Liénard equations
J&S 11.2–3
Tue Stability J&S 8.1–3
9 Mon Stability continued, linear systems J&S 8.4–7, Notes Ch 4
Tue Linearisation at equilibrium points Notes Ch 3, J&S 8.8, 10.10 Corrections: While proving “Grönwall light”, I defined $V(t)=\exp\int_0^t v(\tau)\,d\tau$. The $\exp$ should not have been there.
Also, in Hartman's theorem, the vector field must be $C^2$, not $C^1$ as I wrote.
Addendum: A very brief note on a differential inequality.
10 Mon Stability continued J&S 8.11–11, Thm 8, 9 of the notes
Tue No lecture I was busy with the Abel competition
11 Mon Liapunov methods J&S Ch 10 In my opinion, the book overcomplicates things with its talk of topographic systems. I will aim to keep it much simpler. Also, note that we have already covered parts of Ch 10. (I had neglected to make a note of that; corrected now.)
Tue Liapunov methods
Bifurcations
J&S Ch 10
J&S 12.1, 12.3
12 Mon Bifurcations J&S 12.3, 12.4 Peter Pang lectured. In his words:
We reached into the later part of Chp 12 to mention to centre manifold theorem so that we can have a more systematic development of the transversality conditions that lead to different one-parameter bifurcations, since they have already seen a few bifurcations, rather than the ad-hoc example-by-example way the book went.
We saw how symmetry was the thing that gave us transversality conditions which led to the pitchfork bifurcation. We then worked out the pitchfork bifurcation example. Next we mentioned Hopf's theorem (12.1) without proof, and worked out an example of Hopf bifurcation in which we pointed out that the radial coordinate the Hopf bifurcation would have had a pichfork bifurcation if $r$ could be $<0$.
Tue Various topics J&S 3.1, 13.1, 11.4 Revisit of index theory, looking at two examples (not in the book) in the complex plane $\dot z=z^n$ (index of the origin is $n$) and $\dot z=\bar z^n$ (index is $-n$).
Poincaré sequences.
van der Pol with large parameter. I analysed it as a slightly different system than the book: $\dot x=\beta\bigl(y-(x^3/3-x)\bigr)$, $\beta\dot y=-x$. This makes $x$ a fast variable except when we're near the slow curve $y=x^3/3-x$, and $y$ a slow variable. It is easy to get a rough idea of the nature of the resulting relaxation oscillation; harder to get at the details, which we skipped.
13 I got a bit philosophical, and talked a bit about generic properties. Meaning properties that are true on an open and dense subset of whatever is the natural set for the sort of problems being discussed. For example, a generic matrix is invertible: This is just a more convenient way of saying that invertibility is a generic property of matrices. And generic dynamical system has only hyperbolic equilibria (no eigenvalues with zero real part). Also, all eigenvalues are distinct in the generic situation. This does not make the exceptions uninteresting – on the contrary!
I mentioned that a fixed point at which the matrix of the linearization is invertible, will only move and not disappear or bifurcate under small perturbation of the system (a consequence of the implicit function theorem).
On the Hopf theoreim: Theorem 12.1 in the book is wrong as stated. But if the hypothesis $f(r)>0$ for $r>0$ is replaced by $f'(r)>0$, it is correct.
I gave examples showing that a transcritical bifurcation can becomeS two saddle-node bifurcations, or no bifurcation at all, given small changes in a second parameter. Similarly, a pitchfork bifurcation can become a saddle-node bifurcation (and one equilibrium that does not bifurcate) under a small change. Thus, generic systems with a parameter will not have such bifurcations – it usually takes symmetry or other special constraints to force them into being.
I ended with the fold and the cusp from §12.2, and the declared us done.
14 Summary
15 Last week of lectures Worked through last years' exams (all of the spring exam, and a big part of the August exam)