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 Wellposedness  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 oneparameter bifurcations, since they have already seen a few bifurcations, rather than the adhoc examplebyexample 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: \(\dot x=\beta\bigl(y(x^3/3x)\bigr)\), \(\beta\dot y=x\). This makes \(x\) a fast variable except when we're near the slow curve \(y=x^3/3x\), 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 saddlenode bifurcations, or no bifurcation at all, given small changes in a second parameter. Similarly, a pitchfork bifurcation can become a saddlenode 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) 