I have now posted model answers for problem sheet 8. Please note that questions 7 and 9 are not relevant for the exam (I updated the exam preparation note). I added a boundary condition to simplify question 6 somewhat.

## Tuesday, 29 March 2016

## Tuesday, 22 March 2016

### Notes on synchronisation

My last lecture was loosely based on some lecture notes by Dr Tiago Pereira. This lecture connects my first lecture (where I mainly motivated the study of differential equations) to the material covered in this course. None of this material is examinable, but I hope it is interesting in context.

### Problem class tuesday 22/3, 12-1pm, in Blacket 1004

Room 340 is unfortunately not available. Blackett 1004 is on Level 10 of the Blackett building.

Come out of the lifts and turn left. Prof Turaev will discuss starred questions from the final problem sheet 8.

## Sunday, 20 March 2016

## Thursday, 17 March 2016

### Final problem sheet nr 8 about calculus of variations.

This problem sheet has now been posted in the left-hand side margin.

## Thursday, 10 March 2016

### Update on solutions to problem sheet 4

I corrected a typo in the solution to question 1(a) and reformulated the solution to question 5(a).

## Wednesday, 9 March 2016

### Additional office hour on wednesday 9 March 12-2pm in room 144

Questions will be answered about the material for the class test of tomorrow.

## Tuesday, 8 March 2016

### About the duration of the second test

You will have 1.5 hours to complete the test on thursday (16:00-17:30). I set the test such that this should normally be plenty of time.

## Monday, 7 March 2016

### Proof of Lemma 4.1.1 (and video)

Some of you have been querying me about the proof of Lemma 4.1.1. I wrote in the guidance for the second test that it is important to study this proof. Of course, there will never be a question on the test (or exam) like "prove lemma 4.1.1". First of all it is too long, but secondly I do not believe in remembering such proofs by heart. However, the manipulations and arguments used are very instructive of similar but even more involved estimates of this kind one encounters in the analysis of ordinary and partial differential equations in various contexts, so although I do not want you to learn this proof by heart, I would really appreciate if you understand it when you read it.... I posted a video in the left-hand side margin where I go a bit slower through the proof than in the lecture notes or in the lecture, and I hope it helps you if you have been struggling with some of the steps in the proof.

## Sunday, 6 March 2016

### Problem sheet 6, model answer posted

With apologies for the delay, please find the model answers for problem sheet 6 (except for question 7, I hope to complete this asap) in the left-hand side margin.

## Saturday, 5 March 2016

### Question 4 sheet nr 6

This question is more involved than I originally anticipated, so I am stripping it of the *, as it is not in the category "elementary". This question is also no longer relevant for the test (and in the processes I added a more questions of this sheet not being essential study for the test, see the updated post below). I also added some more detail to this question on the problem sheet 6, and switched the parts (a) and (b), since it is most natural to answer these questions in the opposite order.

### Poincare-Bendixson appendix

I attached a short note extending the Poincare-Bendixson theory including connecting orbits between equilibria, as I presented in the lecture on 23/2.

## Tuesday, 1 March 2016

### How to prepare for the class test of 10 March?

The material to study for the class test is Chapter 3 - Linear autonomous ODEs

Chapter 4 -The flow near an equilibrium and Chapter 5 - Poincare-Bendixson Theorem from the lecture notes, and the problems sheets nr 3, 4 and 5 relating to these chapters.

I now summarize in some detail the main points in this material:

Chapter 3

definition of linearity

solution of autonomous linear ODEs in terms of exponential of matrix (including relevant proofs)

existence and uniqueness of solutions of autonomous linear ODEs

flow map and its computation in elementary examples in the two-and three-dimensional case

geometric interpretation of explicit formulas for the flow map: invariant subspaces, eigenspaces and generalised eigenspaces, projections and the decoupling principle for linear ODEs

Jordan normal form (general result, but not general proof, and ability to determine the jordan normal form in some simple examples); generalised eigenspaces

Jordan Chevalley decomposition; definition, implications for exp(A) and finding the J-C decomposition in some elementary examples.

Lyapunov and asymptotic stability. Application to linear systems; role of eigenvalues and determination of (in)stability based on information about eigenvalues.

Lyapunov functions; proofs and elementary applications

Chapter 4

Linear approximation near an equilibrium point

Lemma 4.1.1, including interpretation (what does the lemma establish?) and proof (with relevant components, like gronwell estimate and variations of constant formula

Theorem 4.1.2 and proof of part (i)

Hartman-Grobman Theorem may be skipped

Hyperbolic equilibria: proposition 4.2.2 and corollary 4.2.3 (inclusive of proof)

prop 4.2.5 & prop 4.2.6 incl proofs

stable and unstable manifolds (only definitions and main results but no proofs - as not given)

simple examples of bifurcations (and use of implicit function theorem in this context)

Chapter 5 (Hirsch, Smale, Devaney chap 10)

limit sets (definitions and identification of limit sets in simple examples) (10.1)

local sections and flow box (10.2)

monotone sequences (10.4)

Poincare-Bendixson Theorem (10.5 and 10.6) + additional note on classification of omega-limit sets for planar ODEs (as discussed in lecture)

Exercise sheets to be studied:

Problem sheet 4 (but nr 6 not important for test).

Problem sheet 5 (but nrs 1 and 6 not important for test).

Problem sheet 6 (but nrs 4, 5 and 7 not important for test)

Chapter 4 -The flow near an equilibrium and Chapter 5 - Poincare-Bendixson Theorem from the lecture notes, and the problems sheets nr 3, 4 and 5 relating to these chapters.

I now summarize in some detail the main points in this material:

Chapter 3

definition of linearity

solution of autonomous linear ODEs in terms of exponential of matrix (including relevant proofs)

existence and uniqueness of solutions of autonomous linear ODEs

flow map and its computation in elementary examples in the two-and three-dimensional case

geometric interpretation of explicit formulas for the flow map: invariant subspaces, eigenspaces and generalised eigenspaces, projections and the decoupling principle for linear ODEs

Jordan normal form (general result, but not general proof, and ability to determine the jordan normal form in some simple examples); generalised eigenspaces

Jordan Chevalley decomposition; definition, implications for exp(A) and finding the J-C decomposition in some elementary examples.

Lyapunov and asymptotic stability. Application to linear systems; role of eigenvalues and determination of (in)stability based on information about eigenvalues.

Lyapunov functions; proofs and elementary applications

Chapter 4

Linear approximation near an equilibrium point

Lemma 4.1.1, including interpretation (what does the lemma establish?) and proof (with relevant components, like gronwell estimate and variations of constant formula

Theorem 4.1.2 and proof of part (i)

Hartman-Grobman Theorem may be skipped

Hyperbolic equilibria: proposition 4.2.2 and corollary 4.2.3 (inclusive of proof)

prop 4.2.5 & prop 4.2.6 incl proofs

stable and unstable manifolds (only definitions and main results but no proofs - as not given)

simple examples of bifurcations (and use of implicit function theorem in this context)

Chapter 5 (Hirsch, Smale, Devaney chap 10)

limit sets (definitions and identification of limit sets in simple examples) (10.1)

local sections and flow box (10.2)

monotone sequences (10.4)

Poincare-Bendixson Theorem (10.5 and 10.6) + additional note on classification of omega-limit sets for planar ODEs (as discussed in lecture)

Exercise sheets to be studied:

Problem sheet 4 (but nr 6 not important for test).

Problem sheet 5 (but nrs 1 and 6 not important for test).

Problem sheet 6 (but nrs 4, 5 and 7 not important for test)

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