Sunday, March 29, 2009
How to study for the 2009 exam.
Please see VERY USEFUL notes on how to study for the exam in the lhs margin.
Some additional reading material (good books in which you may find some useful exercises on material related to the course):
Perko: Differential Equations and Dynamical Systems
Chicone: Ordinary Differential Equations and Applications
Some additional reading material (good books in which you may find some useful exercises on material related to the course):
Perko: Differential Equations and Dynamical Systems
Chicone: Ordinary Differential Equations and Applications
Friday, March 20, 2009
Final week
For the final week of term (23-27 mar) we have the following program:
TUE 23/3 9-10am final lecture
WED 24/3 10-11am final problem class (but I will not be attending)
So NO lectures on wed&thu and NO office hour on wed.
I will soon post here instuctions about "How to study for the exam", which will give you some very useful information on how to approach the material and problems presented in this course, in view of the exam.
TUE 23/3 9-10am final lecture
WED 24/3 10-11am final problem class (but I will not be attending)
So NO lectures on wed&thu and NO office hour on wed.
I will soon post here instuctions about "How to study for the exam", which will give you some very useful information on how to approach the material and problems presented in this course, in view of the exam.
Monday, March 16, 2009
Final problems (sheet nr 7)
The final "problem sheet" consists of exercises 2,3,5,10 from HSD chapter 11 (see also lhs margin under chapter 7 of the course notes), and question 4 of the 2008 exam. Note that the latter was probably a bit (too) hard work as an exam question (see also section 10.7 of HSD).
Chapter 7
Chapter 7 will follow HSD chapter 11 (applications in Biology).
Saturday, March 14, 2009
Lyapunov-Schmidt reduction
Some of you have asked me for some alternative reading on Lyapunov-Schmidt reduction. I would like to suggest you consult section 8.2 of Carmen Chicone's (good!) textbook on Ordinary Differential Equations. (consult this section online in google books)
Thursday, March 12, 2009
12/3 lecture
Apologies for cancelling the lecture so abruptly after having just started. I had been feeling not so well earlier but thought all would be fine. But after just a minute in I realized that I would not be able to give the lecture properly (strong headache). My apologies.
Good luck with the test on monday!
Good luck with the test on monday!
Wednesday, March 11, 2009
Additional corrections to sheet 5 and answers
Some of you pointed out the following typos (thanks for telling me!):
question 5: a should be -a in rhs
question 4 answer: a pm sign missing in the expression of lambda
The corresponding corrections have been made.
question 5: a should be -a in rhs
question 4 answer: a pm sign missing in the expression of lambda
The corresponding corrections have been made.
Thursday, March 5, 2009
Progress test 16 March: exercise sheets
Someone asked me which problems are part of the material of chaps 3,4, and 5. These are
sheets nr 3, nr 4, nr 5 and nr 6 (but not question 8 and 9 which concern omega-limit sets). I have posted model answers for sheet 6 in lhs margin.
Please note again that I will try test your understanding of the material and not whether you have learned things by heart. So I will not ask for literal proofs etc (but still it should be beneficial for you to make sure you understand them!). Also make sure to focus on "basics" more than the "hardest" problems on the exercise sheets.
Good luck!
PS: I am doing my best to get Chap 5 typed up, but content will not differ essentially from the written lecture notes (that you can already download from the lhs margin). So - IN ADDITION TO WHAT I ALREADY TYPED UP AND THE SLIDES - please use the written lecture notes!
sheets nr 3, nr 4, nr 5 and nr 6 (but not question 8 and 9 which concern omega-limit sets). I have posted model answers for sheet 6 in lhs margin.
Please note again that I will try test your understanding of the material and not whether you have learned things by heart. So I will not ask for literal proofs etc (but still it should be beneficial for you to make sure you understand them!). Also make sure to focus on "basics" more than the "hardest" problems on the exercise sheets.
Good luck!
PS: I am doing my best to get Chap 5 typed up, but content will not differ essentially from the written lecture notes (that you can already download from the lhs margin). So - IN ADDITION TO WHAT I ALREADY TYPED UP AND THE SLIDES - please use the written lecture notes!
Prize problem
In the draft version of the lecture notes chapter 5 I suggest in the proof of part (b) of Proposition 5.1.2 that it is possible to estimate the eigenvalues of a matrix (A+B) as a function of the eigenvalues of the matrix A and the norm |B| of matrix B. In fact, I meant to ask you to work this out in Exercise sheet 5, question 1(a). The question is posed slightly less specific there, and on the solution sheet I answer the question without giving an explicit estimate to the change of the eigenvalues of (A+B) as a function of |B|, only arguing that as |B| goes to zero, the eigenvalues of (A+B) converge to those of A.
Prize question: Let A be a real mxm matrix with set of eigenvalues lambda_1,...,lambda_m with real part >= epsilon>0. Find some explicit bound for delta (as a function of epsilon, where delta(epsilon)>0 if epsilon>0) such that the eigenvalues of (A+B) are guaranteed to not intersect the imaginary axis for all B with |B| smaller than delta(epsilon).
Please send your answer(s) to me by e-mail before March 23. A box of chocolates will be made available to the author(s) with the - in my view - best answer (if quality is equal the answer having been sent-in first will prevail).
Prize question: Let A be a real mxm matrix with set of eigenvalues lambda_1,...,lambda_m with real part >= epsilon>0. Find some explicit bound for delta (as a function of epsilon, where delta(epsilon)>0 if epsilon>0) such that the eigenvalues of (A+B) are guaranteed to not intersect the imaginary axis for all B with |B| smaller than delta(epsilon).
Please send your answer(s) to me by e-mail before March 23. A box of chocolates will be made available to the author(s) with the - in my view - best answer (if quality is equal the answer having been sent-in first will prevail).
Wednesday, March 4, 2009
Problem sheet 6
I have made some amendments to problem sheet 6 (new version is found under link in lhs margin). In question 1 we look for equilibria near the equilibrium (x,y)=(0,0) at parameter value lambda=0. In question 2 there is no need to assume that algebraic and geometric multiplicities of eigenvalue zero are equal to each other, so this is a confusing detail that I have deleted from the question. Moreover, in question 6, U is a neighbourhood of x_0 and we need to add the fact that V(x)>0 for all x in U\{x_0}.
Correction solution to sheet 4 question 1(d)
The exact solution written on the answer sheet was wrong and should be t=ln(sec(x)+tan(x)). It can be derived by those of you who are well skilled in doing such special integrals (here of cos(x)^{-1}dx). Do not worry if you would have difficulty to derive this (by hand).
Tuesday, March 3, 2009
Course notes chaps 6 and 7
The two final chapters of the course notes will concern selected material (I will make the precise selection known asap) from chapters 10,11 and 12 of HSD. You find scans of the relevant material under the chapter headings in the lhs margin.
Please note that in the first scan that I posted, some pages were a bit fuzzy (224-227)
so I scanned them again and the version in the lhs margin should now be ok to read.
Please note that in the first scan that I posted, some pages were a bit fuzzy (224-227)
so I scanned them again and the version in the lhs margin should now be ok to read.
Monday, March 2, 2009
Problem sheet 5 question 6
This question was put in a bit too early (as it deals with a concept that only appears in chap 6), so it has been moved from sheet 5 to sheet 6.
Progress test 16 March
This progress test will concern the material of chapters 3,4, and 5.
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