I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 4

On Monday we spoke about cobweb diagrams that are a graphical method of locating fixed points and determining whether they are attracting, repelling or indifferent. At this point you should look at Section 2.4 of these notes (NOT mine).

In Section 3.1, 3.2 & 3.3, the cobweb diagrams suggest that if $|f'|<1$ ($f'$ aka the slope) at the fixed point that the point is attracting and if $|f'|>1$ the fixed point is repelling.

A normal person would have proved this as per page 25 of the notes but I wanted to show you the beautiful contraction mapping principle and do it that way… we did conclude, loosely but correctly,

### Theorem The: Contraction Mapping Principle

If $f:I\rightarrow I$ is a contraction on a closed interval $I$then $f$ has a unique, attracting fixed point in $I$.

I mainly wanted to show ye this for the nice pictures! There is one here that I got off the internet but I think our blobs were better:

O.K.

In the end what I did was a hodge-podge of the (correct) contraction mapping principle we did on Monday with the proof in the notes of the Fixed Point Dynamics Theorem below which isn’t really satisfactory to me. For next year I will probably try and take this approach but as my proof had a few holes, I am just going to say that you need to know the following:

### Theorem: Fixed Point Dynamics

Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ is an iterator function with a fixed point at $x_f\in \mathbb{R}$. If $f$ is differentiable in an interval $I$ containing $x_f$ then

• If $|f'(x_f)|<1$then $x_f$ is an attracting fixed point
• If $|f'(x_f)|>1$then $x_f$ is a repelling fixed point
• If $|f'(x_f)|=1$then we can make no conclusion and we call $x_f$ an indifferent or neutral fixed point

### Week 5

In Week 5 we will begin our study of the Logistic Mapping — rabbits!

## Exercises

I have emailed ye a copy of the exercises and ye should be able to look at these questions for the Week 5 tutorial.

• 19 — a good question for more understanding for test
• 23-25, 27-29 — good basic practise for the test. Do 25 and you can maybe leave the others
• 26 — more theoretical than other questions of that type. You need to understand parts (a) to (c). These are the three theorems whose proofs are examinable.

The rest of these are not going to be on the test but the exam. If you forget them now don’t forget to look at before the exam to boost understanding. The theory is the same as before

• 30-32 — not examinable on the test but have occurred in Q.2 of the exam. Week 4 theory: same as as 23-25, 27-29
• 33 — full analysis of the dynamical system required using theory of Week 4.
• 34-34 — given the orbit. Find the iterator function and analyse using theory of Week 4.
• 36-37 — Newton-Raphson method is a dynamical system. Analyse using Week 4 theory.

As there are a lot of questions it might make sense to allocate so much time and say do (A)s first, then (B)s then (C)s or whatever.

## Test and Other CA

The test will take place on February 12 in Week 6. Everything up to but not including section 3.4 in the typeset notes is examinable: we will have this covered by Februaray 3 but probably January 29. I have emailed ye a copy of a sample test.

The Concept MCQ will still take place in Week 8.

The homework will be given to you towards the end of the semester and I will give ye three weeks to do it. It will probably be on complex numbers and won’t be as long as last year’s homework.

You will be given marks for the best two out of Test, Concept MCQ and Homework.

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question about where the OP didn’t understand why roots of $f(x)-x$ are roots of $f^2(x)-x$.