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 6

We finished our study of the Logistic Family by analysing when zero was attracting/repelling and when \displaystyle \frac{\mu-1}{\mu} was attracting/repelling. We summarised our results in a bifurcation diagram

bifurcation

The green line corresponds to attracting fixed points and the red, dashed line to repelling fixed points. The graph is of fixed points vs \mu.

The fact that there are no attracting fixed points for \mu>3 indicates that the behaviour is more complicated when the growth rate, \mu, gets large. We could have periodic behaviour and perhaps more strange, chaotic behaviour.

We studied therefore the case where \mu=4. We said that for \mu=4Q_\mu is symmetric about x=1/2 and is unimodal. We showed that  Q_4^n has 2^{n-1} branches and hence 2^n period-n points.

Week 7

In Week 7 we will finish of our study of the Tent Mapping and perhaps begin our study of the Doubling Mapping.

Exercises

I have emailed ye a copy of the exercises and ye have looked at questions 38, 39, 42, 43 for the Week 6 tutorial (assuming that ye were ready for Wednesday’s Test). For the Week 7 tutorial you should look at these first and then backtrack and look at questions 30-37.

Test and Other CA

The results of the test will be revealed in time.

The Concept MCQ will take place on Wednesday 26 February Week 8 in WGB G05. It will be a half hour test and starts at 10.25 and runs until 10.55.

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 finding a formula for the iterates of Q_1(x).

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