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.

Test

Yes test this Monday morning at 10:00. Any questions send me an email or even better comment below.

Homework

Ye will be getting a homework assignment a lá MS2002 last year. I hope to have more information on this next week.

Tutorial Venue

I have applied to get this changed to WGB G03… I didn’t get everything. The timetable is

  • This week coming, 21 February WGB G03
  •  28 February LL2
  • 7 March LL2
  • 14 March WGB G03
  • 21 March WGB G03
  • 28 March WGB G03

Week 5

We continued our study of the Logistic Map given by

Q_\mu(x)=\mu x(1-x)

where x\in[0,1] can be interpreted as the proportion of a maximum population with a growth rate \mu\in[0,4]. We found the fixed points of Q_\mu in terms of the growth rate. These were found to be at zero and \displaystyle \frac{\mu-1}{\mu} (the second one only applies when \mu\geq 1; why?). We analysed 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=4, Q_\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.

We then postulated the existence of more complicated behaviour, chaotic behavior. The first thing we needed was a point with a dense orbit: such a point would necessarily not have any pattern or any periodicity:

dense orbit

What next?

We will finish writing off our definition of what a chaotic dynamical system and do a special study of the tent mapping.

Tutorials

Exercises for Thursday 21 February are to look at the following:

Summer 2011 Question 2(c)

Summer 2010 Question 2(b), (c)

Summer 2009 Question 2(b), 3(d), 4(c)

I understand that ye are busy with the test on Monday but after this I would strongly urge you to look at these problems and also the ones from Week 5