MS 3011: Week 9

March 4, 2012 in MS 3011

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 jippo@campus.ie and I will add you to the mailing list.

This Week

We started the section complex dynamical systems. We introduced the complex numbers by looking at the following equations:

$x+5=10$ .

$x+10=5$ .

$5x=6$ .

$x^2=2$ .

$x^2=-1$ .

We said that the ‘need’ for complex numbers is analogous to the fact that we can solve the first equation using natural numbers $\mathbb{N}$ but we require

negative integers to solve the second equation ( $\mathbb{Z}$ )
fractions to solve the third equation ( $\mathbb{Q}$ )
real numbers to solve the third equation ( $\mathbb{R}$ )
and finally, complex numbers to solve the fourth equation ( $\mathbb{C}$ )

We then asked do we need more and more complicated number systems to solve more complicated equations and we mentioned in passing that the complex numbers are algebraically closed and hence we don’t need more complicated number systems to solve polynomial equations:

$a_nx^n+\cdots+a_1x+a_0=0$ .

This is usually where people stop but we asked what about equations like $xy=yx+1$ …

We also showed that complex numbers multiply in the rotating manner: in the notation $z=(\text{modulus},\text{argument})$ ;

$(r,\theta)\times (s,\alpha)=(rs,\theta+\alpha)$ .

On Monday we will talk about roots of unity and then we will start examining the iterates of the complex linear map.

In the tutorial we discussed some aspects of the homework.

Complex Number Exercises

One of the important things about this module is that you get some more exposure to complex numbers. For those of you who hope to go teaching we won’t be doing enough complex number algebra to get ye going great guns so you will either have to get comfortable with the complex number stuff you did in first year or else look at stuff like this.

Problems

2009 Autumn Q. 1(d)

Additional Exercises

Show that $2\text{ Re}(z)=z+\bar{z}$ and 2\text{ Im}(z)=z-\bar{z} for all $z\in\mathbb{C}$ .
Show that $\overline{(w+z)}=\bar{w}+\bar{z}$ and $\overline{(wz)}=\bar{w}\bar{z}$ for all $w,\,z\in\mathbb{C}$ . Hence prove the Conjugate Root Theorem — which exhibits the symmetry between $i$ and $-i$ with respect to the real numbers. In particular $\sqrt{-1}=i$ is imprecise: $i^2=-1$ is better/

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May 17, 2012 at 8:43 am

Student 14

I am a student in your MS3011 Dynamical systems. I am just unsure about the content of the exam or more so the way to approach my studies. I have written out the theorems in the book plus Newtons and De Moivre, will this cover the theory aspect of the exam or is there more? I have also gone through exam papers but feel this is almost irrelevant due to the change in content and lecture? What do you suggest I focus mainly on now to eliminate the prospects of failing again?
Any advice would help a very worried and stressful student.

May 17, 2012 at 8:54 am

J.P. McCarthy

Firstly you must understand all the central concepts:

Dynamical system, iterator function, orbit, fixed point, periodic point, prime period $n$ , attracting, repelling, indifferent, chaos theory, dense

I would say understanding them is far more important than being able to write them out.

Question 1 is on real dynamical systems. This encompasses what we did in the in-class test. You need to understand the relevant theorems (p.12/13) as well as be able to prove them. In fact I believe you can’t expect to prove them without understanding them. If you can do the past papers here and understand the theorems you should be O.K.

Question 2 will either be on the logistic map OR the tent map OR the doubling map. As a hint I told ye that the summer exam is NOT the tent map. Past papers are worthwhile here but not unless you know what you are doing (i.e. if you can answer the questions that is great but you’d want to really know what you’re doing. Learning off a solution will not do the job here.). An intimate understanding of these maps will be rewarded here which means motivating the logistic map, understanding how the doubling mapping is chaotic, etc.

Question 3 is the unseen question but if you have a keen understanding of “Dynamical system, iterator function, orbit, fixed point” you will be rewarded. This question is to reward those who understand the material. Past papers are useless here.

Question 4 is on complex mappings. This will be somewhat similar to the past papers but with more of an emphasis on looking at complex numbers geometrically (as we did on the blackboard by and large).

Regards,
J.P.

May 17, 2012 at 1:49 pm

Conor mangan

Hi Jp,
I’m one of the students doing your MS3011 test tomorrow and there’s been one quesiton that I can’t quite get. I don’t know how many times that I thought I was on the right track and then realised that i was miles off…
Its Q. 4 of the past papers, Summer 2011… All about Julia’s set and the madlebrot set… Help is URGENTLY needed J.P…. I’m lost…

May 17, 2012 at 2:03 pm

J.P. McCarthy

Conor,

Chill out.

Regarding the Newton’s method question from last year I sent this to another student:

“Question 3 will be an unseen question. If you have a keen understanding of the terms dynamical system, iterator function, orbit and fixed point you will be rewarded. This question is to reward those who understand the material.

Past papers are useless for question 3 and as I have dubbed the question “unseen” it is very unlikely that Newton’s Method will be seen in question 3 (considering that I did the question in a lecture). It does not belong anywhere else in the paper neither.”

Julia sets and Mandlebrot sets are not examinable. We stopped at p.6 of the complex numbers section of the notes and instead worked on the board.

Regards,
J.P.

May 18, 2012 at 1:55 pm

Fergal Walsh

who is this loser? julia and mandlebrot sets arent even on the exam man

May 18, 2012 at 1:57 pm

J.P. McCarthy

Fergal,

MS3011 banter: I’ve seen it all now.

Regards.

May 17, 2012 at 2:53 pm

Student 15

I struggle with the term orbit, what does it mean in plain english? I think I have an handle on the rest of the material!

May 17, 2012 at 2:59 pm

J.P. McCarthy

Plain English? Don’t know what you are talking about!!

A dynamical system consists of a set $S$ and a(n) (iterator) function $f:S\rightarrow S$ which maps elements of the set to other elements of the set.

We have to start somewhere and the seed is an element of the set which we usually denote by $x_0$ and is considered the initial state of the system.

The set of iterates of $x_0$ under $f$ is the orbit of $x_0$ :

$\text{orb}(x_0)=\{x_0,f(x_0),f^2(x_0),f^3(x_0),\dots\}=\{f^k(x_0):k\in\mathbb{N}_0\}$ .

Example

Let $S=[0,1]$ and $f:[0,1]\rightarrow [0,1]$ be given by

$f(x)=x-x^2$ .

Suppose we begin at $x_0=1/2$ . Thence:

$1/2,f(1/2)=1/4,f(1/4)=3/16,f(3/16)=39/256$

Then the orbit of $1/2$ is given by:

$\{1/2,1/4,3/16,39/256,\dots\}$ .

Regards,
J.P.

May 17, 2012 at 4:57 pm

Student 18

I have a few questions regarding tomorrow’s exam. Firstly, I am finding it very difficult to prove that there are or there are not eventually periodic points. Do you just try and come up with a point that is eventually periodic or is there a method to it? And what is especially difficult is to show eventually periodic points in binary form??

Also, With regard to chaos theory, one needs to prove sensitivity on initial conditions, that it has a dense orbit and that it has a dense set of periodic points. The sensitivity part if fine, but I am not grasping the other two. Can you please expand it out clearly for me to understand?

How do you convert fractions into binary form?

Will ternary form be coming up on the test? I kind of understand how to expand something into binary form, but the notes have slightly confused me as regards ternary form. Can you please explain?

May 17, 2012 at 5:25 pm

J.P. McCarthy

To find an eventually periodic point (which is not itself period) you need to find points that are sent to fixed points which are not themselves fixed:

Solve $f(x)=x$ for the fixed points ( $x_{f_1},\dots,x_{f_k}$ ).
Solve $f(x)=x_{f_i}$ to find points sent to fixed points.
The fixed points will always be solutions to this equation but the solutions which aren’t themselves are not periodic but are eventually periodic (namely eventually period-1 or fixed).

Proofs of non-existence can be tough. The reason is there is a possibly infinite amount of cases to check. If you believe that there are no eventually periodic points then your best bet is to use a proof by contradiction.

Example
Prove that $L(z)=\alpha z$ for some fixed $\alpha$ has no non-zero periodic points if $|\alpha|=1$ and $\text{arg}(\alpha)=\kappa 2\pi$ with $\kappa\not\in\mathbb{Q}$ .
Proof: Suppose on the contrary that such a non-zero periodic point $z$ exists, say of period $N$ .
Then $L^N_(z)=z$ . Now $L^N(z)=\alpha^Nz$ so we have

$z=\alpha^Nz\Rightarrow \alpha^N=1$ as $z\neq 0$ .

Now $\alpha=e^{2\pi \kappa i}$ and $\alpha^N=(e^{2\pi N\kappa i})$ must equal one hence have an argument of the form $2\pi M$ for some $M\in\mathbb{N}$ :

$2\pi N\kappa =2\pi M\Rightarrow \kappa=N/M$ which is a contradiction as $\kappa\not\in\mathbb{Q}$ .

Hence no such periodic point exists. $\Box$

Now what is a period- $n$ point of the doubling mapping? Well a solution to

$D^n(x)=x$ .

Let $x\in[0,1]$ have binary representation $x=0.a_1a_2a_3a_4\dots$ . All the time when in talking binary the $a\in\in\{0,1\}$ . $D^n$ chops the first $n$ digits off.
Hence

$D^n(x)=0.a_{n+1}a_{n+2}a_{n+3}\cdots\overset{!}{=}0.a_1a_2\cdots a_na_{n+1}\cdots$ .

Hence $x=0.\overline{a_1\cdots a_n}$ recurring.

Eventually periodic points $y$ are points for which some iterate is periodic:

$f^n(y)=x_p$ with $x_p$ periodic.

Now, with respect to the Doubling Mapping (whenever we are talking about the binary representation it will be in relation to the Doubling Mapping), eventually periodic points are just that

$D^n(y)=x_p$ with $x_p$ periodic.

That is

$D^n(y)=$ a recurring binary representation. Hence the eventually periodic points have a finite number of arbitrary digits but then begin to recur. They have the form:

$y=a_1a_2\cdots a_n\overline{b_1b_2\dots b_m}$

for some $m\in\mathbb{N}$ and $b_i\in\{0,1\}$ .

I gave my best explanation of density of periodic points and a dense orbit in the review lecture but I’ll give a more technical explanation here.

A subset $A$ of a set $S$ is said to be dense in set $S$ if for for any point $s\in S$ , there exists a point $a\in A$ arbitrarily close to $s$ ; i.e.

$\forall\,\varepsilon>0$ and $s\in S$ , $\exists\,a\in A$ such that

$|s-a|<\varepsilon$ .

So to show that a dynamical system (a set $S$ together with a function $f:S\rightarrow S$ ) has a dense orbit you must find a seed $x_0$ such that the orbit of $x_0$ ;

$\text{orb}(x_0)=\{x_0,f(x_0),f^2(x_0),f^3(x_0),\dots\}$

is dense in $S$ . That means given any $x\in S$ , you must find an $x_0$ such that $f^n(x_0)$ can be made arbitrarily close to $x$ for some $n\in\mathbb{N}$ .

To show that the periodic points are dense you must, for any $x\in S$ , find a period point $x_p$ such that $x_p$ is arbitrarily close to $x$ .

We don’t need to convert fractions into binary but http://cs.furman.edu/digitaldomain/more/ch6/dec_frac_to_bin.htm .

I should have allowed ternary but I have said it will be the Logistic Map or the Doubling Map. But ternary is very similar to binary and if you can do binary ternary should be no problem at all.

Regards,
J.P.

May 18, 2012 at 1:48 pm

Conor

J.P., your a life saver… Thank god I don’t have to do mandelbrot sets..

May 18, 2012 at 1:59 pm

Fergal Walsh

Conor, J.P. already said this in the lectures if you had bothered to turn up

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MS 3011: Week 9

This Week

Complex Number Exercises

Problems

Additional Exercises

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Categories

J.P. McCarthy Maths

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MS 3011: Week 9

This Week

Complex Number Exercises

Problems

Additional Exercises

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J.P. McCarthy Maths

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