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This follows on from this post.

Recall the Doubling Mapping $D:[0,1)\rightarrow [0,1)$ given by:

$\displaystyle D(x)=\begin{cases} 2x & \text{ if }x<1/2 \\ 2x-1 & \text{ if }x\geq 1/2 \end{cases}$

At the end of the last post we showed that this dynamical system displays sensitivity to initial conditions. Now we show that it displays topological mixing (a chaotic orbit) and density of periodic points.

First we must talk about periodic points.

### Periodic Points

Consider, for example, the initial state $\displaystyle x_0=\frac{1}{9}$. The orbit of $x_0$ is given by:

$\displaystyle \text{orb}(x_0)=\left\{\frac{1}{9},\frac29,\frac49,\frac89,\frac79,\frac59,\frac19,\frac29,\dots\right\}$

Here we see $\frac19$ repeats itself and so gets ‘stuck’ in a repeating pattern:

The orbit of $x_0=1/9$.

The orbit of any fraction, e.g. $\displaystyle x_0=\frac{4}{243}$, must be periodic, because $\displaystyle D\left(\frac{i}{243}\right)$ is either equal to $\displaystyle \frac{2i}{243}$ of $\displaystyle \frac{2i-243}{243}$ and so the orbit consists only of states of the form:

$\displaystyle \frac{i}{243}$,

and there are only 243 of these and so after 244 iterations, some state must be repeated and so we get locked into a periodic cycle.

If we accept the following:

### Proposition

A fraction $\frac{p}{q}$ has a recurring binary expansion:

$\displaystyle \frac{p}{q}=0.b_1\dots b_m\overline{a_1a_2\dots a_n}_2$,

then this is another way to see that fractions are (eventually) periodic. Take for example,

$\displaystyle x_0=0.101,101,101,101,\dots_2=0.\overline{101}_2=\frac{5}{7}$.

## Dynamical Systems

A dynamical system is a set of states $S$ together with an iterator function $f:S\rightarrow S$ which is used to determine the next state of a system in terms of the previous state. For example, if $x_0\in S$ is the initial state, the subsequent states are given by:

$x_1=f(x_0)$,

$x_2=f(x_1)=f(f(x_0))=(f\circ f)(x_0)=:f^2(x_0)$

$x_3=f(x_2)=f(f^2(x_0))=f^3(x_0)$,

and in general, the next state is got by applying the iterator function:

$x_{i}=f(x_{i-1})=f^i(x_0)$.

The sequence of states

$\{x_0,x_1,x_2,\dots\}$

is known as the orbit of $x_0$ and the $x_i$ are known as the iterates.

Such dynamical systems are completely deterministic: if you know the state at any time you know it at all subsequent times. Also, if a state is repeated, for example:

$\text{orb}(x_0)=\{x_0,x_1,x_2,x_3,x_4=x_2,x_5\dots,\}$

then the orbit is destined to repeated forever because

$x_5=f(x_4)=f(x_2)=x_3$,

$x_6=f(x_5)=f(x_3)=x_4=x_2$, etc:

$\Rightarrow \text{orb}(x_0)=\{x_0,x_1,x_2,x_3,x_2,x_3,x_2,\dots\}$

### Example: Savings

Suppose you save in a bank, where monthly you receive $0.1\%=0.001$ interest and you throw in $50$ per month, starting on the day you open the account.

This can be modeled as a dynamical system.

Let $S=\mathbb{R}$ be the set of euro amounts. The initial amount of savings is $x_0=50$. After one month you get interest on this: $0.001\times50$, you still have your original $50$ and you are depositing a further €50, so the state of your savings, after one month, is given by:

$x_1=50+0.001\times 50+50=(1+0.001)50+50$.

Now, in the second month, there is interest on all this:

interest in second month $0.001\times((1+0.001)50+50)=0.001x_1$,

we also have the $x_1=(1+0.001)50+50$ from the previous month and we are throwing in an extra €50 so now the state of your savings, after two months, is:

$x_2=x_1+0.001x_1+50=(1+0.001)x_1+50$,

and it shouldn’t be too difficult to see that how you get from $x_i\longrightarrow x_{i+1}$ is by applying the function:

$f(x)=(1+0.001)x+50$.

#### Exercise

Use geometric series to find a formula for $x_n$.

## Weather

If quantum effects are neglected, then weather is a deterministic system. This means that if we know the exact state of the weather at a certain instant (we can even think of the state of the universe – variations in the sun affecting the weather, etc), then we can calculate the state of the weather at all subsequent times.

This means that if we know everything about the state of the weather today at 12 noon, then we know what the weather will be at 12 noon tomorrow…

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.

## Homework

The closing date is 11 April at 2 pm. Work handed up late will be assigned a mark of zero. Note what the homework says”

Answer the following using geometric arguments.

## Week 12

We finished on Monday by talking a little bit more about taking roots of complex numbers.

## Week 13 Review Week (21-25 April)

I will book a two hour slot and we will first go through the layout of the paper and then I will answer any questions that ye might have.

## 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.

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.

## Homework

The closing date is tomorrow three weeks: that is 11 April at 2 pm. Work handed up late will be assigned a mark of zero.

## Week 11

More on complex numbers including De Moivre’s Theorem and taking roots.

## Week 12

On Monday we will talk a little bit more about taking roots of complex numbers and also do an actual example. This will wrap things up in terms of lectures.

On Wednesday we will have an additional tutorial so that we can get through some more examples. As normal there will be a tutorial on Thursday (and a review lecture after the break).

## Exercises

For the Week 12 tutorial you should look back over the exercises in general.

If you are happy with them then maybe you could have a look at some of the harder applied problems — which are definitely not examinable but will test your understanding.

If you are doing abstract algebra Q.1 is a good, tough exercise. Q. 6 (h) is another good question bringing in complex numbers.

## 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.

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.

## Homework

Will be discussed in class on Monday.

## Week 10

On Monday we showed that to multiply one complex number $z_1$ by another $z_2$ involves stretching $z_1$ by a factor of $|z_2|$ and then rotating through an angle $\text{arg }z_2$. On Tuesday we used this information to find roots of unity. We also spoke about the Conjugate Root Theorem.

## Week 11

More on complex numbers including De Moivre’s Theorem and taking roots.

## Exercises

For the Week 11 tutorial you should look at Q.54 and 61-63.

## 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 conjecture about calculating $i^i$.

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.

## Homework

Will be discussed in class on Monday.

## Week 9

We finished our study of the Doubling Mapping and started talking about complex numbers and how they arise both algebraically and geometrically.

## Week 10

More on complex numbers.

## Exercises

For the Week 10 tutorial you should look at Q.44, 45, 52, 53, 56-60. At some point you need to backtrack and look at questions 30-37. These are questions that are harder than the test but use the same ideas.

## 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 conjecture about solving a complex numbers problem geometrically.

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.

## MCQ Results

As discussed previously, the MCQ results will come out the day you hand in the homework: 11 April 2014. I will send the homework next week.

## Week 8

We finished describing Chaos Theory and began proving that the Doubling Mapping is chaotic.

## Week 9

We will finish our study of the Doubling Mapping and possibly start talking about complex numbers.

## Exercises

I have emailed ye a copy of the exercises. For the Week 9 tutorial you should look at Q.46 and Q. 47. At some point you need to backtrack and look at questions 30-37. These are questions that are harder than the test but use the same ideas.

## 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 conjecture about the iterates of the tangent function.

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.

## MCQ Announcement

I have decided to make the MCQ open book and also ye will have 50 minutes and not 30 minutes as previously stated.

• Do bring in paper for you to tease out questions
• Do expect to do just as good without notes as with notes
• Don’t expect it take you 50 minutes to do the test. You can leave early or if you want get some study in

The Concept MCQ will take place this Wednesday 26 February Week 8 in WGB G05. It starts at 10.05 and runs until 10.55.

## Week 7

In Week 7 we finished of our study of the Tent Mapping and began talking about Chaos Theory.

## Week 8

In Week 8 we will begin our study of the Doubling Mapping.

## Exercises

I have emailed ye a copy of the exercises. For the Week 8 tutorial you should look at Q.48 and Q. 49. At some point you need to backtrack and look at questions 30-37. These are questions that are harder than the test but use the same ideas.

## Test and Other CA

The results and solutions of the test were emailed to the class.

The Concept MCQ will take place on Wednesday 26 February Week 8 in WGB G05.

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 be on complex numbers and won’t be as long as last year’s homework (MS2001 nor MS3011).

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 result about the periodic points of the tent mapping.

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

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.

## 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).

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 5

We began our study of the Logistic Family. We postulated the equation as a model of population growth with two assumptions and ended up with

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

where $x\in[0,1]$ can be interpreted as the proportion of a maximum population with a growth rate $\mu\in[0,4]$.  We began analysing when zero was attracting/repelling and when $\displaystyle \frac{\mu-1}{\mu}$ was attracting/repelling. We were rudely interrupted by the fire drill!

http://kkcb.com/one-of-the-best-scenes-of-the-office-is-dwights-fire-drill/

## Week 6

In Week 6 we will finish of our study of the Logistic Mapping and perhaps begin our study of the Tent Mapping.

## Exercises

I have emailed ye a copy of the exercises and ye should be able to look at these questions 38, 39, 42, 43 for the Week 6 tutorial (assuming that ye are ready for Wednesday’s Test).

## 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 have everything for the test covered. Please find a sample test and the test I gave last year after page 31. in these typeset notes.

The following theorems from the notes are examinable: the very bottom of page 12 and start of page 13. Also the Fixed-Point Factor-Theorem (which was called such on the board. It is not in the notes but is found here. When I say examinable you should be able to

• state the theorem
• prove the theorem
• understand the theorem and the proof

Learning off the proof letter by letter won’t do you!

The Concept MCQ will 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 finding a formula for the iterates of $Q_1$(x).