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.

## Story So Far

In the first three weeks we have defined a dynamical system $(S,f)$. It is a set of states $S$ together with an iterator function/ rule of evolution $f:S\rightarrow S$. We take an initial state/ seed point $x_o\in S$ and examine the orbit of $x_0$:

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

where the states $x_1,x_2,\dots$ are produced iteratively by the iterator function:

$x_1=f(x_0)$ and $x_n=f(x_{n-1})$.

We developed the Logistic Model of Population Growth, and this comprises an important example of a dynamical system which we will examine in more depth a little later on.

We studied fixed points. These are states $x_f\in S$ such that if an orbit of a point hits’ $x_f$ then the orbit will remain fixed at $x_f$. Thus fixed points are points with the property that

$f(x_f)=x_f$.

So the fixed points of a function $f:S\rightarrow S$ are points such that the output of the function equals the input.  Note that when we graph functions, the $x$-axis comprises the inputs and the $y$-axis the outputs, and if we are looking for fixed points/ points such that output equals input we need to look for points such that $y=x$. This means that if we graph $latex$y=f(x)\$ then the fixed points of $f(x)$ are the points on the graph of $f(x)$ that intersect $y=x$.

Similarly periodic points are states/ points $x\in S$ such that if an orbit of a point hits’ $x$ then the orbit will keep returning to $x$ after, say $N$ iterations of $f$; that is $f^N(x)=x$:

$x_0,x_1,x_2,\dots,x,f(x),\dots,f^{N-1}(x),x,f(x),\dots,f^{N-1}(x),x,f(x),\dots.$

We also noted that a period-2 point would also be period-6 for example:

$\{\alpha,\beta,\alpha,\beta,\alpha,\beta,\alpha,\beta,\alpha,\beta,\dots\}$

Here $\alpha$ is period-6 but the lowest period is two. We call this the prime period of $\alpha$. Finally we proved some little theorems about periodicity.

Finding periodic points, say period-2 points means finding points $x\in S$ such that if we apply the iterator function twice, then we get back to $x$:

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

Solving this equation is not necessarily that easy but we proved that if $f:S\rightarrow S$, then the fixed-point factor-theorem applies: $f(x)-x$ divides into $f^2(x)-x$ and this helps immensely.

Also we expect that Theorems 1 & 2 hold and we proved these.

## Exercises

For future weeks I will try and organise these exercises a little better…

## Test

The test will take place on February 20. Everything up to but not including section 3.4 in the typeset notes is examinable: we should have this covered this week or next.