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## Story So Far

In the first three weeks we have defined a *dynamical system* . It is a *set of states * together with an *iterator function/ rule of evolution *. We take an *initial state/ seed point* and examine the *orbit of :*

,

where the states are produced iteratively by the iterator function:

and .

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 such that if an orbit of a point `hits’ then the orbit will remain fixed at . Thus fixed points are points with the property that

.

So the fixed points of a function are points such that the output of the function equals the input. Note that when we graph functions, the -axis comprises the inputs and the -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 . This means that if we graph $latex $y=f(x)$ then the fixed points of are the points on the graph of that intersect .

Similarly *periodic points *are states/ points such that if an orbit of a point `hits’ then the orbit will keep returning to after, say iterations of ; that is :

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

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

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

.

Solving this equation is not necessarily that easy but we proved that if , then the *fixed-point factor-theorem *applies: divides into 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.

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