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.

Course Notes

Have been emailed to you.

Weeks 1 & 2

In the first two 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 looked at some examples of dynamical systems (note there was a small error in the regular savings example but that is not too important).

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.

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.

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.

We will look at this problem next week.

Week 3

In Week 3 we will study periodic points in more depth and introduce the idea of an attracting fixed point.

Exercises

I have emailed ye a copy of the exercises and ye should be able to look at questions 1-9 and 11 for next week’s tutorial.

Test

The test will take place on February 5. Everything up to but not including section 3.4 in the typeset notes is examinable: we should have this covered by January 27. I have emailed ye a copy of a sample test.

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 fixed points.

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