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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.

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