What is the significance of the diagonal line in Cobweb Diagrams?

[mr.tea]

Hi,

I saw at the lecture something that called Cobweb diagram/plot and it was given without too much explanation. We were working on recursive sequences(Math major, first semester). The sequence was $$\frac{1}{2-\frac{1}{2-...}}$$

I tried to find an explanation to this diagram; why did we choose the diagonal line y=x, and why does it even say something(we could also choose y=2x or y=-3.5x, didn't we?)? I would like to know more why does this diagram even have some meaning.

Thanks,
Thomas

 

[RUber]

For recursive functions, I have seen something similar.
Essentially, you have a recursion formula like $x_{n+1} = \frac{1}{(2 - x_n)}$ with x_0 = 1/2.

You can see a detailed video and a plotting tool here: http://mathinsight.org/cobwebbing_graphical_solution

 

[RUber]

This method reminds me of fixed points. That is, the search for a value a, such that f(a) = a. This is why the reference line in the spider web in y=x, because for each iteration, you are not only using the line y=x as a reference line to help you bring f(x_n) back to your x-axis to use it as x_{n+1}, but you will also notice for most limited systems, the plots will converge to the fixed point.
In this case, if you solve for the fixed point where x = 1/(2-x), you will get x^2 -2x +1 = 0, or x = 1.

 

[HallsofIvy]

Yes, the concept of "fixed point" is what you need. If you have the recursion $x_{n+1}= \frac{1}{2- x_n}$, $x_0= 1/2$ we can, temporarily, assume that this sequence has a limit. If that is true then taking the limit, x, on both sides, $x= \frac{1}{2- x}$ so that $x(2- x)= 2x- x^2= 1$ or $x^2- 2x+ 1= 0$, the equation RUber gives. Once you know the putative limit, you can use that information to prove that the sequence does in fact converge.

Here, for example, it is not difficult to prove, by induction, that the sequence is increasing and bounded above by 1 so converges.

 

[mr.tea]

Thank you both HallsofIvy and RUber.

As you said HallsofIvy, the concept of "fixed point" is probably what I need. I can find the limit without the diagram(with the theorem of increasing sequence and bounded above + induction), but this is a nice another point of view on the problem and I would like to know more about it.

What is the importance of "fixed point"? what its contribution for the study of functions?

RUber, I have already seen this very video, but he does not explain why he does that, just how to draw the diagram(I already know to draw the diagram and finding the limit).

Thank you again!
Thomas

 

[RUber]

A fixed point is one where f(x)=x. So in an inductive process, you would have x_{n+1} = x_n, and that would then hold true for all future iterations. It is exactly this reason which makes using y=x as the reference line for the spiderweb plot make sense.
There are other proofs for transformations on spaces that talk about how only this in more general terms, but for most functional forms, as long as you choose an appropriate initial value you will approach a fixed point if one exists through iteration.