Exploring the Meaning of Cobweb Diagrams

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In summary, the conversation discusses the use of Cobweb diagrams in recursive sequences and their connection to fixed points. The diagram is used as a visual aid to help bring the sequence back to the x-axis and it is chosen based on the concept of a fixed point, where the function equals its own output. The importance of fixed points in the study of functions is also briefly mentioned.
  • #1
mr.tea
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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 [tex]\frac{1}{2-\frac{1}{2-...}}[/tex]

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
 
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  • #2
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
 
  • #3
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.
 
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  • #4
Yes, the concept of "fixed point" is what you need. If you have the recursion [itex]x_{n+1}= \frac{1}{2- x_n}[/itex], [itex]x_0= 1/2[/itex] we can, temporarily, assume that this sequence has a limit. If that is true then taking the limit, x, on both sides, [itex]x= \frac{1}{2- x}[/itex] so that [itex]x(2- x)= 2x- x^2= 1[/itex] or [itex]x^2- 2x+ 1= 0[/itex], 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.
 
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  • #5
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
 
  • #6
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.
 
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1. What are cobweb diagrams and how are they used in scientific research?

Cobweb diagrams, also known as spider diagrams, are graphical representations of complex systems or relationships. They are commonly used in scientific research to visualize and analyze data, patterns, and connections between different variables or factors.

2. Can cobweb diagrams be used in any field of science?

Yes, cobweb diagrams can be used in various fields of science, including biology, physics, chemistry, economics, and social sciences. They are particularly useful for studying complex systems and relationships in these fields.

3. How do cobweb diagrams differ from other types of diagrams?

Unlike traditional line graphs or bar charts, cobweb diagrams do not show specific values or numbers. Instead, they focus on the relationships between variables or factors and illustrate how they change over time or under different conditions. They also allow for a more holistic view of the system or relationship being studied.

4. What are the benefits of using cobweb diagrams in scientific research?

Cobweb diagrams offer several benefits in scientific research. They provide a visual representation of complex systems, making it easier to understand and analyze data. They also allow for the identification of patterns and trends within the data. Additionally, cobweb diagrams can help scientists make predictions and test hypotheses about the system or relationship being studied.

5. Are there any limitations to using cobweb diagrams?

While cobweb diagrams can be a useful tool in scientific research, they also have some limitations. They may oversimplify complex systems or relationships, and their interpretation may be subjective. Additionally, they may not be suitable for representing certain types of data or relationships. As with any scientific tool, it is important to consider the limitations and potential biases when using cobweb diagrams.

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