Explaining Catastrophe Theory for an Infinite 2D Body

  • Thread starter boatzanshoe
  • Start date
  • Tags
    Theory
In summary: No prob. Here's the plan. You can also "move" the vase along the edge of the table. As you do so, the plot goes from the top fold, down the front fold, up the back fold and down the bottom fold. Can you see the "history" of the vase below?But there's a "problem" with the bottom fold. It's the "catastrophe" fold. The two plots (top fold and bottom fold) meet at the edge of the cusp (the red curve on your plot) and then "go their separate ways" into infinite space. We have a "problem" with the top fold. Did you notice the "
  • #1
boatzanshoe
2
0
Consider an infinite 2D body, bounded by a parabola. It is made of an non-uniform material, so it's center of mass is finite. Initlaly, the body lies on a horizontal line in a stable position, so that the line that connects the center of mass with the bottom point is normal to the boundary. The center of mass starts moving, and the body changes its position. Sometimes, when the center of mass crosses certain line C (shown in red at the picture), the body makes a sharp swing. Explain this phenomenon, and find the line C.

http://www.math.uiuc.edu/~roitman/m...catastrophe.gif


help me out guys, I am really really

...
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
Please don't duplicate post!
 
  • #3
More importantly, please explain your reasoning first, so we can pinpoint where you got stuck!
 
  • #4
boatzanshoe said:
Consider an infinite 2D body, bounded by a parabola. It is made of an non-uniform material, so it's center of mass is finite. Initlaly, the body lies on a horizontal line in a stable position, so that the line that connects the center of mass with the bottom point is normal to the boundary. The center of mass starts moving, and the body changes its position. Sometimes, when the center of mass crosses certain line C (shown in red at the picture), the body makes a sharp swing. Explain this phenomenon, and find the line C.

http://www.math.uiuc.edu/~roitman/m...catastrophe.gif


help me out guys, I am really really

...

Your website is not connecting so I can't see the problem but what I've often found in Catastrophe Theory is the dynamics can be reduced to how the zeros of a polynomial abruptly change as the plot is "shifted" up or downward. I wouldn't be surprised, even without looking at the problem, if it can be reduced to this phenomenon.
 
Last edited by a moderator:
  • #5
saltydog said:
Your website is not connecting so I can't see the problem but what I've often found in Catastrophe Theory is the dynamics can be reduced to how the zeros of a polynomial abruptly change as the plot is "shifted" up or downward. I wouldn't be surprised, even without looking at the problem, if it can be reduced to this phenomenon.


https://netfiles.uiuc.edu/phsu2/shared/catastrophe.gif?uniq=wdarem


im sorry about the double post.

to tell you the truth, i have no idea where to start. my teacher kind of just threw this project at me.

are there some websites that might help me understand the catastrophe theory better? I've look at a lot of them, but i can't seem to understand much of it.

thank you for your help!
 
Last edited by a moderator:
  • #6
boatzanshoe said:
https://netfiles.uiuc.edu/phsu2/shared/catastrophe.gif?uniq=wdarem


im sorry about the double post.

to tell you the truth, i have no idea where to start. my teacher kind of just threw this project at me.

are there some websites that might help me understand the catastrophe theory better? I've look at a lot of them, but i can't seem to understand much of it.

thank you for your help!

That's got cusp catastrophe written all over it. The red plot (cusp) is the bifurcation set but that's not helpful is it? I tell you what, the best way to study the cusp catastrophe is to study the following cubic differential equation:

[tex]\frac{dy}{dx}=c+ky-y^3[/tex]

In that case, we study the abrupt changes that the roots of a cubic polynomial undergo as the plot is shifted up or down, you know, from 1 to 1 double+another, to three and back again.

Also, check out Saunders, "An Introduction to Catastrophe Theory". Rene' Thom is the father of such. Please, allow me to quote a profound statement he made:

"all creation or destruction of forms or morphogenesis, can be described by the disappearance of the attractors representing the initial forms, and the replacement by capture by the attractors representing the final forms".

The changes are catastrophic . . . you know, the straw that breaks the camel's back. Lots of things in nature are like that right?
 
Last edited by a moderator:
  • #7
Here's the cusp catastrophe. Think about being on the upper fold of the surface and moving to the left. Eventually you fall off and end up on the bottom fold. The points on top where you fall off are the bifurcation points. A 2-D plot of those points is the bufurcation points (red diagram in your figure).

The top surface represents "stable states" like a vase on the top of a table that you move about on the table. Nothing much happens. However, if you move the vase to the very edge of the table. It's now on it's bifurcation curve. Moving it ever so slightly and it will "traject" abruptly and qualitatively change states from being a stable vase on a table to a broken one on the floor.

In the language of Rene' Thom, the table surface is a basin of attraction for the stable state of the vase on the table. Pushing it past it's bifurcation point, and it moves into the basin of attraction of the floor attractor and undergoes "morphogenesis" in passing through the bifurcation point to the new stable state (analogous to the bottom fold of the cusp).

Yea, I know what you're thinking, "nevermind Salty, anyone else up there?".
 

Attachments

  • cusp.JPG
    cusp.JPG
    16.2 KB · Views: 389

Related to Explaining Catastrophe Theory for an Infinite 2D Body

1. What is catastrophe theory?

Catastrophe theory is a branch of mathematics that studies the sudden and unexpected changes that can occur in complex systems, such as the behavior of an infinite 2D body. It seeks to understand how small changes in a system's inputs can lead to large and sudden changes in its output.

2. How does catastrophe theory apply to an infinite 2D body?

An infinite 2D body is a hypothetical object with infinite length, width, and depth. Catastrophe theory can be used to explain how this body would behave under different inputs, such as external forces or changes in its shape or position.

3. What is the main principle of catastrophe theory?

The main principle of catastrophe theory is that small changes in a system's inputs can lead to large and sudden changes in its output. This is known as the "butterfly effect," where a small change in one part of a system can have a significant impact on the overall behavior of the system.

4. What are the main factors that can cause a catastrophe in an infinite 2D body?

There are several factors that can cause a catastrophe in an infinite 2D body, including changes in external forces, changes in the body's shape or position, and interactions with other bodies or systems. These factors can combine in complex ways to create sudden and unexpected changes in the behavior of the body.

5. How is catastrophe theory useful in scientific research?

Catastrophe theory is useful in scientific research because it can help us understand and predict the behavior of complex systems, such as an infinite 2D body. By studying how small changes in a system's inputs can lead to large changes in its output, scientists can better understand and control these systems for various applications, such as in physics, biology, and economics.

Similar threads

  • Introductory Physics Homework Help
2
Replies
36
Views
510
  • Special and General Relativity
Replies
10
Views
892
  • Quantum Interpretations and Foundations
2
Replies
41
Views
3K
  • Special and General Relativity
Replies
14
Views
2K
Replies
11
Views
2K
Replies
20
Views
2K
  • Math Proof Training and Practice
2
Replies
46
Views
5K
  • Special and General Relativity
3
Replies
75
Views
4K
  • Introductory Physics Homework Help
Replies
1
Views
2K
Replies
12
Views
6K
Back
Top