Explaining Catastrophe Theory for an Infinite 2D Body

[boatzanshoe]

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

...

 

[Hurkyl]

Please don't duplicate post!

 

[Theelectricchild]

More importantly, please explain your reasoning first, so we can pinpoint where you got stuck!

 

[saltydog]

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.

 

[boatzanshoe]

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!

 

[saltydog]

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:

\frac{dy}{dx}=c+ky-y^3

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?

 

[saltydog]

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?".