Integrating Swallowtail Catastrophe Theory & Population Dynamic Equation

[zhouyang]

Hi all,
I am new to this forum. I am requesting all, please if you can help me. I want to integrate the Swallowtail catastrophe theory into the population dynemic equation. Anybody who knows about swallowtail catastrophe theory as well as population dynemic equation, please reply me. Or if anybody knows about web resources on these topics please send me.
Thanks!

 

[jackmell]

Can you post your particular DE?

 

[zhouyang]

yes,
i attached it as jpeg file
Thanks

 

[jackmell]

Ok. Looks interesting but unfortunately I don't have time right now to work with it. May I suggest taking a look at "An Introductioon to Catastrophe Theory" by Saunders and try and adapt your equation to the canonical version of the swallowtail:

$$\frac{dx}{dt}=5x^4+3ux^2+2vx+w$$

I did notice when you put yours over a common denominator, the numerator is a quartic but it includes a cubic term which the canonical swallowtail does not include. Not sure how that would effect the bifurcation set. So the general procedure is to then take the derivative of the RHS, then set the RHS and it's derivative to zero and then eliminate x from these two expressions. This then gives an implicit equation in u, v, and w. That surface is the swallowtail bifurcation set. However, your equation has more than three parameters. Not sure about this also but I would start by trying to fit your equation to the canonical version even if I have to simplify it or constrain it.

 

[zhouyang]

Dear jackmell,

Thank you very much. I will go through it. I got the said book from google, but it has only 23 pages. Anyway I would find some other notes, and try it. Your instructions really help me and I hope, if you could do it please post.
Cheers!

 

[jackmell]

Ok, your equation will take quite a bit of time to fully investigate. The best approach is to first spend some time with the canonical cusp catastrophe:

$$\frac{dx}{dt}=4x^3+2ux+v$$

get that one conceptually straight, then spend time with the canonical swallowtail, then adapt your equation to "fit" the canonical form. I' talkin' weeks for that but it's a very interesting field for me, answers many questions about the world in my opinion, and maybe when I'm done with some work I'm working on now, I'll go back and spend some time with your equation but don't wait for me. You try doing this now: just put up your equation for now, and just study the cusp.

 

[zhouyang]

Dear jackmell,

Thank you very much!