Can Quotient Spaces Show Chaotic Behavior?

[Tom Piper]

The logistic equation;
x(n+1) = rx(n)(1-x(n))
shows chaotic behavior under some values of r. It is a discrete chaotic system. Is it possible to translate it to continuous chaotic system? I tried it in the following site;
http://geocities.com/tontokohirorin/mathematics/moduloid/moduloid2.htm

 

[HallsofIvy]

The corresponding continuous system would be.
$$\frac{dx}{dt}= rx(1- x)$$
where x is a continuous function of t.

 

[saltydog]

I wish to clarify something: The continuous counterpart of the logistic equation is not chaotic. It turns out that in order to find chaos in continuous systems, we need to consider at least a three-dimensional system. Such as the Rossler System:

$$x^{'}=-(y+z)$$

$$y^{'}=x+ay$$

$$z^{'}=b+xz-cz$$

or the Lorenz system.

Hey Tom, have you ever studied these two systems? Have you drawn a Feigenbaum plot for either? I hope you have Peitgen's book, "Chaos and Fractals". That's a good reference.

 

[Tom Piper]

The "translation" I got is;
dx/dt = 0
dy/dt = 1
(Do you think they are too simple?) Although the solution of above differential equations is a line in two dimensional Euclidean space, the solution in the following quotient space;
http://geocities.com/tontokohirorin/mathematics/moduloid/fig12.jpg
is thought to have chaotic behaviour.

 

[saltydog]

The "translation" I got is;
dx/dt = 0
dy/dt = 1
(Do you think they are too simple?) Although the solution of above differential equations is a line in two dimensional Euclidean space, the solution in the following quotient space;
http://geocities.com/tontokohirorin/mathematics/moduloid/fig12.jpg
is thought to have chaotic behaviour.

Yep, yep, that's not happening for me. When in doubt . . . Mathworld . . . Quotient Space . . . still didn't happen for me. You got me Tom. And that diagram your cited, what is that? Anyway, I'm not clear at all what a quotient space is and how it can model chaotic behavior. Might you provide a concrete example with pictures. Yea, pictures . . . that would be nice.