Continuous Logistic Map (1 Viewer)

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HallsofIvy

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

saltydog

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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:

[tex]x^{'}=-(y+z)[/tex]

[tex]y^{'}=x+ay[/tex]

[tex]z^{'}=b+xz-cz[/tex]

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.
 
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

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Tom Piper said:
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.:rolleyes:
 

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