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Continuous Logistic Map

  1. Feb 12, 2006 #1
  2. jcsd
  3. Feb 13, 2006 #2

    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.
     
  4. Feb 13, 2006 #3

    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.
     
  5. Feb 14, 2006 #4
    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.
     
  6. Feb 14, 2006 #5

    saltydog

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    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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