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Lorentz attractors and fractals

  1. Jul 14, 2010 #1
    Hello,
    as far as I know a "fractal", by definition should manifest self-similarity or at least statistical self-similarity. This usually takes the form of scale invariance.
    Can anyone point out where is the self-similarity in the plots of Lorentz attractors?

    Thanks.
     
  2. jcsd
  3. Jul 14, 2010 #2

    Pythagorean

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    Viswanath, D. (2004) The fractal property of the Lorenz attractor. Physica D, 190: 115–128.

    http://www.math.lsa.umich.edu/~divakar/papers/Viswanath2004.pdf [Broken]
     
    Last edited by a moderator: May 4, 2017
  4. Jul 15, 2010 #3
    ...I still have troubles understanding where is the self-similarity, especially the scale-invariance.
    I understand that "zooming" into one point will reveal more and more orbits (infinitely many). Still I don´t see how that is similar to the whole.

    Referring to the article you mentioned: where is the resemblance of the plots in figure 2 (a part) with the plot in figure 1 (the whole) ?

    Any hint?
     
    Last edited: Jul 15, 2010
  5. Jul 15, 2010 #4

    Pythagorean

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    You zoom in, you see a pile of sticks. But if you zoom in on each of those piles of sticks, they're piles of sticks... but each of those piles of sticks are piles of sticks.

    You can't see the whole orbit at once because the semimajor axis (approximating it as an oval) is huge compared to the thickness of the "sticks", so we're forced to look at little sections of the orbits, that cuts off at each end, making it look like... well, a pile of sticks.
     
  6. Jul 15, 2010 #5
    Thanks.
    Now it is clear.
     
  7. Jul 15, 2010 #6

    Pythagorean

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    I meant that you zoom in on a stick and it's really a bunch of sticks, them you zoom in one of those sticks and it's really a bunch, etc. But hopefully you saw past my redundancy.
     
  8. Jul 15, 2010 #7
    Yes. don´t worry. It was pretty clear to me that you meant to zoom in on one "stick". The explanation was clear.
     
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