Lorentz attractors and fractals

[mnb96]

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.

 

[Pythagorean]

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

 

[mnb96]

...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?

 

[Pythagorean]

...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?

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.

 

[mnb96]

Thanks.
Now it is clear.

 

[Pythagorean]

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.

 

[mnb96]

Yes. don´t worry. It was pretty clear to me that you meant to zoom in on one "stick". The explanation was clear.