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ker2x

• Fractal Friend
• Posts: 11

« on: February 11, 2018, 01:43:08 AM »
I watche a great video about the lorenz attractor that pretty much explain everything : https://www.youtube.com/watch?v=YS_xtBMUrJg

and i have a few questions.
First, my understanding of the whole stuff :

I was never interested in it because i don't find it aesthetically pleasing.
Never coded anything, never really played with it.

So, you have a few initial condition dx, dy, dz, R.
Depending of R you can have : 1 attractor, 2 attractor, periodic.
In some condition the system stabilise, and in some other it never stabilise.

In case of 2 attractor that stabilize, but value of R still low :
If you start on one side you end up on the attractor of the same side.
same for the other.
And there is a messy boundary inbetween.

With higher value, it become seriously chaotic and the initial condition can be "extremely sensible".

My question : is there a map available ?

eg : a 2 color map (one color per attractor) that show where it will stabilise (which attractor) depending on initial condition ?
is it a nice fractal ? does it look like a known fractal ? (my wild guess is : newton fractal ?)

It's inherently 3D since there is 3 parameters in the initial condition. or even 4 if you consider R.

Thx. if you don't understand my question please tell. it's still a bit messy in my head to be honnest

claude

• Fractal Freak
• Posts: 756

« Reply #1 on: February 13, 2018, 12:21:41 AM »
https://en.wikipedia.org/wiki/Lorenz_system

It has three variables (x, y, z) which change over time and three constants (sigma, rho, beta).  So maybe it's 6D, 7D including time

Traditionally visualisations fix the constants (sigma, rho, beta) and pick a particular starting value (x, y, z) and plot the evolution of that orbit over time, perhaps plotting a few nearby (x, y, z) to show the sensitivity to initial conditions that is characteristic of chaos and failure of long-term prediction.

I guess you are asking about a paramter-space plot (visualizing changes in (sigma, rho, beta)) as opposed to a phase-space plot (visualizing changes in x,y,z over time).

https://en.wikipedia.org/wiki/Lorenz_system#Analysis has some interesting formulas, maybe you could try to visualize isosurfaces of the "Lyapunov dimension (Kaplan-Yorke dimension)" in (sigma, rho, beta) space...

v

• Fractal Fanatic
• Posts: 26

« Reply #2 on: February 15, 2018, 12:13:57 AM »
My question : is there a map available ?

eg : a 2 color map (one color per attractor) that show where it will stabilise (which attractor) depending on initial condition ?
is it a nice fractal ? does it look like a known fractal ? (my wild guess is : newton fractal ?)

I've plotted something similar that is assigning a color to how long it takes for a point (x,y) in (x,y,z) for arbitrary random z to reach the center (not center of the attractor but rather the origin) computed with a crude euler's method and got some interesting fractal-like results.

ker2x

• Fractal Friend
• Posts: 11

« Reply #3 on: February 19, 2018, 09:19:46 PM »
that's close to what i was thinking indeed

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