question about lorenz attractor

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Offline ker2x

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« 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


Offline claude

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« 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 :D

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...

Offline v

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« 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.

Offline ker2x

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


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