
Interactive Lorenz Attractor - webdva
http://www.malinc.se/m/Lorenz.php
======
rsiqueira
There is a collection of animated Lorenz Attractors with interactive code
created with 140 (or less!) characters of JavaScript:
[https://www.dwitter.net/h/lorenz](https://www.dwitter.net/h/lorenz)
Disclaimer: I created some of them :-)

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mikimac
Have you seen this? It's amazing: glChAoS.P / wglChAoS.P real time 3D strange
attractors GPU explorer:
[https://www.michelemorrone.eu/glchaosp/dtAttractors.html#Lor...](https://www.michelemorrone.eu/glchaosp/dtAttractors.html#Lorenz)
Recent "Simulating comet's journey in Lorenz" post on HackerNews:
[https://news.ycombinator.com/item?id=21945598](https://news.ycombinator.com/item?id=21945598)

~~~
BrutPitt
Thank you for appreciation.

The correct glChAoS.P / glChAoSP and wglglChAoS.P / wglChAoSP website is:

[https://www.michelemorrone.eu/glchaosp/](https://www.michelemorrone.eu/glchaosp/)

glChAoSP /wglChAoSP is RealTime 3D strange attractors GPU explorer and
hypercomplex fractals - over 200 Million particles.

It's freeware, Open Source, and both: native Multi Platform and WebGL... and
display not only Lorenz Attractor, but over 100 object types between
Attractors, Hypercomplex Fractals, and DLA/DLA3D (Diffusion Limited
Aggregation)

Thanks again.

BrutPitt / Michele Morrone

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seism
I <3 generative art - this is really well done, the detailed explanations
appreciated. And I can especially enjoy this after playing through Everybody's
Gone to the Rapture
[https://en.wikipedia.org/wiki/Everybody%27s_Gone_to_the_Rapt...](https://en.wikipedia.org/wiki/Everybody%27s_Gone_to_the_Rapture)

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nmc
Excellent!

Just one thing: on a touchscreen interface with Firefox, zooming out works
well but zooming in is sometimes jittery or not registering at all. (Panning
and rotating work fine.)

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m4r35n357
Nice visualization (particularly the butterflies)! If the OP is here, what ODE
solver are you using BTW?

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TheGallopedHigh
Not OP but a real common one is Runge Kutta 4; it’s a balance between
complexity and stability of solution.

[https://en.m.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods](https://en.m.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods)

~~~
ChrisRackauckas
There is no advantage to using a better ODE solver on this example because the
sensitivity to errors is so high. You can measure this using the uncertainty
quantification methods in the Julia ODE solver suite:

[https://docs.juliadiffeq.org/latest/analysis/uncertainty_qua...](https://docs.juliadiffeq.org/latest/analysis/uncertainty_quantification/#Example-3:-Adaptive-
ProbInts-on-the-Lorenz-Attractor-1)

where divergence on the Lorenz attactor tends to occur by t=80 or so even with
accuracy of 1e-16.

But the funny thing about chaotic problems is that the shadowing theorem
holds, which is:

>Although a numerically computed chaotic trajectory diverges exponentially
from the true trajectory with the same initial coordinates, there exists an
errorless trajectory with a slightly different initial condition that stays
near ("shadows") the numerically computed one.

So you might as well just use Euler's method with high error, because it's
backwards stable for this calculation, i.e. it gives a trajectory on the
attractor, just the wrong trajectory. But since every method gives an O(1)
error wrong trajectory after a short finite time, you might as well use the
cheapest most error prone but convergent method.

~~~
m4r35n357
While we are both here, if you are not aware of this paper it might give you a
giggle (3500th order simulations on a supercomputer):
[https://arxiv.org/abs/1305.4222](https://arxiv.org/abs/1305.4222)

They simulated Lorenz reliably up to 10000 time units; I managed 1400 units
using MPFR on my NUC using my own code (500th order, took about 13 hours!).

~~~
ChrisRackauckas
Haha great! This is 100% a "might as well put it on Arxiv since no one
reviewer would ever see this as significant" lol. It's at least very fun. I'm
going to have to save this one.

