The "Lorenz Set"?

[Ode_to_dx]

Hi everyone. I have been thinking of this for some weeks so I decided to ask here.
So I coded* the Lorenz Attractor and when I spawn a particle (i.e. I set the starting conditions x,y,z and use the euler method to get the next position, eventually drawing an orbit) near the origin everything is fine.
However, when I spawn a particle too far away from the origin it seems to diverge towards infinity instead of orbiting around the two "butterfly wings".

My question is, if we kept track of all the points that don't diverge, would the resulting 3d shape be a fractal?

It makes sense since orbits are chaotic. The Mandelbrot set is created like this, except that it uses a function that tells you the next "position" of the original number or "particle" instead of a differential equation that tells you the "speed" of that particle, so the accuracy of the render will be determined by the chosen step size when numerically integrating the equations, not only how many times you iterate (in this case how many steps you take).

Finally, does this mean that other sets of differential equations can generate fractals too? I have not found examples of this on the internet so I don't know if this is a dumb idea or a new branch of fractals yet undiscovered.

*I implemented it in Scratch, a visual programming language (you drag and drop blocks), but rendering the actual thing in this browser-based program would take forever so I am learning c++ to try to render the actual thing. Anyways if someone knows a mathematical approach to determining wether or not this set of points will be a fractal or a way to code this feel free to research this topic, I lack the knowledge to make any breakthrough out of this.

[HazardHarry]

I'm no expert in attractors, but I did some googling and I think what you're looking for are called "Basins of Attraction".

http://www.scholarpedia.org/article/Basin_of_attraction
"For each such attractor, its basin of attraction is the set of initial conditions leading to long-time behavior that approaches that attractor. Thus the qualitative behavior of the long-time motion of a given system can be fundamentally different depending on which basin of attraction the initial condition lies in (e.g., attractors can correspond to periodic, quasiperiodic or chaotic behaviors of different types)."

http://sprott.physics.wisc.edu/technote/tristab.htm
The examples here are only in 2D planes. I would be very interested to see their structure in 3D.

[JosLeys]

Points should not diverge to infinity. If they do, your calculation method is too coarse.
The set of equations defines an attractor: points converge to it.
Try a Runge-Kutta method instead of Euler.

And I see no connection to fractals here...

[Chillheimer]

And I see no connection to fractals here...

If this is because points don't escape to infinity, ok.

Else, how can you see no connection to fractals?
"An attractor is called strange if it has a fractal structure" wikipedia

[JosLeys]

Well, we could have a long boring debate about the meaning of the word fractal, but I suggest we don't start that!

[DarkBeam]

Then start a short debate... 
Seriously it is an interesting topic because chaos is a big family

[Sockratease]

Then start a short debate...  ...

Short debate?

OK...

No.

Nothing is truly Fractal since there is no firm definition for the word!

I know...

I've been saying that for years, but it's partly to encourage the effort to devise a Universally Accepted definition of Fractal so such debates can be settled.

And partly because I still believe Fractals don't exist outside of our imaginations.

[Chillheimer]

Well, we could have a long boring debate about the meaning of the word fractal, but I suggest we don't start that!

 

(but I'll be offline the next 3-4 days)

[vinecius]

Interesting idea, I decided to plot it with the conventional attractor on (x,y,z) using Euler's method and an escape time algorithm on (x,y) with a random z initial condition. first is 100 iterations, second 1000.  Might be some fractal like structures near the edges.  I'll play with it more, find the best parameters and zoom in on it. Interesting structure but I don't see it growing in complexity, it's a similar idea to the newton fractal.

[vinecius]

So the spikes on the edges which result from a large step size with Euler's method are recursive when zoomed upon, the same spiky pattern repeats forever, much like the spirals on a madelbrot.  The solid colored bits in the middle do not grow in complexity when zoomed on.  Not terribly interesting, but "fractal" i suppose.

The plot is generated by seeing how fast it takes a point on the screen (x,y) to get to a certain distance to the center (between the attractors, not the centers of the attractors themselves, which is why you're seeing the two disembodying lobes).  All points should eventually reach the center (the transition between one attractor to the other) but black colored are those that fail to under the amount of iterations.

edit: upon closer inspection, this thing does grow with complexity, it's the same swirly sort of look but the spiked shapes seem to change when you zoom in to certain random regions.

[Ode_to_dx]

Wow thanks to everyone. This is my first post in this forum and I am surprised by how quickly people replied. Sorry I was not present I have been on holidays and I thought I would get notified by email when someone replied (now I found the setting).

Points should not diverge to infinity. If they do, your calculation method is too coarse.
The set of equations defines an attractor: points converge to it.

That's right I should have realised it was because of the step size, which was pretty big due to the lack of power of Scratch.

So the spikes on the edges which result from a large step size with Euler's method are recursive when zoomed upon, the same spiky pattern repeats forever, much like the spirals on a madelbrot.  The solid colored bits in the middle do not grow in complexity when zoomed on.  Not terribly interesting, but "fractal" i suppose.

The plot is generated by seeing how fast it takes a point on the screen (x,y) to get to a certain distance to the center (between the attractors, not the centers of the attractors themselves, which is why you're seeing the two disembodying lobes).  All points should eventually reach the center (the transition between one attractor to the other) but black colored are those that fail to under the amount of iterations.

edit: upon closer inspection, this thing does grow with complexity, it's the same swirly sort of look but the spiked shapes seem to change when you zoom in to certain random regions.

Since points are not supposed to diverge, it is certainly a good idea to use a scape time algorithm, thanks for implementing it. I think what causes that "swirl" when you zoom in is sensitivity to initial conditions: as orbits go on, some of them are more "straightforward" than others, so the valleys and mountains represent the points that take the fastest and slowest (respectively) paths.
If this was the case, that would explain why you have to zoom in (at first points that are close to one another don't spread much), but if you recorded the 2nd, 3rd, etc. time that points reach the center instead of the first one, the orbits may have more time to spread and sensitivity to initial conditions would cause this differences to show up exponentially earlier (i.e. at a larger scale).