# Is this a new strange attractor?

I recently made some experiments in programming strange attractors, and I found this (very simple) equations, which create a nice strange attractor:

xn=x+dt*(z-y)
yn=y+dt*(x/2-1)
zn=z+dt*(-xy/2-z)


You can see it in action on my Youtube channel: https://youtu.be/Bm_M6mUGjtg

My question: Is this a variation of the Lorenz- or Rössler attractor - or did I stumble upon something new?

EDIT:

Meanwhile I programmed a little 3D View for this attractor:   You can see/move the view on this applet (Java needed): https://cerumen.de.cool/attractor/index.html (Here you also find the source code for Processing)

And here the Javascript-Version: https://cerumen.de.cool/attractor/js/index.html (with processing.js... bit slow)

Perhaps my question was also asked too amateurishly. I was simply surprised by the simplicity of the system of equations I have found.

Therefore I would like to know if this strange attractor is a descendant of one of the well known ones (Lorenz / Rössler).

Edit 2:

I have now brought the system of equations into a more general form:

xn=x+dt*(z-y)
yn=y+dt*(ax-b)
zn=z+dt*(-axy-z)


with a in range [0 to 1], b in range [0.5 to 1]

This makes it more interesting. Here some sample images for different values for a and b:

Edit 3:

Here a video with the generalized equations and constantly changing parameters a and b: https://youtu.be/gxusM8pmNwU

I think you can see here quite well how the system goes from order through bifurcation into chaos...

• I believe it is not that difficult to cook up new fractals - there is plenty of software products out there that does this, Flamethyst for example (although the method is a bit different). Feb 18 '20 at 8:10
• It's certainly very close to the Rossler system: two coordinates with linear differential equations, one 'slightly' nonlinear one, with a quadratic perturbation term. While the specific form of the linear piece is slightly different, I wouldn't be surprised if the two are fundamentally equivalent in some fashion. Feb 18 '20 at 20:29
• Thank you, Steven, I kind of suspected that too. But so far I have not found a way to transform the Rössler equations into my system of equations. I think this is the core of my question. Feb 18 '20 at 20:43
• As Per Alexandersson suggests, strange attractors are actually very common for nonlinear systems in three variables. You can literally make up new ones just by making up a system of nonlinear differential equations and then using some graphing software to check whether it has a strange attractor. There are certain "famous" strange attractors which are famous mostly for historical or scientific reasons, but there are lots of unnamed strange attractors associated with various systems. This is a particularly nice one and you've made some great pictures of it, but it's not really very novel. Feb 19 '20 at 11:42
• Thank you, @Jim, for your answer. It seems that the answer to my question is: This is just another new strange attractor. But I still love the simplicity of the equation system, which really surprised me when I programmed this thing. Feb 19 '20 at 12:42

I just figured I added some Mathematica code and a picture for the attractor in the question. With[{dt = 0.001},
iter[{x_, y_, z_}] := {x, y, z} +
dt {(z - y), x/2 - 1, -x y/2 - z}
];
pts = NestList[iter, {0.1, 0.1, 1/2}, 500000];
ListPlot[{#1, #2} & @@@ pts[[1 ;; ;; 5]], PlotRange -> All,
Axes -> False, PlotStyle -> {Opacity[0.9], PointSize[Tiny], Orange},
AspectRatio -> 1, Background -> Gray, ImageSize -> {800, 600}]


I only plotted every fifth of the points, as it is a bit quicker, and the image is a bit more pleasant with this variant.

EDIT:

With some more creative edits, $$(x_{n+1}, y_{n+1}, z_{n+1}) =(x_n, y_n, z_n)+dt (z - y, -1 + x + 6 \sin(\pi/4 + 10 x/ z), -x y/2 - z)$$ one can produce the following picture. Adding any non-linear disturbance, and making sure that it does not diverge, or converge to something boring, it is rather easy to cook up exotic variations that give rise to chaotic behavior. • Per, thank you for your detailed explanations. But my question is not how to create chaotic systems with increasing complexity. What I want to know is whether my really simple system of equations is some sort of derivative of already known attractors. Feb 18 '20 at 23:07

In another forum a user drew my attention to the publication of

This shows that there are many very simple chaotic systems of equations.

I guess this answers my question...

You can see some of the equations here, and here are the corresponding graphs.

But thank you all for your interest.