### Lyapunov diagrams

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#### pauldelbrot #### Re: Lyapunov diagrams

« Reply #75 on: February 17, 2019, 03:01:29 PM »
The whole thing is symmetric in the bottom-left-to-top-right diagonal, so if one allows rotations and reflections, there are pairs of identical cards.

#### marcm200 #### Re: Lyapunov diagrams

« Reply #76 on: February 17, 2019, 03:12:25 PM »
Thanks! Should've seen it myself. But I was so concentrated on rotating individual tiles in my mind (and got stuck) that I guess, it was the thing with the forest and the tree.

#### marcm200 #### Re: Lyapunov diagrams

« Reply #77 on: February 19, 2019, 11:18:54 AM »
My girlfriend suggested to use the branch heat value as a filter for the sectionally defined images. I took a look at that and found quite an interesting example. Taking only those values into the final image that spent a specific amount of time in the ELSE-part of their computation (and putting the rest to black) changed the nature of the image quite strongly. One can get a closer look at the edges of some of the creatures - and even new structures appear (lower left corner) which haven't been prominent in the full colour scheme. I guess it's the 2D equivalent of the 3D carving I use.

x0=0.5
Trajectory function f(x):
Initial 500 iterations: if 0.3 <= x <= 0.8: f(x)=-2.5*sin²(x+r) else f(x)=-.5*sin(x+r*cos(x+r)
Computing 1000 iterations: if -1.00000 <= x <= 0.00000: same
Sequence AABB. Center (0.00/0.00) size=12

#### marcm200 #### Re: Lyapunov diagrams

« Reply #78 on: February 22, 2019, 10:31:27 AM »
Here's a nice example of the power of sectionally defined functions. Although the function used spent less than 0.1% in the ELSE-part, the resulting image proved to be quite different from one only deriving from the THEN-part (logistic equation). I doubt that those 0.1% have much impact on the actual summation value, but I suspect that they change the trajectory of the x values ccmpletely. So in one sense, the function is dependent on its starting value x0. Usually I don't see much influence there (I'm not examining it everytime I use a new function), but here clearly it has some major impact.

Has anyone taken a closer look into the x0-dependency of functions and has come to some genereal findings?

I'm currently checking whether x0 can be used as a third axis to generate 3D Lyapunov spaces.

x0=0.5
Trajectory function f(x):
Initial 500 iterations: if -1 <= x <= 0: f(x)=r*x*(1-x) else f(x)=-2.75*sin(x)*sin(x+r)*cos(-2.75*x)*cos(r*x)
Computing 1000 iterations: if 0 <= x <= 1: same
Sequence AABB. Center (3.50/3.50) size=0.73

#### marcm200 #### Re: Lyapunov diagrams

« Reply #79 on: February 25, 2019, 11:09:36 AM »
"Let's play: I spy with my little eye..."

Often one can tell by a look on the image, when the function used was sectionally defined. There are some sudden direction changes of structures, geometric shapes appearing out of nowhere (like the Vienna image in the gallery), but here's an image that does not show any aberration. It looks like some orange-y creatures on a plane. But when I looked at the branchheat histogram, I was surprised to see that two of the central structures were actually mainly derived from the ELSE-part - opposite to all the others. It's quite interesting how big the range of changes is, sectionally defined functions show.

x0=0.5
Trajectory function f(x)
Initial 500 iterations: if -1 <= x <= -0.4: f(x)= exp(-2.25*sin(x+r)) else f(x)= -2.25*atan((x+r)*sin(x+r))
Computing 1000 iterations: if -0.4 <= x <= 0: same
Sequence AABB. Center (-1.31/3.38) size=0.75

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