Dynamical Systems Examples and Tips

Here are some belated examples and a few tips...

The first example is a good starting point for exploration. Note that the Drawing Method on the Formula tab is set to One-pass Linear and that Repeat Gradient on the Outside tab is un-checked. The gradient is greyscale. You can try other types of gradients but the results will vary.

20181013aDynamicalSystemsExample#1 {
fractal:
  title="20181013a Dynamical Systems Example #1" width=800 height=800
  layers=1 credits="Mark Townsend;10/13/2018"
layer:
  caption="Background" opacity=100 method=multipass
mapping:
  center=0/0 magn=0.15
formula:
  maxiter=100 filename="mt.ufm" entry="mt-pixel" p_inside=no
inside:
  transfer=none
outside:
  transfer=linear repeat=no filename="mt5.ucl"
  entry="mt-dynamical-systems-r" p_sample_mode=Grid
  p_sample_center=0/0 p_sample_width=20 p_h=0.1 p_max_iterations=50
  p_resolution=500 p_circular_mask=no p_dual_functions=no
  p_XFuncParam="mt.ulb:MT_VineFunction" p_XFuncParam.flavor=-1
  p_XFuncParam.a=7 p_XFuncParam.b=2.0
  p_YFuncParam="mt.ulb:MT_DynamicalFunction" p_add_noise=no
  p_noise_amount=1 p_noise_scale=1 p_noise_texture.v_generic=100
  p_noise_texture.v_coloring=100
  p_noise_texture.v_gradientcoloring=100
  p_noise_texture.p_texture="mt.ulb:MT_PerlinNoiseTexture"
  p_noise_texture.p_texture.v_generic=100
  p_noise_texture.p_texture.v_trapshape=101
  p_noise_texture.p_texture.octaves=7
  p_noise_texture.p_texture.persistence=0.5
  f_noise_texture.p_texture.fn1=ident
  p_noise_texture.p_texture.seed=1234567 p_add_transform=no
  p_transform="common.ulb:UserTransform" p_transform.v_generic=100
  p_transform.v_transform=100 p_transform.v_usertransform=100
  p_seed=1234567
gradient:
  comments="Use with Lighting coloring algorithm." smooth=yes index=0
  color=16777215 index=399 color=0 index=399 color=0
opacity:
  smooth=no index=0 opacity=255
}

For then next one I changed the Vine parameters and zoomed into one of the patterns. Because I'm zooming in I can decrease the Width parameter because I want to concentrate the calculation on the visible area. I copied Center from the Location tab and pasted it into the Center parameter of the coloring. For this one I've changed the Sample Mode to "Random". I've included an image of the Vine fractal formula in the same location so you can see the differences.

20181013bDynamicalSystemsExample#2 {
fractal:
  title="20181013b Dynamical Systems Example #2" width=800 height=600
  layers=1 credits="Mark Townsend;10/13/2018"
layer:
  caption="Background" opacity=100 method=multipass
mapping:
  center=-3.1333333335/-3.09999999985 magn=0.6
formula:
  maxiter=100 filename="mt.ufm" entry="mt-pixel" p_inside=no
inside:
  transfer=none
outside:
  transfer=linear repeat=no filename="mt5.ucl"
  entry="mt-dynamical-systems-r" p_sample_mode=Random
  p_sample_center=-3.1333333335/-3.09999999985 p_sample_width=8
  p_h=0.1 p_max_iterations=20 p_resolution=1000 p_circular_mask=no
  p_dual_functions=no p_XFuncParam="mt.ulb:MT_VineFunction"
  p_XFuncParam.flavor=0 p_XFuncParam.a=7 p_XFuncParam.b=2.0
  p_YFuncParam="mt.ulb:MT_DynamicalFunction" p_add_noise=no
  p_noise_amount=1 p_noise_scale=1 p_noise_texture.v_generic=100
  p_noise_texture.v_coloring=100
  p_noise_texture.v_gradientcoloring=100
  p_noise_texture.p_texture="mt.ulb:MT_PerlinNoiseTexture"
  p_noise_texture.p_texture.v_generic=100
  p_noise_texture.p_texture.v_trapshape=101
  p_noise_texture.p_texture.octaves=7
  p_noise_texture.p_texture.persistence=0.5
  f_noise_texture.p_texture.fn1=ident
  p_noise_texture.p_texture.seed=1234567 p_add_transform=no
  p_transform="common.ulb:UserTransform" p_transform.v_generic=100
  p_transform.v_transform=100 p_transform.v_usertransform=100
  p_seed=1234567
gradient:
  comments="Use with Lighting coloring algorithm." smooth=yes index=0
  color=16777215 index=0 color=16777215 index=259 color=0 index=399
  color=0 index=399 color=0
opacity:
  smooth=no index=0 opacity=255
}

Next I add a Transform. I tend to go straight for my "Trap Shape" transform plug-in because it allows me to load my texture plug-ins and it has a Strength parameter. Notice that I've added a new control point to the gradient and dragged it to the left to darken the image. This one takes a bit longer to appear. That's a combination of the higher Resolution and the fact that the transform is calculated for every iteration.

20181013cDynamicalSystemsExample#3 {
fractal:
  title="20181013c Dynamical Systems Example #3" width=800 height=800
  layers=1 credits="Mark Townsend;10/13/2018"
layer:
  caption="Background" opacity=100 method=multipass
mapping:
  center=-2.1607881435/3.1891277375 magn=0.50420168 angle=44.2834
formula:
  maxiter=100 filename="mt.ufm" entry="mt-pixel" p_inside=no
inside:
  transfer=none
outside:
  transfer=linear repeat=no filename="mt5.ucl"
  entry="mt-dynamical-systems-r" p_sample_mode=Grid
  p_sample_center=0/0 p_sample_width=10 p_h=0.1 p_max_iterations=50
  p_resolution=1000 p_circular_mask=no p_dual_functions=no
  p_XFuncParam="mt.ulb:MT_VineFunction" p_XFuncParam.flavor=-1
  p_XFuncParam.a=7 p_XFuncParam.b=2.0
  p_YFuncParam="mt.ulb:MT_DynamicalFunction" p_add_noise=no
  p_noise_amount=1 p_noise_scale=1 p_noise_texture.v_generic=100
  p_noise_texture.v_coloring=100
  p_noise_texture.v_gradientcoloring=100
  p_noise_texture.p_texture="mt.ulb:MT_PerlinNoiseTexture"
  p_noise_texture.p_texture.v_generic=100
  p_noise_texture.p_texture.v_trapshape=101
  p_noise_texture.p_texture.octaves=7
  p_noise_texture.p_texture.persistence=0.5
  f_noise_texture.p_texture.fn1=ident
  p_noise_texture.p_texture.seed=1234567 p_add_transform=yes
  p_transform="mt.ulb:MT_TrapShapeTransform" p_transform.v_generic=100
  p_transform.v_transform=100 p_transform.v_usertransform=100
  p_transform.mode="Return Value" p_transform.strength=0.03
  p_transform.p_trapshape="reb.ulb:REB_TrapShapeLissajous"
  p_transform.p_trapshape.v_generic=100
  p_transform.p_trapshape.v_trapshape=101
  p_transform.p_trapshape.v_Lissajous=101
  p_transform.p_trapshape.a=0.2 p_transform.p_trapshape.b=1
  p_transform.p_trapshape.c=1 p_transform.p_trapshape.scale=1
  p_transform.p_trapshape.partype="cabs(z)" p_seed=1234567
gradient:
  comments="Use with Lighting coloring algorithm." smooth=yes index=0
  color=16777215 index=146 color=0 index=399 color=0 index=399 color=0
opacity:
  smooth=no index=0 opacity=255
}

For the next one I've checked the Add Noise parameter. Gnarl with Noise is currently the only function plug-in that uses noise. It took me awhile to find something that showed how adding noise is different to adding a transform. If you duplicate the image and un-check Add Noise you'll see that the overall shape doesn't change much but the noise adds a nice texture.

20181013dDynamicalSystemsExample#4 {
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  +WQER6q4KLvBoK90JhjhcphpstofUaFMdiu39YVCuCYGiB6AmDBeW8Kj7rFRe8HjZz+6xMbo
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  5maMTZQDs35pcXVTPTU12rqDm6zDeT/xHRRflZYyQ2Tc7rvZV1nUMj4LauF+4oFRnhNmRzfo
  9x3SzJtnKoVlStc1DFm5F6t+Byzp1aL6yf+kU6QT+BGFIArppS5ZKdH4te4cFTiNmOac5BJW
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  emO2sC==
}

Finally, I left the noise but checked Dual Functions and chose Popcorn for the second function.

20181013eDynamicalSystemsExample#5 {
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  z6OZZyPS+RKSqWLr2zEvb8ICxz9CgmUklvIPLfKQ+4OFTyrZCyN7ceQ6IXtlJNCg8myEyv5N
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  Eipirc8Ggq0jHFPGrjWmy1i+RpVw4R6O/TVJ4KgZJWwAMPC8k4Vm2VLSC4PEzm93FTcx7iJ2
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  LKxm4/Q170tk
}

Nice. Some of them stay as lack field after pasting

Very nice. They need lots of time to open up.

Yes, that's because all the calculation needs to be done in the coloring algorithm. Ultra Fractal can't show you anything until all the work is finished. One of the images I posted on Facebook takes about 40 minutes before anything appears. Work with a low Resolution until you're ready to have a look at a higher resolution render. You probably don't need to increase Iterations all that much, it will mostly affect how much "outgrowth" there is at the edges of the sample area.

Mark,

Is there a reason why the gray density changes from high to low when I put the resolution higher and higher. I suppose it is the same fractal that becomes bigger and bigger without changing in reality.

Yes there's a reason, and I'm sorry if I didn't mention it (I meant to). The image is made up of a fixed number of dots (Resolution squared times Iterations). When you increase the image size the same number of dots has to cover a larger area. There's probably a formula that could estimate a new value for Resolution to get a similar image (and hence increase render time) but I can't work it out at the moment.

DynSys {
::7y5pfhn2tj1SvxtNQ47Gw/HE0dLz3iKB6SRQylk2AUni2LdBjWu7qYJRVJtJe/33h6BFXvat
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  pa9FSvfIke4f4kef0UnZaqOPvXdK97gJbwdfysnlJ7MMRzAeXYyuwk9TDTmYiKhJFT8FMMNG
  KSDyBLmpSSSYSgNbCQXORLRkU+sLzylAUAOh5B1TmiCmKkxIgnZUhwpwGfijKi6HfMjD1Fi8
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  C8vAHTIKUB==
}

Dynamical Function, plus a little bit of Popcorn and Vine. Reminiscent of something produced by the strange attractor software Chaoscope.

OK, thanks for the answer. Very clear now.