Strange Attractors – An Alternative Approach

Strange attractors are a category of fractal that have not featured in Saturn and Titan. For version 4.0.0 a new fractal type called Attractors has been introduced which is a generalised formula which has been extended so that it can be calculated using the Mandelbrot algorithm or the Julia algorithm and the resulting orbits plotted, it can also be coloured as though it were just any other escape time (or bailout fractal). In short Saturn treats the Attractors formula in the same way as all the other defined formulae with the exception of Lyapunov fractals.

The usual way of producing Strange Attractors is to start at a random location and then calculate a single very long orbit (typically 10s of thousands of locations in length) and plot each location in the orbit. Saturn plots a very large number of orbits of a very short length (typically 15) with the starting location of each orbit taken from a grid, all points on the grid are used to calculate an orbit.

The Attractors formula is based on two named Strange attractor types: de Jong and Clifford. The de Jong formula is as follows:

x_n+1 = sin(a*y_n) – cos(b*x_n)
y_n+1 = sin(c*x_n) – cos(d*y_n)

The Clifford formula is:

x_n+1 = sin(a*y_n) + c*cos(a*x_n)
y_n+1 = sin(b*x_n) + d*cos(b*y_n)

Saturn implements both these formula as follows:

The parameters A to H are ordinary “real” numbers, function parameters f1 to f4 can be assigned to any of the functions defined for Saturn, z and α are complex numbers. To produce de Jong and Clifford attractors the parameters are assigned the appropriate values, α is set to zero and z0 is set to the location in the complex plane.

As Attractors is a Saturn formula transforms can be applied to z for extra variations. The Attractors fractal can be calculated using the Mandelbrot algorithm by setting α to the location in the complex plane.

The Attractors fractals produced by Saturn and their equivalent Strange Attractors can differ. The differences appear to be an artefact of the method used to plot the fractal, mostly the differences can be removed by omitting the first few points in an orbit. There are other programs that produce plots of de Jong and Clifford attractors and there are plenty of example pictures elsewhere on the web, sometimes the images are upside down in relation to those produced by Saturn.

Time for some pictures:

Clifford Attractor

The picture above shows a Clifford attractor, it looks distinctly different to the image produced using the usual method for plotting strange attractors, an example how it usually looks can be found here. So why the difference? The calculating area used for orbit plotted fractals defaults to 9 times the display area which helps to display Pickover Popcorn correctly, for this image the entire plot is in the display area so the calculating area can be reduced to same as the display area which changes the image:

Clifford Attractor – calculating area equals display area

As can be seen a significant proportion of the plots have vanished but it is still far from the usual plot of this particular strange attractor. Since it can take a few iterations before the Strange attractor settles down, omitting a number of the initial points for every orbit calculated will reduce the number of plots that aren’t really part of the strange attractor. The following pictures show how the image changes as the number of omitted initial points in the orbits is increased.

Omit 1

Omit 2

Omit 3

Omit 4

All the preceding pictures used an orbit length of 13. To get an image that looks like the usual plot of this attractor several adjustments where necessary, the orbit length was increased to 40 and the number of omitted plots per orbit was increased to 20. A further adjustment is also available and that is the plot density, default density is one so for an 500×500 image 250000 orbits are calculated, the density can increased and for the final version of this attractor the density has been increased to 6 so that 1500000 orbits are calculated for the same area. The colours are also different as the colour map has been changed, the colour method is to use the logarithm of the number of times a location has been visited by an orbit and then scaled, the resulting value is used to look up a colour in the assigned colour map. The background, i.e. areas not visited by any orbits, is set to white.

Final version

Some attractors can be very sparse and consequently not that suitable for works of “art”. The extra plots that aren’t really part of the attractor that are calculated by Saturn’s method can produce pleasing results. The usual method starts at a random location and only one orbit is plotted so the effect of the initial plots in the orbit has virtually no affect on the final image, the method used by Saturn has thousands upon thousands of orbits so the initial plots build up significantly affecting the final image.

Here is Saturn’s plot of a de Jong Attractor, again with default calculating area that is 9 times the size f the display area.

A de Jong Attractor

And here it is stripped of its initial plots and it is very sparse indeed.

The bare Attractor

The bare attractor and the over plotted Saturn version are both unsatisfactory so some balance between the two would be preferred.

Calculating area equals Display Area

Density increased to 16

So that concludes an explanation of how Strange Attractors have been implemented for Saturn and Titan, the formula can also used to produce pictures that aren’t strange by setting the extra parameter α to something other than zero and for no orbit plotted fractals. I will cover those variations in a future post.