Finding strange attractors worth being rendering is probably the most
time consuming activity on Chaoscope, and is also what creating fractal
- or chaos based - images is about : looking for the best, unexplored
spot.
You can compare searching for a strange attractor to digging for gold.
Your map is the search domain, or the default limit assigned to all the
parameters (the minimum and maximum values of the parameter sliders).
Pressing the F3 key would be equivalent to throw a dart on the map,
blindfolded, drill at the location where the dart landed, until you find
a gold deposit. This is what Chaoscope does : each parameter is given a
random value within its own range, the equation is then iterated 20,000
times and the resulting attractor is analyzed. If there's no attractor
or if it doesn't qualified as chaotic (in our case, fractal dimension
greater than 1.5 and positive Lyapunov exponent) then the parameters are
randomized again, and the search process carries on until a strange
attractor is found or the search is stopped by the user.
This is a "brute force algorithm", only practical because computers are
faster at doing math than we are at throwing darts. The downside of this
method is nothing says you will hit a huge deposit or find just a single
nugget of gold.
Two new features were added in 0.2 to improve the control over the
search : Randomness and parameter exclusion. Reducing the Randomness
factor limits the size of the map further so the search is concentrated
around the current finding. Excluding a parameter would be similar to
limit the mining to a strip of land instead of the whole area.
Depending on the equation, the ratio of chaotic solutions for the entire
search domain varies. What matters to us is most of the equations will
yield very similar looking attractors from one search to another,
especially Lorenz and Unravel, and a very little proportion will deserve
a render.
Clearly, the search is only indispensable for equations with many
parameters like IFS and high order Polynomial Sprott. For the remaining
equations, there is a more productive way to look for attractors.
Let's use the gold mining comparison again : If you struck gold at one
location, the chances are you may find more gold nearby. The same
applies for strange attractors. If you've found an unlikely Pickover,
there are probably more to find just by changing a couple of parameters
slightly. Keep in mind that two completely different shapes might
coexist within a very small parameter space. Drag one of the sliders
around and keep an eye on the view, there could be a beautiful specimen
hiding at the reach of your mouse. Guess how many images, out of the 28
displayed on Chaoscope gallery, are Unravels?
With some practice, you will be able to tell, looking at the view, if
you're close to find a nice attractor or not. Despite being
intrinsically different, all the equations behave roughly the same way,
because they're all producing chaos. The same patterns occur when you
shift the sliders, a circle splitting into two like a replicating cell,
then circles become loops, and loops turn into millions of intertwined
rings.
Like with gold mining, chance plays a part in the game. Some nights will
be rich in great discoveries, some other you will wish I'd never
released Chaoscope. :-)
Remember, when you find a strange attractor you've never seen before,
save the parameters, it will be useful later on as a starting point. And
don't forget to share your findings on the mailing-list!
Nicolas Desprez ====================================================== The Chaoscope mailing-list Archives : //www.freelists.org/archives/chaoscope Admin contact : chaoscope@xxxxxxxxxxxxxx Web site : http://www.chaoscope.org ======================================================
