Fractalforums
Fractal Related Discussion => Fractal Mathematics And New Theories => Topic started by: marcm200 on March 29, 2020, 08:13:22 PM

An interesting article about strange attractors:
Robust Visualization of Strange Attractors using Affine Arithmetic
Afonso Paiva, Luiz Henrique de Figueiredo, Jorge Stolfi, 2006
full text http://lhf.impa.br/ftp/papers/sa.pdf (http://lhf.impa.br/ftp/papers/sa.pdf)
using cellmapping. The crucial observation here is that the strange attractor is contained in the union of all strongly connected components.
One starts with the Nsquare containing (most of) the attractor and analyzes every pixel as in the Julia set TSA algorithm, identifying a pixel as a complex interval, calculating its bounding box (for classic IA as I use here) or via more sophisticated methods (like affine arithmetics in the article) and follows the intersected pixels to build a cell graph consisting of all pixels and a directed edge from the starting pixel B to every one of its bbx intersected pixels.
Then using Tarjan's algorithm gives one the strongly connected components, with one special case: As Tarjan's algorithm enumerates the vertices to get the SCCs, it eventually returns every pixel, so even those that are not contained in an SCC at all.
Therefore a potential SCC of size 1 is analyzed whether it actually has a path to itself (hence a fix point), if so, it is kept, if not, that SCC is discarded.
Finally, pixels that do not belong to any SCC are transformed to white.
Below are the first results (blue is the SCC union, white is the nonattractor containing part of the 2square):
Left, Henon's map with a=46976204 * 2^25 = approx. 1.4, b=10066329 * 2^25 = approx. 0.3
Middle: a Henonlike map with b=0.5
Right: my favourite test subject: the z²1 "basilica".
I especially like the basilica image, as it shows both invariant sets: The Fatou period2 cycle (singular dots at 1 and 0) and the Julia set itself as the latter is totally invariant under z²1.
Currently I use a recursive algorithm by Brett Bernstein, but am planning on converting it to a nonrecursive one to have lower overhead, especially for the Holmes map (1.5*xx^3+0.95*y,x) or higher refinement levels in general.

"Holmes map"
After derecursifying the classic Tarjan SCC algorithm, I was able to compute the Holmes map to a reasonable refinement (the recursive version crashes my system at level 9).
\[
f(x,y) = (a\cdot x  x^3 + b\cdot y, x).
\]
with a=1.5 and b=0.95.
As 0.95 is not floatingpoint representable and I currently have not yet implemented intervals for the parameters, I computed the closeby Holmeslike map at b=31876710 * 2^25 (approx. 0.94999999)
Depicted is the 4square, image is trustworthily downscaled 4fold from level L14. The union of all blue tiles covers the strange attractor.

"Quadratic general case"
A quadratic strange attractor of the general form:
\[
x_{new} := a_1 + a_2 x+a_3 x^2+a_4 xy+a_5 y+a_6 y^2 \\
y_{new} := a_7 + a_8 x+a_9 x^2+a_{10} xy+a_{11} y+a_{12} y^2
\]
(as from the article in post #1).
using the values { 3355445,26843542,23488103,36909875,36909875,23488103,13421774,20132656, 20132660,10066331,40265314,20132656 } * 2^25 similar to fig. 8.
At L15R2 (my current computational limit) the attractor is different from the pointsampled version (right part, L13R2, 500 skip iterations, 200 drawing iterations)  the latter however changes quite a bit with differing iteration settings, so I don't know how informative that shape already is.
Both parts depict roughly [0..2] x [2..0]*i cropped and downscaled from the fully computed 2square.