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

Title: Strange attractors: True shape
Post by: marcm200 on March 29, 2020, 08:13:22 PM
An interesting article about strange attractors:

Code: [Select]
Robust Visualization of Strange Attractors using Affine Arithmetic
Afonso Paiva, Luiz Henrique de Figueiredo, Jorge Stolfi, 2006
full text (

using cell-mapping. The crucial observation here is that the strange attractor is contained in the union of all strongly connected components.

One starts with the N-square 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 non-attractor containing part of the 2-square):

Left, Henon's map with a=46976204 * 2^-25 = approx. 1.4, b=10066329 * 2^-25 = approx. 0.3
Middle: a Henon-like map with b=0.5
Right: my favourite test subject: the z²-1 "basilica".

I especially like the basilica image, as it shows both invariant sets: The Fatou period-2 cycle (singular dots at -1 and 0) and the Julia set itself as the latter is totally invariant under z²-1.

Currently I use a recursive algorithm by Brett Bernstein, but am planning on converting it to a non-recursive one to have lower overhead, especially for the Holmes map (1.5*x-x^3+0.95*y,x) or higher refinement levels in general.

Title: Re: Strange attractors: True shape
Post by: marcm200 on June 06, 2020, 07:14:45 AM
"Holmes map"

After de-recursifying 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 floating-point representable and I currently have not yet implemented intervals for the parameters, I computed the close-by Holmes-like map at b=31876710 * 2^-25 (approx. 0.94999999)

Depicted is the 4-square, image is trustworthily downscaled 4-fold from level L14. The union of all blue tiles covers the strange attractor.
Title: Re: Strange attractors: True shape
Post by: marcm200 on June 09, 2020, 09:39:05 AM
"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 point-sampled 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 2-square.