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.pdfusing 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.
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