doi:10.1016/j.cag.2006.08.016 
Copyright © 2006 Elsevier Ltd All rights reserved.
Chaos and Graphics
Robust visualization of strange attractors using affine arithmetic
aDepartamento de Matemática, PUC-Rio, Rio de Janeiro, Brazil
bIMPA—Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
cInstituto de Computação, UNICAMP, Campinas, Brazil
Available online 2 October 2006.
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Abstract
We propose the use of affine arithmetic in cell-mapping methods for the robust visualization of strange attractors and show that the resulting cellular approximations converge faster than those produced by cell-mapping methods based on classical interval arithmetic.
Keywords: Strange attractors; Cell mapping; Interval arithmetic; Affine arithmetic; Discrete dynamical systems
Fig. 1. The Hénon strange attractor.
Fig. 2. Cell graph of the Hénon map based on a 3×3 rectangular subdivision. The strongly connected component is shown in green.
Fig. 3. Cellular approximation for the Hénon attractor.
Fig. 4. Hénon strange attractor for a=1.4 and b=0.3 (left), 0.4 (center), and 0.5 (right). Top row: point-sampling method. Bottom row: cell-mapping method using interval arithmetic.
Fig. 5. Attractor of the Holmes map using sampling method (red) and cell mapping using interval arithmetic (gray).
Fig. 6. Attractor of the logistic map computed with the sampling method (a) and with the cell-mapping method using IA (b).
Fig. 7. Approximating f(A) with the sampling method (blue), with cell mapping using IA (red), and cell mapping using AA (green).
Fig. 8. Strange attractor of the quintic map QBKSKIXQMKEOVVMAHXLBOQQJXEYMBUMBOEFVDBAPWU in [-0.260287,0.952145]×[-0.994131,0.968682]. Point-sampling method shown in red. Cell-mapping method shown in gray using IA (left) and AA (right). From the top to bottom, pictures have resolution 642, 1282, and 2562 cells.
Fig. 9. Strange attractor of the quadratic map LUFBBFISGJYS in [0.0,2.0]×[-1.8,-0.2]. (a) Cell mapping based on IA (blue). (b) Cell mapping based on AA (gray). (c) Sampling method (red). (d) All together. Cell mapping done for a 1024×1024 cell grid.
Fig. 10. Strange attractor of the cubic map OVFKWKEIBPGNYPVKWCYU in [-0.707942,0.537318]×[-0.874148,0.680792]. (a) Cell mapping based on IA (blue). (b) Cell mapping based on AA (gray). (c) Sampling method (red). (d) All together. Cell mapping done for a 1024×1024 cell grid.
Fig. 11. Strange attractor of the quintic map HVOIEGIDJCSFUFJCQGRUGMCLHEPWKRCCYFIRQPYAPH in [-1.14556,0.411485]×[-0.294105,1.10915]. (a) Cell mapping based on IA (blue). (b) Cell mapping based on AA (gray). (c) Sampling method (red). (d) All together. Cell mapping done for a 512×512 cell grid.
Table 1.
Cell graph data for the quadratic map of Fig. 9
Table 2.
Cell graph data for the cubic map of Fig. 10
Table 3.
Cell graph data for the quintic map of Fig. 8
Table 4.
Cell graph data for the quintic map of Fig. 11
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