Visualization Of Chaotic Attractors With Planar Crystallographic Symmetry

Chaos lias been the subject of much recent study. Remarkably symmetry may appear even in the presence of chaos. Carter have illustrated chaotic attractors with symmetries from each of the frieze arid planar crystallographic groups, based on which, many people have illustrated effective techniques to find more visually interesting attractors. Such as, Jeffrey et.al have observed different ways that the symmetries evolve and the degenerate behavior. Cater and Clifford et.al have converted the images of crystallographic symmetries into a polar coordinate planar, transforming fric/c and crystallographic, patterns into fascinating imagines.Also based on Carter, this paper will create effective techniques to find more visually interesting attractors assisted by computers. The proposed method provides a novel approach for the devise of the patterns of planar tilings. Our paper will consider from the flowing four.1. Consider a dynamical system f (x,y) with planar crystallographic symmetries, let the function f(x,y) to function the dots (x,y) and j(x,y), which stastify crystallographic geometry symmetries. Their orbits are fn(x,y] and fn("/(x,y)). The two orbits have the same characters. The distance of the two adjacent dots determine the color of the dots in the plane(see the detail in 4.1 section), then the dots (x,y),j(x,y) have the same color, so we can achive a colorized pattern with planar crystallographic symmetries.2. the iterated function systems play a great role in chaos research. This paper will add the symmetries of one group to another group to put out a novel attractors with hybrid symmetries, this attractors will have more symmetries locally, so this proposed method provides a novel approach for the devise of patterns of planar tilings.3. this paper will combine the attractors of planar crystallgraphic symmetry groups with self-similar characteristic of fractals. Select a fundamental compute region, then copy it alone the horizontal direction, however in theperpendiculai- direction double or halve it, which is determined by whether the fundamental region is moved up or down. By this approach, the upper half plane is self-similar, then by transformations, square limit, circle limit can bo constructed.4. Draw chaotic attractors on three small squares and rotate the three squares, then they will construct a cuboid, on the same time, the patterns on the squares also chang accordingly. So through this, we map attractors of planar crystallgraphic symmetry groups onto the three-demention subjcctcs, then we can construct beautiful tessellations.