Construction Of Spherical Fractal Images With Planar IFS

In the study of the dynamic properties and the visualization about the nonlinear dynamic system,an amount of research achievements have been made in recent years, such as the methods of the construction of the various symmetrical dynamic systems and the generating patterns in the 2d and 3d spaces, the application of the fractal ideas in the actual life and some algorithms from the theory of fractal characteristics.In the research of the visualization of the dynamic systems in the 3D space, there are a lot of research results published, in which, some are about how to generate spherically symmetric chaotic attractors with randomly searching parameters automatically, some are about how to construct the spherical dynamic systems with regular tetrahedron symmetry group, the hyperbolic symmetry group in the three dimensional space and the inscribed regular polyhedron symmetry group. In 2012, our research group proposed a method of constructing the chaotic attractors on the hexahedral surface and spherical surface from the planar mappings. The above research papers were mostly related to dynamic systems, however, the research work about how to construct patterns with IFS was less published, the patterns generated by the 1FS in the 3D space was even more rare. In this thesis, the main research objective is to construct spherical symmetric strange attractor by the planar iterated function systems with the D3 symmetry, which is one of the investigating tasks in the project supported by the Natural Science Foundation of China.Because the thing to construct the spherical symmetric fractal or the strange attractor with the planar IFS is realized by the relationship of the sphere and its inscribed regular polyhedron, furthermore, because the research work in this thesis involves how to construct the spherical patterns by the planar IFS with D3 symmetry properties, the icosahedron is chosen as the intermediate carrier. By studying the inscribed icosahedron's geometric features in the unit sphere, the Sierpinski triangle fractal on the icosahedron is successfully constructed by the planar IFS with D3 symmetry properties. On the surface of the icosahedron, the Sierpinski triangle is been substitued of the general strange attractors with D3 symmetry by using n basic contracted affine transformations and D3 symmetry group. According to the relationship between the icosahedron and the its unit sphere, this thesis realizes the construction of the spherical symmetric strange attractor by the planar IFS with D3 symmetry. In this thesis, the main research results are as follows:(1) Construct the Sierpinski triangle on the surfaces of unit spherical inscribed icosahedron by the planar IFS with the D3 symmetry.(2) Propose the concept of the basic IFS which is made up of n basic linear contracted affine transformations; propose the method of construction of the IFS with D3 symmetry according to the requirement that a transformation is equivariant with D3 symmetry group.(3) Propose the method of generating spherical strange attractors by the IFS with D3 symmetry.(4) Choose randomly parameters to construct planar IFS with D3 symmetry and generate a large number of the strange attractors on the surfaces of the sphere and its inscribed icosahedron.