二维平面与三维球面对称动力系统模型的构造及其图形化方法研究

This research project is the project belonging to the intersection field of multi-domains in the science research, such as the computer science, mathematics and nonlinear science and so on.We take the computers as our experiment platforms. We do our research work based on the theories of chaotic dynamics, fractal geometry and computer graphics. The main tasks include that the construction of new symmetrical iteration mappings and IFSs in the Euclidean plane and the hyperbolic plane or on the spherical surface, the investigation of the estimate arithmetic of the dynamic characteristic of the symmetric mappings on the sphere surface, the investigation of the construction of the chaotic attractors and the filled-in Julia sets on the upper half spherical surface in the any projected plane containing the origin; the investigation of the construction of the chaotic attractors, the filled-in Julia sets and fractals on the spherical surface from the plane mappings and IFSs, the investigation of the construction of the hyperbolic limit disks by asymmetric pictures and the investigation of the construction of plane tilings with different structures by the mappings with the non-equidistant cyclic windows under the same set of the parameters. The research production is not only useful to develop the research of the visualization of dynamic systems, but also useful to develop the applications of the patterns of the chaotic attractors and filled-in Julia sets in the plane and on the spherical surface with our own patent right. We will make several artworks and product samples with the chaotic and fractal patterns. So, the research project is of important value to both the academic research and the industry applications.

本课题在混沌动力学、分形几何学和计算机图形学理论指导下，以计算机为平台，探索对称动力系统计算机图形化构图规律。深入研究：新的欧氏平面、双曲平面以及三维球面上对称动力系统或迭代函数系的构造；球面对称迭代映射动力学特性的数值计算方法；通过空间坐标变换在过原点的任意投影面的上半球面构造混沌吸引子和充满Julia集；利用2维平面迭代映射和IFS迭代函数系构造球面混沌吸引子、充满Julia集和球面分形图的新方法；用非对称图源构造双曲极限圆图形；用变周期窗口迭代映射构造在同一组参数下不同结构的平面排列图形方法。建立有自主知识产权的平面对称与球面对称混沌分形图图库，开展混沌分形图的艺术设计，制作艺术作品和产品样品，可望在理论上和应用层次上取得多点原创成果。本课题属于计算机科学、数学科学和非线性科学等多学科相交叉的应用基础研究，对于动力系统图形化的理论研究具有重要的科学意义，同时，具有明显的应用价值。

在国家自然科学基金的支持下，本课题组全体成员经过4年的不懈努力，完成了课题申请时所提出的各项研究工作任务。在2维平面、3维球面以及双曲平面的对称动力系统构造和相应的构图机制研究、构图方法研究、软件著作权申请和混沌分形图的应用研究及样品制作等方面进行了大量的踏实工作，取得了相应的研究成果，发表论文19篇，其中发表国际学报论文3篇（SCI）、发表国内计算机EI源学报《计算机研究与发展》1篇（EI）、发表国内计算机图形学EI源学报报《计算机辅助设计与图形学学报》2篇（EI 1篇，待印刷1篇）、发表中文核心期刊《小型微型计算机系统》6篇（其中待印刷1篇）、发表艺术类中文核心期刊《美苑》1篇、发表国际会议论文3篇（EI）；申请国家发明专利7项：授权3项、公布实审3项、受理1项；培养硕士研究生毕业10人；在与多家合作的基础上实现了多个分形样品制作；本课题组负责人与香港城市大学数学系开展了互访合作研究。