具有P3对称六边形格子平面排列动力系统图形化研究

Visualizing Research on Dynamic System of Plane Tiling of Hexagonal Lattice with inner P3 Symmetry

目的为构造六边形格子平面排列动力系统的参数空间,从理论分析和计算机实现技术方面进行了P3模型广义M集的构造.方法剖析了六边形格子平面排列的几何特征,讨论由平行四边形格子构造六边形格子的可能性,证明P3模型动力系统所具有的六边形格子的内部对称特性,确定构造P3模型广义M集的图形生成元.结果确定了P3模型六边形格子平面排列的计算机图形化的基本计算域,并因此构造出了生成广义M集所需的初始迭代点集.结论该基本域上的动力学特性决定着P3模型动力系统在整个动力平面上的动力学特性,基于这个基本域上点的轨道的Lyapunov指数可以构造P3模型的参数空间的广义M集,为深入研究P3模型以及其他六边形格子平面排列模型中的参数对图形结构变化的影响打下了基础.

On the research of visualizing research on dynamic system of crystallographic groups, the dynamic system of the plane tiling of the hexagonal lattice is the most complex one. It is difficult to construct the general M set in its parameter space. In this paper, the geometry character of the plane tiling of the hexagonal lattice is analyzed; the possibility presenting the hexagonal lattice with the parallelogram lattice is discussed;the inner symmetry of a parallelogram lattice from the P3 dynamic system is proved; and the generator constructing the generate M set of the P3 dynamic system is determined. The generator is called the fundamental computing region, and the initial iterating point set is chosen from the region. The general M set of the parameter space of P3 model is constructed based on the Lyapunov exponent of the orbits of the points in the initial iterating point set. The investigation of the relationship between the parameter of P3 or other hexagonal lattice models and the image construction can be performed according to this kind of M set. Key Words： chaos, crystallographic group, general M set, filled-in Julia set