超复数系统中的高维广义M-J集

摘要

研究超复数算法优化问题,由于超复数系统中构造曼德勃罗-茱莉亚集(M-J集)的算法目前还停留在三元数和四元数水平,且仅实现了低维M-J集在2-D和3-D截面上的仿真.为了更深入研究分形结构的特征性质,使用倍增和截去方法建立了任意维的超复数系统,讨论了超复数系统中的加法和乘法运算是闭的前提条件,并给出了超复数系统中高维(任意偶数维)广义M-J集的定义及构造算法.通过选取不同参数,实现并绘制了不同维度广义M-J集的2-D截面,对2-D截面的分形结构特征进行分析,结果表明高维广义M-J集在不同水平上的自相似性,并理论证明了2-D截面的对称性.有关对称性的分析将有助于进一步研究超复数动力学.%Hypercomplex algorithm optimization problem has been studied in this paper.In order to carry out a in -depth study on the characteristics and properties of the fractal structure,doubling and truncation techniques were used to generate a hypercomplex number system of any dimension,and the precondition of that addition and multiplication closed in hypercomplex number system was discussed,and the definition and constructing arithmetic of the hyperdimensional(any even-dimensional) generalized Mandelbrot-Julia sets in hypercomplex number system were listed out.Through choosing different parameters,the authors realized and drew the 2-D cross sections of the different dimensions generalized M-J sets,and studied the fractal feature of 2-D cross sections.The results show that the hyperdimensional generalized Mandelbrot-Julia sets were self similar in the different level.By using theories,the authors also have proved the symmetry of 2-D cross sections.The analysis of symmetry in this paper can help to study the dynamics of hypercomplex number.