An Intrinsically Three-Dimensional Fractal

被引：3
|
作者：
Fernandez-Guasti, M. [1 ]
机构：
[1] Univ Autonoma Metropolitana Iztapalapa, Dept Fis, Lab Opt Cuant, Mexico City 09340, DF, Mexico
来源：
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS | 2014年 / 24卷 / 06期
关键词：
3D bifurcations; hyper-complex numbers; 3D hyperbolic numbers; real scators; quadratic iteration; Mandelbrot set; discrete dynamical systems; MANDELBROT SET; SPHERE;
D O I：
10.1142/S0218127414300171
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
The quadratic iteration is mapped using a nondistributive real scator algebra in three dimensions. The bound set S has a rich fractal-like boundary. Periodic points on the scalar axis are necessarily surrounded by off axis divergent magnitude points. There is a one-to-one correspondence of this set with the bifurcation diagram of the logistic map. The three-dimensional S set exhibits self-similar 3D copies of the elementary fractal along the negative scalar axis. These 3D copies correspond to the windows amid the chaotic behavior of the logistic map. Nonetheless, the two-dimensional projection becomes identical to the nonfractal quadratic iteration produced with hyperbolic numbers. Two-and three-dimensional renderings are presented to explore some of the features of this set.
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页数：13
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