An Intrinsically Three-Dimensional Fractal

作者：Fernandez Guasti M^*

来源：International Journal of Bifurcation and Chaos, 2014, 24(6): 1430017.

DOI：10.1142/S0218127414300171

• 3D bifurcations
• hyper-complex numbers
• 3D hyperbolic numbers
• real scators
• quadratic iteration
• Mandelbrot set
• discrete dynamical systems

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.

• 出版日期2014-6