On a visualization of the convergence of the boundary of generalized Mandelbrot set to \((n-1)\)-sphere. (English) Zbl 07251874
Summary: In this article we analyze the generalized Mandelbrot set in higher-order hypercomplex number spaces following both the Cayley-Dickson construction algebraic spaces and the spaces defined by Clifford algebras. The particular case of the generalized 3D quasi-Mandelbrot set was also considered. In particular, we investigated the increase of power of the iterated variable and proved that when this power tends to infinity, the Mandelbrot set is convergent to the unit circle. The same is true for the generalized Mandelbrot sets in higher-dimensional hypercomplex number spaces, i.e. when the power of iterated variable tends to infinity, the generalized Mandelbrot set is convergent to the unit (n-1)- sphere. The results of our investigation were visualized for the generalized Mandelbrot set in a complex number space and the generalized quasi-Mandelbrot set in a 3D Euclidean space.
MSC:
| 37F45 | Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) |
| 28A80 | Fractals |
References:
| [1] | Gujar, U. G.; Bhavsar, V. C., Fractals from \(z \leftarrow{z^\alpha } + c\) in the complex c-plane, Comput. Graph., 15, 441-449 (1991) |
| [2] | Gujar, U. G.; Bhavsar, V. C., Fractals images from \(z \leftarrow{z^\alpha } + c\) in the complex z-plane, Comput. Graph., 16, 45-49 (1992) |
| [3] | Wang, X. Y.; Liu, X. D.; Zhu, W. Y.; Gu, S. S., Analysis of c-plane fractal images from \(z \leftarrow{z^\alpha } + c\) for \(\left( {\alpha < 0} \right)\), Fractals, 8, 307-314 (2000) · Zbl 1211.37059 |
| [4] | Garant-Pelletier, V.; Rochon, D., On a generalized Fatou-Julia theorem in multicomplex spaces, Fractals, 17, 241-255 (2009) · Zbl 1177.30068 |
| [5] | Norton, A., Julia sets in quaternions, Comput. Graph., 2, 267-278 (1989) |
| [6] | Griffin, C. J.; Joshi, G. C., Octonionic Julia sets, Chaos Soliton. Fract., 2, 11-24 (1992) · Zbl 0788.30038 |
| [7] | Katunin, A., On the symmetry of bioctonionic Julia sets, J. Appl. Math. Comput. Mech., 12, 2, 23-28 (2013) · Zbl 1364.37107 |
| [8] | Wang, X. Y.; Jin, T., Hyperdimensional generalized M-J sets in hypercomplex number space, Nonlinear Dyn., 73, 843-852 (2013) |
| [9] | Lakhtakia, A.; Varadan, V. V.; Messier, R.; Varadan, V. K., On the symmetries of the Julia sets for the process \(z \to{z^p} + c\), J. Phys. A: Math. Gen., 20, 3533-3535 (1987) |
| [10] | Dixon, S. L.; Steele, K. L.; Burton, R. P., Generation and graphical analysis of Mandelbrot and Julia sets in more than four dimensions, Comput. Graph., 20, 451-456 (1996) |
| [11] | Blancharel, P., Complex analytic dynamics on the Riemann sphere, Bull. Am. Math. Soc., 11, 88-144 (1984) |
| [12] | Liu, X.; Zhu, Z.; Wang, G.; Zhu, W., Composed accelerated escape time algorithm to construct the general Mandelbrot set, Fractals, 9, 149-153 (2001) · Zbl 1046.37030 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
