Wang, Xingyuan; Chang, Peijun Research on fractal structure of generalized M–J sets utilized Lyapunov exponents and periodic scanning techniques. (English) Zbl 1102.37031 Appl. Math. Comput. 175, No. 2, 1007-1025 (2006). Summary: In order to show details of fractal structure of Mandelbrot sets precisely, Lyapunov exponents and periodic scanning techniques have been brought forward by K. Shirriff [Fractals from simple polynomial composite functions, Comput. Graph. 17, No. 6, 701–703 (1993)] and S. T. Welstead and T. L. Cromer [Coloring periodics of two-dimensional mappings, Comput. Graph. 13, No. 5, 539–543 (1989)]. This paper generalizes these two techniques and puts forward periodicity orbit search and comparison techniques which can be used to discuss the relationship among the generalized Mandelbrot-Julia sets (the generalized M-J sets). Adopting the techniques mentioned above and the experimental mathematics method of combining the theory of analytic functions of one complex variable with computer aided drawing, this paper researches on the structure topological inflexibility and the discontinuity evolution law of the generalized M-J sets generated from the complex mapping \(z\rightarrow z^{\alpha } + c\), \(\alpha \in \mathbb R\), and explores structure and distributing of periodicity “petal” and topological law of periodicity orbits of the generalized M sets, and finds that the generalized M set contains abundant information of structure of the generalized J sets by founding the whole portray of the generalized J sets based on the generalized M set qualitatively. Furthermore, the physical meaning of the generalized M-J sets is expounded. Cited in 23 Documents MSC: 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) Keywords:Lyapunov exponent; periodic scanning; topological inflexibility; discontinuity evolution law; fractal; physical meaning; generalized Mandelbrot-Julia sets PDFBibTeX XMLCite \textit{X. Wang} and \textit{P. Chang}, Appl. Math. Comput. 175, No. 2, 1007--1025 (2006; Zbl 1102.37031) Full Text: DOI References: [1] Mandelbrot, B. B., The Fractal Geometry of Nature [M] (1982), Freeman WH: Freeman WH San Fransisco, 5-47 [2] Peitgen, H. O.; Saupe, D., The Science of Fractal Images [M] (1988), Springer-Verlag: Springer-Verlag Berlin, 137-218 [3] Wang, Xingyuan, Chaos in the Complex Nonlinearity System [M] (2003), Electronics Industry Press: Electronics Industry Press Bejjing, 1-32 [4] Lakhtakia, A., On the symmetries of the Julia sets for the process \(z\)←\(z^p+c\) [J], J. Phys. A: Math. Gen., 20, 3533-3535 (1987) [5] Gujar, U. G.; Bhavsar, V. C.; Vangala, N., Fractals images from \(z\)←\(z^α+c\) in the complex \(z\)-plane [J], Comput. Graph., 16, 1, 45-49 (1992) [6] Dhurandhar, S. V.; Bhavsar, V. C.; Gujar, U. G., Analysis of \(z\)-plane fractals images from \(z\)←\(z^α+c\) for \(α<0\) [J], Comput. Graph., 17, 1, 89-94 (1993) [7] Wang, Xingyuan; Liu, Xiangdong; Zhu, Weiyong, Analysis of \(c\)-plane fractal images from \(z\)←\(z^α+c\) for \(α<0\) [J], Fractals, 8, 3, 307-314 (2000) · Zbl 1211.37059 [8] Wang, Xingyuan; Liu, Xiangdong; Zhu, Weiyong, Researches on general Mandelbrot sets from complex map \(z\)←\(z^α+c(α<0)\) [J], Acta Mathematica Scientia, 19, 1, 73-79 (1999) · Zbl 0922.28008 [9] Wang, Xingyuan, Fractal Mechanism of the Generalized M-J set [M] (2002), Press of Dalian University of Technology: Press of Dalian University of Technology Dalian, pp. 82-116 [10] Hooper, K. J., A note on some internal structures of the Mandelbrot set [J], Comput. Graph., 15, 2, 295-297 (1991) [11] Philip, K. W., Field lines in the Mandelbrot set [J], Comput. Graph., 16, 4, 443-447 (1992) [12] Lakhtakia, A., Julia sets of switched processes [J], Comput. Graph., 15, 4, 597-599 (1991) [13] Wang, Xingyuan, Fractal structures of the non-boundary region of the generalized Mandelbrot set [J], Prog. Natural Sci., 11, 9, 693-700 (2001) [14] Wang, Xingyuan, Switched processes Generalized Mandelbrot Sets for complex index number [J], Appl. Math. Mech., 24, 1, 73-81 (2003) · Zbl 1076.37513 [15] Romera, M.; Pastor, G.; Alvarez, G., Growth complex exponential dynamics [J], Comput. Graph., 24, 1, 115-131 (2000) [16] Wang, Xingyuan; Shi, Qijiang, The generalized Mandelbort-Julia sets from a class of complex exponential map [J], Prog. Natural Sci., 14, 6 (2004) · Zbl 1136.65118 [17] Beck, C., Physical meaning for Mandelbrot and Julia sets [J], Physica D, 125, 171-182 (1999) · Zbl 0988.37060 [18] Wang, Xingyuan; Meng, Qingye, Research on physical meaning for the generalized Mandelbrot-Julia sets based on Langevin problem [J], Acta Physica Sinica, 53, 2, 388-395 (2004) · Zbl 1202.37070 [19] Shirriff, K. W., Fractals from simple polynomial composite functions [J], Comput. Graph., 17, 6, 701-703 (1993) [20] Welstead, S. T.; Cromer, T. L., Coloring periodicities of two-dimensional mappings [J], Comput. Graph., 13, 5, 539-543 (1989) [21] Blancharel, P., Complex analytic dynamics on the Riemann sphere [J], Bull. Am. Math. Soc., 11, 88-144 (1984) [22] Wang, Xingyuan, Research on the relation of chaos activity characteristics of the cardiac system with the evolution of species [J], Chinese Sci. Bull., 47, 24, 2042-2048 (2002) 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.