Additive perturbed generalized Mandelbrot-Julia sets

Abstract Adopting the experimental mathematics method combining complex variable function theory with computer aided drawing, this paper researches on the structural characteristic and the fission-evolution law of additive perturbed generalized Mandelbrot–Julia sets (generalized M–J sets in short). The corresponding relationship between point coordinates in generalized M set and the general structure of generalized J sets has been founded qualitatively and the physical meaning of the generalized M–J sets has been expounded. The following conclusions are deduced: (1) Chaotic patterns of fractal structure of generalized J sets may emerge out of double-periodic bifurcation, which shows that Brownian movement can be chaotic. (2) Experimental evidence of Li–Yorke theorem is given out. (3) The additive perturbed generalized M set contains abundant information on the construction of generalized J sets. (4) Resemble logistic map, in the process of a series of double-periodic bifurcation coming into chaos, generalized J sets also present self-similarity in parameter space.

[1]  Heinz-Otto Peitgen,et al.  The science of fractal images , 2011 .

[2]  Uday G. Gujar,et al.  Fractal images from z <-- z alpha + c in the complex z-plane , 1992, Comput. Graph..

[3]  J. Argyris,et al.  On the Julia sets of a noise-perturbed Mandelbrot map , 2002 .

[4]  J. Argyris,et al.  The influence of noise on the correlation dimension of chaotic attractors , 1998 .

[5]  Leon O. Chua,et al.  EXPERIMENTAL SYNCHRONIZATION OF CHAOS USING CONTINUOUS CONTROL , 1994 .

[6]  Uday G. Gujar,et al.  Analysis of z-plane fractal images from z <-- z alpha + c for alpha < 0 , 1993, Comput. Graph..

[7]  Akhlesh Lakhtakia,et al.  Julia sets of switched processes , 1991, Comput. Graph..

[8]  V. Varadan,et al.  COMMENT: On the symmetries of the Julia sets for the process z==>zp+c , 1987 .

[9]  J. Yorke,et al.  Period Three Implies Chaos , 1975 .

[10]  C. Beck.  Physical meaning for Mandelbrot and Julia sets , 1999 .

[11]  Joshua C. Sasmor,et al.  Fractals for functions with rational exponent , 2004, Comput. Graph..

[12]  M. Feigenbaum.  Quantitative universality for a class of nonlinear transformations , 1978 .

[13]  Uday G. Gujar,et al.  Fractals from z <-- z alpha + c in the complex c-plane , 1991, Comput. Graph..

[14]  Ioannis Andreadis,et al.  On perturbations of the Mandelbrot map , 2000 .

[15]  Gonzalo Álvarez,et al.  External arguments of Douady cauliflowers in the Mandelbrot set , 2004, Comput. Graph..

[16]  J. Argyris,et al.  On the Julia set of the perturbed Mandelbrot map , 2000 .

[17]  Fausto Montoya Vitini,et al.  Chaotic bands in the Mandelbrot set , 2004, Comput. Graph..

[18]  Xingyuan Wang,et al.  ANALYSIS OF C-PLANE FRACTAL IMAGES FROM z ← zα + c FOR (α < 0) , 2000 .

[19]  Xingyuan Wang,et al.  Research on fractal structure of generalized M-J sets utilized Lyapunov exponents and periodic scanning techniques , 2006, Appl. Math. Comput..

[20]  Dietmar Saupe,et al.  Chaos and fractals - new frontiers of science , 1992 .

[21]  Young Ik Kim,et al.  Accurate computation of component centers in the degree-n bifurcation set , 2004 .

[22]  M. Klein.  Mandelbrot set in a non-analytic map , 1988 .

[23]  Earl F. Glynn.  The evolution of the gingerbread man , 1991, Comput. Graph..