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Additive perturbed generalized Mandelbrot-Julia sets. (English) Zbl 1126.37031

Summary: Adopting the experimental mathematics method combining complex variable function theory with computer aided drawing, this paper studies 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 the generalized M set and the general structure of the generalized J sets are found qualitatively and the physical meaning of the generalized M-J sets are expounded. The following conclusions are deduced: (1) Chaotic patterns of fractal structures of generalized J sets may emerge out of double-periodic bifurcation, which shows that Brownian motion can be chaotic. (2) Experimental evidence of the Li-Yorke theorem is given. (3) The additive perturbed generalized M set contains abundant information on the construction of generalized J sets. (4) Resemble the logistic map, in the process of a series of double-periodic bifurcation coming into chaos, the generalized J sets also present self-similarity in parameter space.

MSC:

37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
60J65 Brownian motion
28A80 Fractals
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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