Additive perturbed generalized Mandelbrot–Julia sets

摘要

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.