Research on fractal structure of generalized M–J sets utilized Lyapunov exponents and periodic scanning techniques

Abstract

In order to show details of fractal structure of Mandelbrot set precisely, Lyapunov exponents and periodic scanning techniques have been brought forward by Shirriff and Welstead. This paper generalizes these two techniques and puts forward periodicity orbit search and comparison technique which can be used to discuss the relationship of 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 function 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 → z α + c( α ∈ 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 have been expounded.