The Study Of The Fractal Structure Of The Generalized Mandelbrot Set

The nonlinear theory is a new developing frontier science which describes the complex systematic structure shape. It contains three important concepts: Fractal, Chaos and Soliton. Here we lay a particular emphasis on the studying of the fractal structure of the generalized Mandelbrot set (simplified by the generalized M set). Also we give out some important research results.Some basic fractal algorithms and Newton's method are introduced here. These fractal algorithms are used to construct the generalized M set, and Newton's method is a strong tool to analyze and compute the important points in the generalized M set. With these algorithms and methods, the structure of the generalized M set can be understood well.Firstly one-dimensional multiple mappings are studied by the structure of the M set and the generalized M set. The period-doubling bifurcations in the chaotic region are mainly analysed and a set of new rules to graphically determine the period of a midget or a hyperbolic component are provided.The regularity of period-doubling bifurcation is analysed.Secondly the parameter space of two-parameter complex mappings is analyzed. By Taylor expansion a method to calculate the positions, the sizes and the orientations of those small copies of generalized M-like set in the parameter space of two-parameter complex mappings and the comparison to the one-parameter complex mappings are shown.Finally a rendering method based on the escape-time method to draw the generalized M set in different colors is offered. The rendering method chooses different colors according to the distance and the escape-time N, so it can effectively render the generalized M set, which can show us the 3-D view of the generalized M set. According to amplify the part of the generalized M set drawn by the rendering method, some cirques with one heart like the equipotential lines can be found. The center point of the cirques with one heart whose color will be rendered by black has a period, and only those center points, whose periods are the divisors of the escape-time N, will show us the cirques with one heart like the equipotential lines around them. Those center points are named the divisor periodic point of the escape-time N.