J Sets Of The Newton Transform For Solving Some Complex Exponential Equation And The Generalized M-J Sets

Fractal, chaos and soliton. are called as the three important concepts of the nonlinear theory. Based on the fractal theory of the nonlinear theory, here we lay a particular emphasis on the studying of Julia sets (abbreviated to J sets) of the newton transform for solving some complex exponential, the generalized Mandelbrot-Julia sets (abbreviated to M-J sets) for bicomplex numbers, the contorl of Julia sets and the contorl of Mandelbrot sets (abbreviated to M sets) and so on.Firstly, this paper extends Kim's complex exponential function, and find that when the parameters α > 0, |ζ| = 1, w = a + bi, |w| > 1, the generalized complex exponential function have many roots, and the roots are distributed on the distortion circle. When b= 0, the roots are symmetry with the x-aixs. The attraction's basins of the generalized complex exponential function is bounded, this means that the J sets are bounded. The generalized complex exponential function has many roots, in this paper proves the topology distribution structure of basins of attraction.Secondly, this paper gives a research into the generalized M-J sets for the bicomplex numbers, and explained the theory about bicomplex numbers, discussed the precondition of that addition and multiplication are closed in bicomplex number mapping of constructing generalized Mandelbrot-Julia sets, and listed out the definition and constructing arithmetic of the generalized Mandelbrot-Julia sets in bicomplex numbers system. Then we study the connectedness of the generalized M-J Sets, the feature of the generalized Tetrabrot and the relationship between the generalized M sets and its corresponding generalized J sets for bicomplex numbers in theory. Using the generalized M-J sets for bicomplex numbers constructed on computer, the author not only studied the relationship between the generalized Tetrabrot sets and its corresponding generalized J sets, but also studied their fractal feature, finding that: (1)The bigger the value of the escape time is, the more similar the 3-D generalized J sets and its corresponding 2-D J sets are; (2)The generalized Tetrabrot sets contain a great deal information of its corresponding 3-D generalized J sets; (3)Both the generalized Tetrabrot sets and its corresponding cross section make a feature of axis symmetry; (4)The bigger the value of the escape time is, the more similar the cross section and the generalized Tetrabrot sets are.