The Mandelbrot-Julia Sets Of Newton's Method

The nonlinear theory is a new developing frontier science which describes the complex systematic structure shape which has a random structure. The nonlinear theory contains three important concepts: Fractal, Chaos and Soliton. Here we lay a particular emphasis on the studying of the theory and methods of the M-J sets of Newton's method of fractals. Also we will give out some important research results.Newton's method is an important method which is used to find the solutions of nonlinear equation or equations. It transforms the solution of the equation f(x) = 0 into a dynamical process and the solution process is related to the initial value. The use of the Newton's method is not limited in the complex plane. The advantage of the using of the complex value is that there are already well-developed mathematical theories i.e. the iteration theory of Julia and Fatou which will give us a well guide in our study. In this article, we firstly apply the Newton's method to the nonlinear equations to construct and study the Julia sets of multivariable Newton's method for real exponent power. And we discover that as β increases, there is a sudden change in the Julia sets which expresses by the addition of the number of the attraction areas, and the structure of the Julia sets depends on the main range of the phase angle.Then we give a research into the orbit trap technique in Newton's method. And the images generated by IFS system are used as the orbit trap in our method with which we construct the pseud 3D M-J sets of the Newton's method of the complex polynomial. It always is found that there is standard Mandelbrot structure in the 3D M-set, which is formed by "bad" points. In generalized M-J set, there are various 3D self-similarity color cells that correspond with the shape of trap unit. And the work of this part has been accepted by the journal.Finally, a gradual changed color plan is adopted in generalized Newton' method to show the inner structure of the attraction area. And we study on how the distances between the roots and the order of the roots affect the distribution of the attraction area. By our study, we discover that when the orders of the roots are the same, the distances between the roots is the only factor determining the attraction area; when the distances are the same, the size of the attraction depends only on the order of the roots.