A Class Of Multi-parameter Fractal Interpolation Surface On Non-Linear Iterated Function System

Fractal interpolation is a new method of approximation of experimental data, It can provide nature deterministic approximation of complex phenomena. Thus, fractal interpolation can simulate the shape of material object distinctly. Compared to the traditional research which focusing on the linear aspect, this article gives a try at the nonlinear aspect. In this paper, a class of nonlinear iterated function system is constructed and a class of more general fractal function will be discussed. The main research works described in this article are as follows:In the introduction of this paper, we briefly review the fractal naissance, the development of fractal geometry and the current situation of this project. Furthermore the major content of the article is summarized.In Chapter 1, we mainly study the fundamental theory and knowledge of the fractal geometry including two definitions of dimension and the theory of iterated function system(IFS).In Chapter 2, we discusse the properties of amplitude and variation of the binary continuous funtion, and argue about the relationship of the variation and the dimension.In Chapter 3, we mainly construct a class of nonlinear iterated function system, including two arbitrary parameters, which can control the fractal interpolation surface better and more effective. We prove the uniqueness of the attractor and the existence of the fractal interpolation surface. Moreover, we discuss the continuity of the fractal interpolation surface. At the same time, a sufficient and necessary condition is presented. Under the boundary contour, the parameter-dependent of the surface is obtained. Numerical experiment demonstrate the effectiveness of the parameter control.In Chapter 4, using the properties of variation of the continuous funtion,We give and prove the calculation formula of the box-dimension of the constructed fractal interpolation surface. Finally, based on MATLAB, numerical experiments of dimension calculation of the interpolation surfaces with different parameters are given.