Reasearch On Nonlinear Iterated Function Systems From The Complex Mapping F(z)=z~n+c

Since fractal was clearly put forward in the 1970s,the research and practical application of fractal rapidly development.At present,People have been familiar with fractal.The research of fractal needs the application of computer graphics method.One of the reasons why fractal concept can be accepted by people rapidly is the help of the computer graphics method,so there are such a large number of beautiful fractal patterns.By giving a deep research on the application of computer graphics in fractal rendering,we can increase the comprehension of fractal and extend fractal application filds.Meanwhile,the nature of complex structure of fractal needs the fractal rendering method to be improved.Symmetric Chaos and Fractal research has been mainly in two directions:1.The visualization of dynamic system;2.Using the approach of iterated function system.Researche on the chaotic fractal symmetry has also been mostly starting from this two directions to construct symmetric function or symmetric IFS in two-dimensional or three-dimensional space to achieve mapping here and abroad.Iterated function system,as important methods of fractal computer graphical,is still based on the traditional linear compression affine transformation.Chaos dynamics and fractal geometry is the important part of nonlinear science,and also a frontier of scientific research.In the research of planar dynamical system of computer visualization,many scholars became interested in the research of the visualization of all kinds of analytical complex mappings,and made a lot of research results.We can see that there still many unsolved problems in computer graphical technology of fractal.This dissertation tries to innovate and break through the construction method of iterated function system,put forward new fractal rending method.Through the research on the complex polynomial mapping,we present a method which can be used to construct a nonlinear IFS.The innovations and research work of this dissertation are listed as follows:(1)We research on the features and rules of iteration of the complex mapping f(z)= z2 + c,including its iterative orbit,attraction and fixed point.We made a comparison between the linear compression affine transformation and the complex mapping,and compression condition of the complex mapping was put forward.(2)We researched the conditions the how to use the iteration of complex mapping system fz)= z2+c,to construct a nonlinear iterated function system,and researched the properties of nonlinear iterated function system.(3)We have researched the actuating range of the nonlinear iterated function system,how to choose the initial point before the the iteration,and we also researched the random iterative orbit of the nonlinear iterated function system.(4)We present a method of choosing parameters of the nonlinear iterated function system,First,we construct the 1-period parameter set in the parameter plane;then,randomly choose more than 2 parameters in the set and build a set of the contract functions;next,build a nonlinear IFS with these functions;last,continuously iterate the fixed point of a contract function by randomly choosing a function in the IFS to construct a fractal or a strange attractor.The method to construct the nonlinear IFS presenting in this paper is valid to construct the strange attractors or fractals in the plane,which have the new structures differring from the fractals coming from the linear IFS.(5)We researched how to construct a nonlinear IFS by the complex mapping f(z)=zn+c.We construct the nonlinear IFS from the 1-period basic symmetric area in the parameter plane.The nonlinear IFS we constructed in this dissertation,not only can be used to construct the Dn symmetrical fractals,but also to construct the Zn symmetrical fractals.(6)We researched the relationship between fixed point and strange attractor,and also researched the range of strange attractor and the influence of mapping changes on the strange attractor.(7)This dissertation has constructed a large number of new structure of strange attractor graphics,and build a new gallery of strange attractors.