The Research On Nonlinear Iterative Function System Constructed By Complex Analytic Mapping F(z)=?cos~n(z)

Chaos science and fractal theory are very important parts in the nonlinear science,and have been widely recognized by the scientific community.The ideas of chaos and fractal have been applied in the fields of physics,biomedicine,astronomy,social sciences and computer science,which have promoted the development of various disciplines.New research results have been obtained in the research of fractal theory with the rapid advancement of computer technology.At the same time,through the new method of constructing fractals,a large number of fractal patterns with novel structure and great artistic value have been constructed.The use of IFS(iterated fanction system)to construct fractals is an important task in the fractal research.With the extensive application of the technologies of the construction of the fractals in in various fields,the research results about the method of constructing IFS and the mathematical properties of various IFS itself are presented.The construction of IFS has been made from the linear compression affines to the non-linear compression maps.In this paper,we try to use the complex cosine mapping family to discuss the arrangement rules about the periodic buds of the M-set in the parameter plane,and the rules of construction of IFS with the different period parameters,and to find out the constructing method of valid IFS to generate fractals by means of researching the relationship between the parameter positions in the M set,the dynamic characterastics of their filled-in Julia sets and their graphical structuresThe main research work and innovations are as follows:(1)Study the relationship between constructing IFS and the M sets and the filled-in Julia sets of the complex cosine mapping family f(z)=?cosn(z).Analyze the mathematical properties of the complex cosine map;in the central periodic window on the dynamic plane,examine whether the orbits of the critical points of the iterative mapping for the pointed parameters are bounded.(2)Randomly select parameters in the high-period buds of a M set and construct the IFS.The fractals in the common attraction domain of all iterative mappings for the IFSs are constructed.(3)According to the D2 geometric symmetry feature of the generalized M set of the complex mapping family in the parameter plane,an IFS with both rotational symmetry and reflection symmetry is constructed,and a fractal with this kind of symmetry is generated.(4)IFSs from the selected parameters in the 1-period parameter region of M sets with different exponents n is realized,and the corresponding fractals are constructed.(5)The convergence characteristics of the iterative mapping are different due to different parameters,so the fractal image obtained by the equal probability random iterative function is not clear.To get clear fractals,two solutions are presented in this paper:The compression ratios of the filled-in Julia sets are taken to as the iterative probability of the iterative mapping in the IFS,by which,the the fractal patterns from the IFS constructed with the parameters in one-cycle buds is clear;the other solution is to calculate the periodic orbital attraction rate analytically to get the iterative probability,which is useful to get the clear fractals from the IFS with the mixed parameters in the 1-cycle bud and high cycle bud(6)A great number of the images of the M sets,the filled-in Julia sets and the fractals of IFSs for the complex cosine mapping falily are built.