Research On Construction Of Nonlinear IFS From Complex Polynomial Function Family With(n-1)Extreme-Points

The visualization of dynamics systems began in the early 20th century.With the development and improvement of computer hardware and software step by step,many scholars have begun to study fractal theory with computer.Researchers have proposed various methods of the visualization of dynamic systems for different iteration maps,and generated a large number of fractals and chaotic attractors in various forms.This research has formed a branch of non-linear science research,and its application extends to various fields,including computer science itself.In the visualization of nonlinear dynamical systems,it is an important research content to construct fractals by using linear affine compression iteration function system?IFS?.With the more and more extensive application of IFS,the research on the fractals has been developed from the linear iterated function system into the non-linear iterated function system.This paper combines the traditional method of constructing linear iteration function system with the method of generating generalized M sets and filled-in Julia sets in the visualization of dynamic system.By selecting two or more parameters of M sets to construct the iteration maps to form a nonlinear iteration function system,the fractal is constructed.In this paper,a single-parameter polynomial complex mapping family f?z?=zn+cz with multiple extremum points is studied in detail on how to select parameters to construct effective IFS and its fractal in high-period parameters in M set periodic bud of parameter plane.The main research work and innovations of this paper are as follows:?1?The geometric characteristics and the construction of IFS to generate fractals about the M sets and the filled-in Julia sets of the mapping f?z?=Zn+cz and f?z?=zn+c of are compared and studied.?2?Investigate the reason why the complex mapping family f?z?=zn+cz cannot be used to construct an effective iterative function system with the parameters from the 1-period bud parameter region of parameter |c|<1 in the M set.?3?Study how to construct an effective nonlinear iterative function system with the parameters chosen in the 1-period bud parameter region of parameter |c|>1 in the M set of the complex mapping family f?z?=zn+cz.A double random iterative algorithm is proposed to generate a large number of?n-1?rotationally symmetric fractals on the dynamic plane.?4?Find out the reasons why the effective iterative function system IFS cannot be constructed by selecting parameters in 1₁ bud and?n-1?₁ bud parameter area in M set.?5?The rule of construction of fractals with the parameters located in the Zn₁ rotation symmetry places in the M sets was studied.?6?For the parameters from the high periodic buds of a M set,if all chosen parameters can be used to construct the iterating mappings with only one high period ordit,the relating IFS canbe used to generate the fractals by the traditional random iteration method;and if all chosen parameters can be used to construct the iterating mappings with more than one high period ordits,the relating IFS must be used to generate the fractals by the double random iteration method presented in this paper.?7?A graph library of the strange attractors for the nonlinear iterative function system of parame-ters |c|>1 based on the complex mapping family f?z?=zn+cz is established.