Research On Construction Of Fractals With IFS From Complex Function Family Of F(z)=z~n+c With Multiperiodic Orbits

Chaos and fractal theories are very important parts in the field of nonlinear science,and has been widely recognized by the scientific community.The ideas of the chaos and the fractal have been applied in physics,biology,astronomy,social science,computer science and many other fields,and have promoted the development of various disciplines.With the help of rapid development of computer technology,the research on fractal theory has continuously made new progress,at the same time,a large number of aesthetic fractals with highly artistic appealing are constructed by the new methods.Construction of fractals by IFS(Iterated fanction system)is an important work in the research of fractal in recent years.With the widely use of IFS in the diverse areas of science and engineering,more and more methods to construct IFS and mathematical characteristics of IFSs have been found.The study of IFS is extended to nonlinear functionsThis paper investigate the construction of nonlinear IFS using the complex mapping family fz=zn?c n?2 and consider how to choose c-values from period-land higher period regions of Mandelbrot set to construct valid IFS.The main researches and innovations of this paper as follows:(1)Investigate the complex mapping family f2?zn ?c when n?2 and discuss the differences of filled-in Julia sets,fixed points or multiperiodic attracting-orbits when c-values from different period bulbs of M set.(2)Study and conclude 4 situations which could not generate fractals when construct IFS by choosing 2 or more than 2 c-values from the period-1 bulb of M set;find out the valid parameter-range in the period-1 bulb of M set to construct the valid IFS.(3)Investigate the rules of constructing IFSs by choosing c-values from period-k bulbs when k is larger than 1,and in what situation fractals are not able to be generated.Find out the valid parameter ranges in the high attracting-period buds to construct the valid IFSs.(4)Comparing with the research of the characters of the filled-in Julia sets and their common attracting-basins for the complex mapping family fz =zn+c when c-values from period-1 bulb,we study the escape orbits in the common attracting-basins of the Julia sets with high attracting-period orbits and present a method by using topological conjugate to generate fractals with Zn symmetry when c-values chosen from k-period bulbs and k is large than 1.(5)A new gallery of the fractal strange attractors is built by the nonlinear IFS with the parameters,which can be used to construct the iterating functions with high attracting-period orbits,based on the complex mapping of fz = z~n+c.