五次单参复多项式映射构造具有高周期吸引轨道的IFS
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摘要
为了采用复解析多项式映射的广义M集的高周期参数构造非线性迭代函数系,提出利用复映射f(z)=z~5+c的高周期参数构造生成分形或奇怪吸引子的非线性迭代函数系构造方法.在参数平面上的M集中,在每个高周期参数区域的变形4瓣结构的周期芽苞上构造一个连结4个花瓣区域顶点的椭圆;在椭圆附近或椭圆内挑选k≥2个参数;在每个参数下的充满Julia集内由拓扑共轭关系定义5个迭代映射;由5k个迭代映射定义了具有旋转对称特性的迭代函数系,在其公共吸引域内随机迭代生成分形.实验结果表明,在复映射f(z)=z~5+c的M集高周期参数区域的椭圆附近或椭圆内挑选参数,可以用于构造有效的非线性迭代函数系;采用文中方法可以生成结构各异的具有5旋转对称特性的分形或奇怪吸引子.
To construct the new fractals or new strange attractors, we present a method which can be used to construct a nonlinear IFS composed of the quintic complex polynomial mappings with the single complex parameter and the multiperiodic orbit. In the M set of the parameter plane, an ellipse is constructed, which connects the 4 vertexes of the periodic bud with the 4 transmutative petal-structure in every high-periodic parameter region;(k ≥2) parameters are chosen in or near the ellipse; 5 topological conjugate mappings are defined in the filled-in Julia set for every parameter; an IFS is composed of the 5k mappings; a fractal is constructed by randomly iterating the IFS in the common attracting basin of the 5k mappings. The results show that the parameters, chosen in or near the ellipse of the high-periodic parameter region from the mapping of f(z)=z~5+c, can be used to construct the valid IFSs. The fractals or strange attractors with 5 rotational symmetry and the different structures can be generated by the method presented in this paper.
To construct the new fractals or new strange attractors, we present a method which can be used to construct a nonlinear IFS composed of the quintic complex polynomial mappings with the single complex parameter and the multiperiodic orbit. In the M set of the parameter plane, an ellipse is constructed, which connects the 4 vertexes of the periodic bud with the 4 transmutative petal-structure in every high-periodic parameter region;(k ≥2) parameters are chosen in or near the ellipse; 5 topological conjugate mappings are defined in the filled-in Julia set for every parameter; an IFS is composed of the 5k mappings; a fractal is constructed by randomly iterating the IFS in the common attracting basin of the 5k mappings. The results show that the parameters, chosen in or near the ellipse of the high-periodic parameter region from the mapping of f(z)=z~5+c, can be used to construct the valid IFSs. The fractals or strange attractors with 5 rotational symmetry and the different structures can be generated by the method presented in this paper.
引文
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[15]Chen Ning,Feng Dongdong.Nonlinear IFS from complex quadratic polynomials with single complex parameter[J].Journal of Computer-Aided Design&Computer Graphics,2016,28(2):255-259(in Chinese)(陈宁,冯冬冬.由2次单参复解析多项式构造非线性迭代函数系[J].计算机辅助设计与图形学学报,2016,28(2):255-259)
[16]Chen Ning,Wu Xiaochen,Li Ming.Research on parameters of construction of IFSs from non-analytical complex mappings[J].Journal of Computer-Aided Design&Computer Graphics,2017,29(7):1267-1274(in Chinese)(陈宁,吴晓辰,李明.非解析复映射构造IFS的参数研究[J].计算机辅助设计与图形学学报,2017,29(7):1267-1274)
[17]Chen Ning,Li Ming,Wu Xiaochen.Construction of fractals from the analytical complex mapping family with negative integer powers[J].Journal of Chinese Computer Systems,2017,38(9):2166-2170(in Chinese)(陈宁,李明,吴晓辰.负整数次幂复解析映射构造分形[J].小型微型计算机系统,2017,38(9):2166-2170)
[18]Chen Shihua,Lu Jun’an.Introduction of chaotic dynamics[M].Wuhan:Wuhan Water Conservancy and Electric Power University Press,1998(in Chinese)(陈士华,陆君安.混沌动力学初步[M].武汉:武汉水利电力大学出版社,1998)
[2]Adcock B M,Jones K C,Reiter C A,et al.Iterated function systems with symmetry in the hyperbolic plane[J].Computers and Graphics,2000,24(4):791-796
[3]Chen N,Ding H,Tang M.Automatic generation of symmetric IFSs contracted in the hyperbolic plane[J].Chaos,Solitons and Fractals,2009,41(2):829-842
[4]Sahu D R,Chakraborty A,Dubey R P.K-iterated function system[J].Fractals,2010,18(1):139-144
[5]Lau K S,Ngai S M,Rao H.Iterated function systems with overlaps and self-similar measures[J].Journal of London Mathematical Society,2001,63(2):99-116
[6]Dekking F M,van der Wal P.The boundary of the attractor of a recurrent iterated function system[J].Fractals,2002,10(1):77-89
[7]Lau K S,Wang X Y.Iterated function systems with a weak separation condition[J].Studia Mathematica,2004,161(3):249-268
[8]Stenflo O.Iterated function systems with a given continuous stationary distribution[J].Fractals,2012,20(3/4):197-202
[9]Lesniak K.Homoclinic attractors in discontinuos iterated function systems[J].Chaos,Solitons and Fractals,2015,81(9):146-149
[10]Vrscay E R,Weil D.“Missingmoment”and perturbative methods for polynomial iterated function systems[J].Physica D,1991,50(2):478-492
[11]Wang X Y,Li F P.A class of nonlinear iterated function system attractors[J].Nonlinear Analysis,2009,70(2):830-838
[12]van Loocke P.Non-linear iterated function systems and the creation of fractals patterns over regular polygons[J].Computers and Graphics,2009,33(2):698-704
[13]Fan Shen.Nonlinear iterated function systems and fractal properties of the spectrum of Schr?dinger operator[D].Beijing:Tsinghua University,2010(in Chinese)(范申.非线性迭代函数系与Schr?dinger算子的谱的分形性质[D].北京:清华大学,2010)
[14]Liu Shuqun,Liu Luosheng.Iterative function system based on polynomial transformation[J].Journal of Lanzhou University of Technology,2011,37(1):81-85(in Chinese)(刘树群,刘罗生.基于多项式变换的迭代函数系[J].兰州理工大学学报,2011,37(1):81-85)
[15]Chen Ning,Feng Dongdong.Nonlinear IFS from complex quadratic polynomials with single complex parameter[J].Journal of Computer-Aided Design&Computer Graphics,2016,28(2):255-259(in Chinese)(陈宁,冯冬冬.由2次单参复解析多项式构造非线性迭代函数系[J].计算机辅助设计与图形学学报,2016,28(2):255-259)
[16]Chen Ning,Wu Xiaochen,Li Ming.Research on parameters of construction of IFSs from non-analytical complex mappings[J].Journal of Computer-Aided Design&Computer Graphics,2017,29(7):1267-1274(in Chinese)(陈宁,吴晓辰,李明.非解析复映射构造IFS的参数研究[J].计算机辅助设计与图形学学报,2017,29(7):1267-1274)
[17]Chen Ning,Li Ming,Wu Xiaochen.Construction of fractals from the analytical complex mapping family with negative integer powers[J].Journal of Chinese Computer Systems,2017,38(9):2166-2170(in Chinese)(陈宁,李明,吴晓辰.负整数次幂复解析映射构造分形[J].小型微型计算机系统,2017,38(9):2166-2170)
[18]Chen Shihua,Lu Jun’an.Introduction of chaotic dynamics[M].Wuhan:Wuhan Water Conservancy and Electric Power University Press,1998(in Chinese)(陈士华,陆君安.混沌动力学初步[M].武汉:武汉水利电力大学出版社,1998)