由2次单参复解析多项式构造非线性迭代函数系
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摘要
为了构造新形式的分形,提出利用单参复解析2次多项式映射构造非线性迭代函数系.首先构造出2次多项式复映射在参数平面上的1周期参数集合;然后在该集合上随机挑选2个以上的参数,由这些参数建立一组迭代映射,用这组迭代映射构造出一个由单参的2次复解析压缩映射构造的非线性压缩IFS迭代函数系;最后对迭代函数系中的一个迭代映射在平面上的压缩不动点连续迭代,构造出相应的奇怪吸引子或分形.实验结果表明,该方法可以用于大量构造平面上的奇怪吸引子或分形,图形结构新颖.
To construct the novel fractals, we present a method which can be used to construct a nonlinear IFS composed of the complex quadratic polynomials with single complex parameter. 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 result shows that 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 differing from the fractals coming from the linear IFS.
To construct the novel fractals, we present a method which can be used to construct a nonlinear IFS composed of the complex quadratic polynomials with single complex parameter. 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 result shows that 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 differing from the fractals coming from the linear IFS.
引文
[1]Peitgen H O,Richter P H.The beauty of fractals[M].Heidelberg:Springer,1986
[2]Field M,Golubitsky M.Symmetry in chaos[M].New York:Oxford University Press,1992
[3]Barnsley M F.Fractals everywhere[M].2nd ed.Boston:Academic Press,1993
[4]Zeng Wenqu,Wang Xiangyang.Fractal theory and its computer simulation[M].Shenyang:Northeast University Press,1993(in Chinese)(曾文曲,王向阳.分形理论与分形的计算机模拟[M].沈阳:东北大学出版社,1993)
[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]Lau K S,Wang X Y.Iterated function systems with a weak separation condition[J].Studia Mathematica,2004,161(3):249-268
[7]Llorente M,Moran M.An algorithm for computing the centered Hausdorff measures of self-similar sets[J].Chaos,Solitons and Fractals,2012,45(2):246-255
[8]He Jin,Zhang Guofeng,Dai Shuling.Methods of obtaining IFS parameters and its realization based on iterated function system[J].Computer Simulation,2010,27(8):222-226(in Chinese)(何瑾,张国峰,戴树岭.迭代函数系统IFS码的获取方法及实现[J].计算机仿真,2010,27(8):222-226)
[9]Adcock B M,Jones K C,Reiter C A,et al.Iterated function systems with symmetry in the hyperbolic plane[J].Computers&Graphics,2000,24(4):791-796
[10]Chen N,Ding H,Tang M.Automatic generation of symmetric IFSs contracted in the hyperbolic plane[J].Chaos,Solitons and Fractals,2009,41(4):829-842
[11]Van Loocke P.Non-linear iterated function systems and the creation of fractal patterns over regular polygons[J].Computers&Graphics,2009,33(6):698-704
[2]Field M,Golubitsky M.Symmetry in chaos[M].New York:Oxford University Press,1992
[3]Barnsley M F.Fractals everywhere[M].2nd ed.Boston:Academic Press,1993
[4]Zeng Wenqu,Wang Xiangyang.Fractal theory and its computer simulation[M].Shenyang:Northeast University Press,1993(in Chinese)(曾文曲,王向阳.分形理论与分形的计算机模拟[M].沈阳:东北大学出版社,1993)
[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]Lau K S,Wang X Y.Iterated function systems with a weak separation condition[J].Studia Mathematica,2004,161(3):249-268
[7]Llorente M,Moran M.An algorithm for computing the centered Hausdorff measures of self-similar sets[J].Chaos,Solitons and Fractals,2012,45(2):246-255
[8]He Jin,Zhang Guofeng,Dai Shuling.Methods of obtaining IFS parameters and its realization based on iterated function system[J].Computer Simulation,2010,27(8):222-226(in Chinese)(何瑾,张国峰,戴树岭.迭代函数系统IFS码的获取方法及实现[J].计算机仿真,2010,27(8):222-226)
[9]Adcock B M,Jones K C,Reiter C A,et al.Iterated function systems with symmetry in the hyperbolic plane[J].Computers&Graphics,2000,24(4):791-796
[10]Chen N,Ding H,Tang M.Automatic generation of symmetric IFSs contracted in the hyperbolic plane[J].Chaos,Solitons and Fractals,2009,41(4):829-842
[11]Van Loocke P.Non-linear iterated function systems and the creation of fractal patterns over regular polygons[J].Computers&Graphics,2009,33(6):698-704