四元数M-J集的构造及其分形结构的研究
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摘要
Mandelbrot集(以下简称M集)和Julia集是分形发生学中的经典集合。借助于计算机的运算和模拟手段,人们对M集和Julia集进行了大量的分析和研究,实现了对动力系统的许多构想,同时其研究结果还涉及交叉发展的诸多学科领域,如Klein群理论、拟共形映照理论、Teichm(u|¨)ller空间理论、拓扑学、复分析、计算方法、遍历性理论和符号动力学等。
目前高维分形是分形领域中研究的一个热点,但是高维分形因高维空间的复杂性和图像显示的困难性还有很多问题亟待探讨。本文构造了对四元数映射f:z←z~α+c(α∈Z)的广义M集和Julia集,并对其特性进行了深入的研究,探讨了四元数广义M-J集的裂变演化规律。
本文利用n维参数L系统描述了超复数空间广义M集和Julia集,给出了描述算法和实验所得图形,并对四元数广义M-J集的动力学特征进行了理论上的分析和探讨。本文研究发现了四元数广义M集的特殊拓扑结构以及四元数广义Julia集的参数不变性,推广了Bogush的相关结论。实验结果表明,n维参数L系统字母表较为简洁,包含信息量大,可以有效的描述诸如广义M-J集和四元数广义M-J集的分形集。
本文绘制了四元数广义M-J集的三维分形图,并采用逃逸时间算法或Lyapunov指数法与周期点查找法相结合,构造了四元数广义M-J集的周期域。计算了四元数M集的周期域的边界条件,并从理论上对四元数广义M-J集的动力学特征进行了分析和探讨。通过在四元数M集取点构造四元数Julia集,定性建立了四元数M集上点的坐标与四元数Julia集整体结构之间的对应关系。
采用逃逸时间算法与周期点查找结合法,构造了四元数映射f:z←z~2+c的多临界点M集,探索了多临界点情况下四元数M集的拓扑结构和裂变演化规律,计算了四元数M集的周期域的中心位置和边界条件。提出了改进的逃逸时间算法,采用该算法构造四元数广义M集可以观察到,周期域中心点的分布随临界点不同产生了迁移甚至分化。通过构造分岔图和计算M集的盒维数,讨论了不同临界点对M集的影响。研究结果表明,四元数映射f:z←z~2+c的M集由所有临界点决定的四元数M集的并集组成。
构造了受动态噪声和输出噪声扰动的四元数M集,并分析了噪声扰动下M集的拓扑结构演变规律。通过构造周期域和分岔图观察了噪声对M集的影响。研究结果表明,加性动态噪声并没有从本质上改变四元数M集的结构,噪声使得M集的周期稳定域产生了偏移。乘性动态噪声使得M集的稳定域产生了收缩,而收缩的比例是由噪声的强度决定的。另外,受干扰的M集始终相对实轴保持着对称。输出噪声并没有改变M集的周期域的边界,但是影响了区域的内部结构。加性和乘性输出噪声都使得周期域产生缺失。乘性输出噪声扰动的M集相对实轴保持对称,而加性输出噪声则完全破坏了M集的对称性。
Mandelbrot sets(abbreviated as M sets) and Julia sets are classic fractal sets in fractals. People have made deep investigation on M sets and Julia sets utilizing the computation and simulation of computer,which realize many supposes of dynamic systems.These researches involve many interdisciplinary subjects in science,such as Klein group,quasi-conformal mappings,Teichm(u|¨)ller spaces theory,analysissitus,complex analysis,numerical computation method,ergodic theory and symbolic dynamics,etc.
At present high-dimensional fractal is a focus in fractal field.However,due to the complexity of high-dimension space and difficulty in shape representation,high-dimensional fractal still has much work to be done.The disseriation constructs the general M sets and Julia sets on the quatemion mapping f:z←z~α+c(α∈Z),researches their properties and explores the fission evolution rules of quaternion M-J sets.
The disseriation describes the general hypercomplex M sets and Julia sets utilizing parametric n-dimensional L systems.The algorithm and the fractal images produced are presented,and the dynamic properties of the general quatemion M-J sets are discussed.It is found that the quatemion M sets have special topology structures and quaternion Julia sets have invariability in the selection of parameter,which extends the conclusion of Bogush.It can be concluded that n-dimensional L systems,which have pithily alphabet but convey plentiful information,could depict such fractals as general hypercomplex M-J sets well.
The disseriation presents the 3-D projections of general quaternion M-J sets and constructs the cycle regions of quaternion M-J sets utilizing the time-escape algorithm or Lyapunov exponent method combining cycle detecting method.The boundary of the cycle region is calculated,and their dynamic characters are theoretically analyzed.The quaternion Julia sets are constructed with the parameter selected from quatemion M sets,which establishes relationship between the whole portray of the quaternion Julia sets and the point coordinates of the quaternion M sets qualitatively.
The disseriation constructs the quaternion M sets on the mapping f:z←z~2+c utilizing the time-escape algorithm combining cycle detecting method,which has multiple critical points.The new M sets' topology structures and fission evolution rules are investigated;the center positions and the boundaries of the cycle regions are calculated.An improved time-escape algorithm is presented,by which the quatemion M sets are constructed.It can be observed that the center of the cycle region appear displacement(even differentiation) under different critical point.The bifurcation diagrams are constructed and the box-dimension is calculated for observing the changes which take place on the M sets.Plenty of experiments show that the collection of the quaternion M sets with different critical points make up of the complete M sets on the mapping f:z←z~2+c.
The disseriation constructs the quaternion M sets perturbed by the dynamic and output noises and analyses the topology structures affected by the noises.The cycle regions and bifurcation diagrams are constructed for observing the influence which the noises bring to the M sets.The experimental results show that the additive dynamic noise does not change the M sets essentially,which effects displacements of the stability regions.The multiplicative dynamic noise makes the stability regions of the M sets shrink acutely with the proportion determined by the strength of the noise and the perturbed M sets keep symmetrical around the real axis.The output noise does not change the area of the stability regions of the M sets but affects the inner structure substantially.Both the additive and multiplicative output noises bring the absence of the stability areas.The M sets with multiplicative output noises keep symmetric around the real axis while the additive output noises destroy the M sets' symmetries completely.
目前高维分形是分形领域中研究的一个热点,但是高维分形因高维空间的复杂性和图像显示的困难性还有很多问题亟待探讨。本文构造了对四元数映射f:z←z~α+c(α∈Z)的广义M集和Julia集,并对其特性进行了深入的研究,探讨了四元数广义M-J集的裂变演化规律。
本文利用n维参数L系统描述了超复数空间广义M集和Julia集,给出了描述算法和实验所得图形,并对四元数广义M-J集的动力学特征进行了理论上的分析和探讨。本文研究发现了四元数广义M集的特殊拓扑结构以及四元数广义Julia集的参数不变性,推广了Bogush的相关结论。实验结果表明,n维参数L系统字母表较为简洁,包含信息量大,可以有效的描述诸如广义M-J集和四元数广义M-J集的分形集。
本文绘制了四元数广义M-J集的三维分形图,并采用逃逸时间算法或Lyapunov指数法与周期点查找法相结合,构造了四元数广义M-J集的周期域。计算了四元数M集的周期域的边界条件,并从理论上对四元数广义M-J集的动力学特征进行了分析和探讨。通过在四元数M集取点构造四元数Julia集,定性建立了四元数M集上点的坐标与四元数Julia集整体结构之间的对应关系。
采用逃逸时间算法与周期点查找结合法,构造了四元数映射f:z←z~2+c的多临界点M集,探索了多临界点情况下四元数M集的拓扑结构和裂变演化规律,计算了四元数M集的周期域的中心位置和边界条件。提出了改进的逃逸时间算法,采用该算法构造四元数广义M集可以观察到,周期域中心点的分布随临界点不同产生了迁移甚至分化。通过构造分岔图和计算M集的盒维数,讨论了不同临界点对M集的影响。研究结果表明,四元数映射f:z←z~2+c的M集由所有临界点决定的四元数M集的并集组成。
构造了受动态噪声和输出噪声扰动的四元数M集,并分析了噪声扰动下M集的拓扑结构演变规律。通过构造周期域和分岔图观察了噪声对M集的影响。研究结果表明,加性动态噪声并没有从本质上改变四元数M集的结构,噪声使得M集的周期稳定域产生了偏移。乘性动态噪声使得M集的稳定域产生了收缩,而收缩的比例是由噪声的强度决定的。另外,受干扰的M集始终相对实轴保持着对称。输出噪声并没有改变M集的周期域的边界,但是影响了区域的内部结构。加性和乘性输出噪声都使得周期域产生缺失。乘性输出噪声扰动的M集相对实轴保持对称,而加性输出噪声则完全破坏了M集的对称性。
Mandelbrot sets(abbreviated as M sets) and Julia sets are classic fractal sets in fractals. People have made deep investigation on M sets and Julia sets utilizing the computation and simulation of computer,which realize many supposes of dynamic systems.These researches involve many interdisciplinary subjects in science,such as Klein group,quasi-conformal mappings,Teichm(u|¨)ller spaces theory,analysissitus,complex analysis,numerical computation method,ergodic theory and symbolic dynamics,etc.
At present high-dimensional fractal is a focus in fractal field.However,due to the complexity of high-dimension space and difficulty in shape representation,high-dimensional fractal still has much work to be done.The disseriation constructs the general M sets and Julia sets on the quatemion mapping f:z←z~α+c(α∈Z),researches their properties and explores the fission evolution rules of quaternion M-J sets.
The disseriation describes the general hypercomplex M sets and Julia sets utilizing parametric n-dimensional L systems.The algorithm and the fractal images produced are presented,and the dynamic properties of the general quatemion M-J sets are discussed.It is found that the quatemion M sets have special topology structures and quaternion Julia sets have invariability in the selection of parameter,which extends the conclusion of Bogush.It can be concluded that n-dimensional L systems,which have pithily alphabet but convey plentiful information,could depict such fractals as general hypercomplex M-J sets well.
The disseriation presents the 3-D projections of general quaternion M-J sets and constructs the cycle regions of quaternion M-J sets utilizing the time-escape algorithm or Lyapunov exponent method combining cycle detecting method.The boundary of the cycle region is calculated,and their dynamic characters are theoretically analyzed.The quaternion Julia sets are constructed with the parameter selected from quatemion M sets,which establishes relationship between the whole portray of the quaternion Julia sets and the point coordinates of the quaternion M sets qualitatively.
The disseriation constructs the quaternion M sets on the mapping f:z←z~2+c utilizing the time-escape algorithm combining cycle detecting method,which has multiple critical points.The new M sets' topology structures and fission evolution rules are investigated;the center positions and the boundaries of the cycle regions are calculated.An improved time-escape algorithm is presented,by which the quatemion M sets are constructed.It can be observed that the center of the cycle region appear displacement(even differentiation) under different critical point.The bifurcation diagrams are constructed and the box-dimension is calculated for observing the changes which take place on the M sets.Plenty of experiments show that the collection of the quaternion M sets with different critical points make up of the complete M sets on the mapping f:z←z~2+c.
The disseriation constructs the quaternion M sets perturbed by the dynamic and output noises and analyses the topology structures affected by the noises.The cycle regions and bifurcation diagrams are constructed for observing the influence which the noises bring to the M sets.The experimental results show that the additive dynamic noise does not change the M sets essentially,which effects displacements of the stability regions.The multiplicative dynamic noise makes the stability regions of the M sets shrink acutely with the proportion determined by the strength of the noise and the perturbed M sets keep symmetrical around the real axis.The output noise does not change the area of the stability regions of the M sets but affects the inner structure substantially.Both the additive and multiplicative output noises bring the absence of the stability areas.The M sets with multiplicative output noises keep symmetric around the real axis while the additive output noises destroy the M sets' symmetries completely.
引文
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