【作者】 王春梅；

【导师】 王兴元；

【作者基本信息】 大连理工大学 ， 计算机软件与理论， 2006， 硕士

【摘要】 非线性理论是描述具有无规则结构的复杂系统结构形态的一门新兴边缘科学。它包含了分形、混沌和孤子这三个非常重要的概念。本文侧重研究了分形学中具有重要意义的复合复映射M-J集(简称M-J集)的相关理论方法和一类混沌系统的投影同步问题，并取得了一些重要的研究成果。 (1) 将复合复映射z←(z~2+c)~2+c推广为z←(z~α+c)~β+c(α，β∈R)，研究了由该映射所构造的广义J集的分形结构和裂变演化规律，通过研究发现：α为整数的广义J集具有旋转对称性；α和β为小数时，相角θ主值范围的不同选取，将导致广义J集的不同演化；计算了广义J集之间的Hausdorff距离，这对研究广义J集之间的匹配程度提供了一种新的研究方法。 (2) 研究了由复合复映射z←(z~α+c)~β+c(α，β∈R)所构造的准广义Mandelbbrot集(简称M集)。通过对准广义M集分形图的绘制发现：当α和β为整数时，准广义M集关于x轴对称，并且α和β为为奇数时，准广义M集不仅关于x轴对称，还关于y轴对称，并在理论上给出了证明；利用DeMoivre理论，当α和β为小数时，对相角θ的主值范围的不同选取，准广义M集将出现错动和断裂，出现了雏瓣和部分卫星群。最后将Hausdorff距离应用到准广义M集，得出了不同准广义M集之间的匹配程度。 (3) 实现了一种新混沌系统——变形耦合发电机系统的投影同步。利用状态观测器理论，设计了与该混沌系统相应的投影同步观测器，利用该观测器理论上使得变形耦合发电机系统达到投影同步。最后，对变形耦合发电机系统进行了数值仿真，仿真结果进一步验证了该方法的有效性。更多还原

【Abstract】 The nonlinear theory is a new developing frontier science which describes the complex systematic structure shape which has a random structure. The nonlinear theory contains three important concepts: Fractal, Chaos and Soliton. Here we lay a particular emphasis on the studying of the theory and methods of the M-J sets (abbreviated to M-J sets) of Compound complex mapping and Projective Synchronization of a class of chaos System. Also we will give out some important research results.Scholars have made deep researches on the Mandelbrot-Julia sets generated from the complex mapping z← z~α + c(α∈R). Entwistle had studied the J sets of compound complex mapping z←(z~2 +c)~2+c. We generalize complex mapping as z←(z~α+ c)~β+ c(α,β∈R), and research on the structure topological inflexibility and the discontinuity evolution law of the generalized J sets of this mapping. The researches as below: generalized J sets have α-fold rotation symmetry and its center is the origin when α is integer; the different choices of angle lead to the different evolution of generalized J sets; computer the Hausdorff distance between two generalized J sets.Then we give a research into the generalized M sets of z←(z~α+c)~β+ c(α,β∈R), By plotting the Quasi-M sets, we find that: when α and β are integers, the Quasi-M sets are symmetrical about x axis, when α and β are odd, the Quasi-M sets are symmetrical about x and y axis, this has been illustrated theoretically; Making using of DeMoivre theory, when α and β are decimal fractions, the different choices of angle 9 lead to the different evolution of generalized M sets. Finally, Hausdorff distance is applied to the Quasi-M sets, offer the mathing of two Quasi-M sets.Finally, we study projective sysnchronization of a new chaos system—a modified coupled dynamos system. Using techniques from the state observer design, we present a systematic, design procedure to synchronize a modified coupled dynamos system by a scaling factor., and have proved the validity of the design. Finally, feasibility of the technique is illustrated for a modified coupled dynamos system.更多还原