In order to show details of fractal structure of Mandelbrot set precisely, Lyapunov exponents and periodic scanning techniques have been brought forward by Shirri and Welstead. This paper generalizes these two techniques and puts forward periodicity orbit search and comparison technique which can be used to discuss the relationship of the generalized Mandelbrot–Julia sets (the generalized M-J sets). Adopting the techniques mentioned above and the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, this paper researches on the structure topological inflexibility and the discontinuity evolution law of the generalized M-J sets generated from the complex mapping z→zα+c(α∈R), and explores structure and distributing of periodicity "petal" and topological law of periodicity orbits of the generalized Msets, and finds that the generalized Mset contains abundant information of structure of the generalized J sets by founding the whole portray of the generalized J sets based on the generalized Mset qualitatively. Furthermore, the physical meaning of the generalized M-J sets have been expounded.

照非线性理论，对健康人冠心病人和成年犬进行了心电取样，在对人和犬的大量心电信号的功率谱分析关联维数和Lyapunov指数计算的基础上，通过对比研究，得出以下结论：（ⅰ） 心电信号的功率谱关联维数和Lyapunov指数反映了心脏的总体动态特征，它们可作为一种定量研究心电图的新方法进行心血管疾病的早期诊断。（ⅱ）在正常的生理状态下心脏的运动是混沌的，而在病理状态下则趋于有序。（ⅲ）基于人犬对比，揭示出混沌是衡量生物体制进化的一个定量指标。

阐述了构造复映射z←zα+c（α<o）所产生的广义Mandelbrot-集（简称M-集或M-分形图）的逃逸时间算法。通过改变参数α，作出了一系列有趣的分形图这些分形图为若干卫星群环绕中央恒星的星群结构。定量地研究了恒星和卫星群的几何结构，并对α取非整数时分形图的结构特点和卫星群胚胎出现的原因进行了分析。最后给出几点结论。

推广了Baker, Devaney和Romera等的工作，并构造出一系列得指映射的广义Mandel-brot-Julia集（简称广义M-J集）。采用实验数学的方法，做如下工作：给出了复数阶广J集发生突变的理论依据；众理论上分析了广义M-J集的对称性和周期性；给出了复数阶广义M集周期花瓣分布的新的相邻规则；发现了复数阶广义M集包含了广义J集构造的大量信息；复数阶广义M-J集的分形生长速度要快于实数阶广义M-J集的分形生长速度，参数λo的值决定了广义J集的分形生长速度，复数阶广义M集的分形生长指向多分岔点和Misiurewicz点。

给出了NMIFS（nonlinear markov iterated function system）理论与构造NMIFS吸引子的方法，讨论了一类NMIFS吸引子的平衡向量测度和“矩”分析了NMIFS吸引子的结构物特征. 研究结果表明：对于MIFS，可以通过递归方法来计算矩M（i)（I=1，2，…）；而对于NMIFS，因M（i)（j≥i），故不能直接计算M（i)，而只能计算其近似值。

按照非线性理论，作者设计并实施了用于心脏系统非线性研究的深低温停循环（Profound Hy-pothermaia and Circulatory Arrest, 简称PHCA）实验。通过对PHCA实验中几个温度台阶上所采集的心电信号的功率谱分析和Lyapunov指数的计算，得出以下结论：（1）心电信息的功率谱和Lyapunov指数反映了心脏的总体动态特征，它们可用以估计心脏的健康状态；（2）在正常的生理状态下心脏的运动是混沌的，而在病理状态下则趋于有序。

基于对一典型Langevin问题-在双势井和变化的磁场中并受-恒冲量不断作用的运动带电子粒子的动力学分的，利用频闪采样法，给出了描述粒子速度变化规律的复差分方程。选限适当的磁场强度和时间间隔（采样周期）。将这一差分方程简化为用来构造广义M_J（Mandelbrot_Julia）集的复映身，并基于粒子的动力学特征探讨了广义M_J集的物理意义。结果发现：1）广义M_J集的分形结构特征可形象地反映出粒子速度的变化规律；2）选限的时间间隔有、无意义、决定了广义M_J集的分形结构是否具有连续性；3）广义M_J集的演化，即粒子速度的变化规律依赖于相角主值范围的不同和选限；4）若改变磁场强度和时间间隔的选限，如选限一随机波动的磁场，则此时广义J集可能会出现内部被填充的结构特征，即在速度空间中粒子的不稳定周期轨道的闭包出现“爆炸”现象。

为探讨Mandelbrot集（简称M集）的内部结构，推广了Pickover和Hooper的“ε正交”法和“星迹”法；及Philip的“区域分解”法和“角度分解”法并提出了用于研究M集外部结构的等势线法和色彩调配法利用上述方法，构造了一系列广义M集非边界区域的结构图研究表明整数阶广义M集的非边界区域具有分形特征，小数阶广义M集的非边界区域出现了错动和断裂，且其演化过程依赖于相角主值范围的选取。

According to the swiched complex mapping proposed by the author, the method constructing the switched processes generalized M (Mandelbrot) sets was elaborated, and a series of the switched processes generalized M sets for complex index number were constructed. The construction charaeteristies of the generalized M sets were expounded according to the analysis of teh algorithm constructing the generalized M sets. On the basis of what has already been achieved, the trajectories of a starting point in the complex C plane under the switched mapping were researched into, The results show the switched processes generalized M sets have the fractal feature, the construction characteristics of the switched processes generalized M sets are dependent on the complex number W and the switched variable ro, and the reaso which results in the discontinuity of switched processes generalized M sets is the discontinuity of choice of the pricipal range of phase angle.

In orderlo stody the iatefior strcture of the Mandelbrot (M) set. Pickover and Hooper have ad-wwed the epsilon cross method and the star trails method. respectively. In order to reveal the exterior structure of the Mset. Philip has proposed the regional decomposition method and the angle-slicing decompoaition method. In this pa-per. these methods-are extended. and the equipolenfial line method and the color adjustment method are proposod to study the exterior struture of the M set. A series of the non-boundary region structure images of the generalized M sets is generated by using the above-mentioned methode. The results show that the non-boundary regions of the generalized M sets for intexer index number have the fraetal feature; while those for decimal index number have diseontlnuity and col-lapse, and their erolutions depend on the choice of the principal range of the phase angle.