复合Logistic映射中的逆分岔与分形

利用分岔图，揭示出复合Logistic映射可按倍周期分岔走向混沌，且混沌区中存在混沌危机及逆分岔现象．同时，分析了复合Logistic映射临界点的轨道，给出了复合Logistic映射Mandelbrot-Julia集(简称M-J集)的定义，推广了Welstead和Cromer所提出的周期点查找技术，并利用该技术，构造出一系列复合Logistic映射的M-J集．在此基础上，研究了M-J集的对称性；探索了M集周期区域分布的拓扑不变性；通过定性地建立M集上J集的整体刻画，发现M集包含了J集构造的大量信息．

Reverse bifurcation and fractal of a compound logistic map

Using the bifurcation diagram, the paper reveals that chaotic patterns of the compound Logistic map may emerge out of double-periodic bifurcation, and the chaotic crisis phenomena and the reverse bifurcation are found in the chaotic region. At the same time, we analyze the orbit of critical point of the compound Logistic map and put forward the definition of the Mandelbrot-Julia set of the compound Logistic map. We generalize the Welstead and Cromer's periodic scanning technology and use this technology to construct a series of the Mandelbrot-Julia sets of the compound Logistic map. We investigate the symmetry of the Mandelbrot-Julia set and study the topological inflexibility of distributing of period region in the Mandelbrot set, and find that the Mandelbrot set contains abundant information of structure of the Julia sets by founding the whole portrait of the Julia sets based on the Mandelbrot set qualitatively.