广义M-集周期芽苞Fibonacci序列的拓扑不变性

Researches on the Topological Invariable Fibonacci Sequence of the Periodic Buds in the General Mandelbrot Sets

研究了复映射z←zα+c(α <0 )所产生的广义Mandelbrot集 ,利用逃逸时间算法绘制广义M 集混沌分形图谱 ,经大量计算机数学实验 ,得知逃逸区嵌于稳定区中 ,并由此得出稳定区的周期数·同时利用代数方程解出周期芽苞的数量及位置 ,为更好的了解M 集的结构提供了理论依据·另外作者发现M 集周期芽苞的Fibonacci序列的拓扑不变性 ,并在目前公认的通向混沌的三种途径的基础上 ,阐述了Fibonacci序列是通向混沌的又一途径 ,为建立新的数据加密、压缩。

The inner structure of the general Mandelbrot sets generated by the complex map z←z α+c (α<0)was studied.A series of families of chaos fractal images were generated by using the escape time algorithm. The escaping area was embedded in stable area by making many computational mathematic experiments .Periodic numbers of stable area and the numbers and position of the periodic buds were got by solving algebraic equations. This presents a better understanding on the structure of the Mandelbrot sets.Furthermore,the topological invariance on the Fibonacci sequence of the periodic buds was discovered. The Fibonacci sequence is another way to the chaos except three commonly accepted ways to the chaos. The Fibonacci sequence can be used in the encryption,compression and storage of data.