广义高斯和分形序列及其M-J集研究

推广了Lakhtakia 和Berndt等的工作,分析了分形(自相似)序列的生成规则,给出了二次高斯和所生成的分形序列的标度及维数.利用逃逸时间算法,构造了广义高斯和的Mandelbrot-Julia集(M-J集),并从理论上分析了M-J集的周期性和结构特征.结果表明:M-J集由许多螺旋状的花束构成,这种结构在不同水平上嵌套出现,体现了明显的自相似分形特性;随指数值增大,M-J集中的精细花瓣结构增多并趋于复杂;J集在x轴方向上具有周期性.本研究成果有助于理解广义高斯和的动力学性?

Research on generalized Gauss sums fractal sequences and their M-J sets

The studies brought forward by Lakhtakia and Berndt,et al.are extended.The form rules of the fractal sequences(self-similar) and the scale and dimension of fractal sequences generated by quadratic Gauss sums are presented.The generalized Mandelbrot-Julia sets(M-J sets) of generalized Gauss sums are constructed by the escape-time algorithm and the periodicity and structure of them are analyzed in theory.The result shows the following phenomena: The M-J sets consist of many spiry bouquets,such overlapping embedment structure appears at different levels,and it shows self-similar fractal characteristic.The fine bouquet structure grows and gets complicated with the increase of exponent.The J sets are periodic on x-axis.The research can contribute to the comprehension of dynamics of generalized Gauss sums.