Newton迭代法的吸引域及其Julia集

Basins of Attraction and Julia Set with Newton's Iteration Method

用Newton迭代法讨论f(z)=zα-1在复平面上零点的吸引域及其Newton迭代函数的Julia集随α的不同的变化.当α是整数时f(z)=zα-1的零点的吸引域及Julia集是次旋转对称的;当α是非整数时,不具有旋转对称性,是一种过渡状态,并且这种过渡状态对α从奇数变成偶数与从偶数变成奇数的过渡形式是不同的,而且从奇数变成偶数时情况更为复杂.在某一范围内的α值,在复平面上存在一个包含开区域的点集,其Newton迭代不收敛于任何不动点,从而进一步说明:即使简单的复迭代系统还有许多复杂和未知现象需要我们去探讨.

Newton's Iteration method is applied to f(z)=z α-1 in the complex plane And its basins of attraction for zero points and its Julia set are discussed With the α change, the basins of attraction and Julia set change If α is integer,the basins of attraction and Julia set is α spin symmetry If α is non-integer, it has not spin symmetry It is a transitional state The transitional state for α change from odd to even is different to from even to odd, it is more complicated If α is with in a certain range there is a set containing open region of starting points, the Newton's method does not converge to any fixed points Therefore, even the simple complex iterated systems, there are much complicated and unknown phenomena to be discussed