论文标题
Mandelbrot套件和朱莉娅分数订单集
Mandelbrot set and Julia sets of fractional order
论文作者
论文摘要
在本文中，引入了（0,1）$的$ q \ in（0,1）$的$ q $ - th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-th-like离散分数差异，并且在分析和数值研究中进行了几个属性。揭示了分数模型的一些有趣的特性。因此，对于$ q \ uparrow1 $，与期望相反，它并不是Integer Order的Mandelbrot的已知形状，而是以$ Q \ downarrow0 $的形式获得。此外，我们猜想对于$ q \ downarrow0 $，分数级的mandelbrot集类似于整数订购的mandelbrot集，而对于$ q \ downarrow0 $和$ c = 0 $，这是基础分数的朱利亚集合之一，类似于Integer-rorder mandelbrot集合。为了支持我们的猜想，进行了几项广泛的数值实验。为了绘制分数顺序的mandelbrot和Julia集，使用了基本初始值问题的数值积分，而为了绘制集合，使用了针对分数订单案例的逃生时间算法。该算法表示为伪代码。
In this paper the fractional-order Mandelbrot and Julia sets in the sense of $q$-th Caputo-like discrete fractional differences, for $q\in(0,1)$, are introduced and several properties are analytically and numerically studied. Some intriguing properties of the fractional models are revealed. Thus, for $q\uparrow1$, contrary to expectations, it is not obtained the known shape of the Mandelbrot of integer order, but for $q\downarrow0$. Also, we conjecture that for $q\downarrow0$, the fractional-order Mandelbrot set is similar to the integer-order Mandelbrot set, while for $q\downarrow0$ and $c=0$, one of the underlying fractional-order Julia sets is similar to the integer-order Mandelbrot set. In support of our conjecture, several extensive numerical experiments were done. To draw the Mandelbrot and Julia sets of fractional order, the numerical integral of the underlying initial values problem of fractional order is used, while to draw the sets, the escape-time algorithm adapted for the fractional-order case is used. The algorithm is presented as pseudocode.
