Mandelbrot set and Julia sets of fractional order

Abstract

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 $q\uparrow 1$, contrary to expectations, it is not obtained the known shape of the Mandelbrot of integer order, but for $q\downarrow 0$. Also, we conjecture that for $q\downarrow 0$, the fractional-order Mandelbrot set is similar to the integer-order Mandelbrot set, while for $q\downarrow 0$ 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.

Appendices

Appendices

1.1 A Escape-time algoritm for $FO\mathcal {M}$ set and $FO\mathcal {K}_c$ sets

There are several algorithms to plot the sets as the escape-time method, the boundary scanning method, the inverse iteration method. Also there are several optimizations to increase the speed and images accuracy (see, e.g., [5]). For the exposition clarity, in this paper only black/white coloring scheme is used (for color schemes see, e.g., [5]).

In this paper the Mandelbrot and Julia sets, of IO or FO, are obtained with the slow, but easy to understand escape-time algorithm based on the theorem which states that iterating $f_c$, with starting value $z_0$, only one of the following possibilities happens: either the obtained orbit remains bounded by 2, or diverges to $\infty$. For Mandelbrot set $z_0=0$, and c is varied within a complex parametric domain, usually rectangular, while for Julia sets $z_0$ is varied and c fixed. This well known algorithm is considered in order to facilitate the presentation of the algorithm for the $FO\mathcal {M}$ and $FO\mathcal {K}_c$ sets. For simplicity, in this paper the complex parametric plane of c and the plane of the complex variable z are considered similar.

• 1. To generate the $FO\mathcal {M}$ set with the escape-time algorithm, consider in the cartesian plane the image of a rectangular domain of complex numbers c, ${\mathcal {L}}=\{c_x,c_y| c_x\in [c_{x_{min}},c_{x_{max}}], c_y\in [c_{y_{min}},c_{y_{max}}],c_x,c_y\in {\mathbb {R}}\}$, with an equidistant grid of $m_x\times m_y$ points $(c_x,c_y)$, $m_x,m_y\in \mathbb {N^*}$. The exploration of numbers c within the considered complex domain ${\mathcal {L}}$, can be realized with two nested loops, while a third, inner cycle (steps (9)-(13), Fig. 6), the core of the algorithm, makes the escape-time verification. The inside cycle implements the integrals (11) and (13). As for $IO\mathcal {M}$, for the $FO\mathcal {M}$ ${\mathcal {L}}$ is taken ${\mathcal {L}}=\{c_x,c_y| c_x\in [-2.5,0.5],c_y\in [-1.5,1.5]\}$. The domain ${\mathcal {L}}$ is explored with the steps $step_{c_x}=(c_{x_{max}}-c_{x_{min}})/m_x$, and $step_{c_y}=(c_{y_{max}}-c_{y_{min}})/m_y$. To generate the $FO\mathcal {M}$, to each c within ${\mathcal {L}}$, one applies the recurrence (11) until, either after a chosen finite number of iterations, N (in this paper $N=30$), |z(n)|, $n=1,2,\ldots$, $z_0=0$, remains less than 2 and the underlying point c belongs to $FO\mathcal {M}$ being plotted black, or |z(n)| becomes greater or equal to 2 (escape radius), when $c\not \in$ $FO\mathcal {M}$ and c is not plotted. Note that because of the symmetry of the $FO\mathcal {M}$, if one intends to generate the entire $FO\mathcal {M}$ set, one might save about $50\%$ of drawing time if the algorithm is run only on the superior half of the complex plane, with $y_{min}=0$ and plotting $(c_x,\pm c_y)$.

  The pseudocode is presented in Fig. 6.

• 2. To generate $FO\mathcal {K}_c$ sets, one iterates the map $f_c$, but with c fixed and $z_0$ variable inside ${\mathcal {L}}$ with the recurrence (13). The initial condition, variable, is denoted $z_0:=x+\i y$. If after N iterations, |z(n)|, remains less than 2, the underlying point $z_0$ (of coordinates x and y) belongs to $FO\mathcal {K}_c$ and is plotted black. If |z(n)| becomes greater or equal to 2, $z_0\not \in$ $FO\mathcal {K}_c$ and $z_0$ is not plotted. The input data are N and data defining ${\mathcal {L}}$ (${x_{min}},{x_{max}}, {y_{min}}, {y_{max}}$ and $m_x,m_y$), and c. For most of $FO\mathcal {K}_c$ sets, ${\mathcal {L}}=\{x,y| x\in [-2,1],y\in [-1.5,1.5]\}$, not ${\mathcal {L}}=\{x,y| x\in [-1.5,1.5],y\in [-1.5,1.5]\}$ as for $IO\mathcal {K}_c$ sets. The exploration of the domain ${\mathcal {L}}$ is realized with $step_x=(x_{max}-x_{min})/m_x$, and $step_y=(y_{max}-y_{min})/m_y$.

  The pseudocode is presented in Fig. 7. variables xx and yy are designed for the inner loop.

Several speed improvements can be done, such as calculating $x(n)\cdot x(n)+y(n)\cdot y(n)$ instead $x^2(n)+y^2(n)$, or calculating only once the expressions $\frac{\varGamma (n-i+q)}{\varGamma (n-i+1)}$ and $\frac{1}{\varGamma (q)}$, or plotting after the domain ${\mathcal {L}}$ is explored and so on. Also, the algorithm can be written y using, e.g., the vectorial calculus, such as the performing matrix calculus of Matlab. Regarding the implementation in Matlab, a solution for the zero index can be found in [18].