Mandelbrot set and Julia sets of fractional order

The study of the dynamics of complex maps was initiated by P. Fatou and G. Julia in the early of twentieth century, before being revived by B. Mandelbrot. As know, in the complex plane $\mathbb{C}$, the Integer Order (IO) Mandelbrot set represents the set of complex numbers (parameters) $c$ for which the quadratic map $f_{c}(z)=z^{2}+c$ (Mandelbrot map) does not diverge to infinity when it is iterated with $f_{c}$ from $z=0$. This set, has been first defined and drawn by Robert W. Brooks and Peter Matelski in 1978 [1], and later made famous by Benoit Mandelbrot (see e.g. [2]). The dynamics generates by $f_{c}$ represent a huge sources of fractal structures (see e.g. [3, 4, 5, 6]).

Julia sets are made of all points which under iterations do not go to an attractor which may be at infinity. Compared to Mandelbrot set, where $c$ is variable in the parametric plane $\mathbb{C}$, Julia sets are obtained for fixed $c$ and the origin of iterations variable in $\mathbb{C}$.

The Mandelbrot set is known as the set of all points $c$ for which Julia sets are compact and connected.

Details and background on Mandelbrot set and Julia sets can be found in the following works [2, 7, 8, 9, 10, 3, 11], to cite only few of them.

Due the description of memory and hereditary properties, Fractional Order (FO) difference equations still receive increasing attention. However, as mentioned in [12], overall, fractional calculus, closely related to classical calculus, is not direct generalization of classical calculus in the sense of rigorous mathematics. One of the first definitions of a fractional difference operator has been proposed in 1974 [13]. While there are many works on fractional differential equations, there still are only few works in the theory of the fractional finite difference equations. In [14, 15, 16] problems related to Caputo fractional sums and differences can be found, while in [29] Initial Value Problems (IVPs) in fractional differences are studied. For stability of fractional differences compare [17, 18], while properties of fractional discrete logistic map, weakly fractional difference equations, symmetry-breaking of fractional maps can be found in [19, 20, 21]. The nonexistence of periodic solutions in fractional difference equations is analyzed in [22].

Notations utilized in this paper:

• •

  $IO\mathcal{M}$: Mandelbrot of IO;

• •

  $IO\mathcal{K}_{c}$: Filled Julia set of IO;

• •

  $IO\mathcal{J}_{c}$: Julia set of IO;

• •

  $FO\mathcal{M}$: Mandelbrot set of FO;

• •

  $FO\mathcal{K}_{c}$: Filled Julia set of FO;

• •

  DEM: Distance Estimator Method;

• •

  IIM: Inverse Iteration Method

In this section, few of the most important characteristics of $IO\mathcal{M}$ set, and $IO\mathcal{K}_{c}$ sets, which will be revealed analytically or numerically in the cases of Mandelbrot and Julia sets of FO, are briefly presented.

The iteration of $f_{c}$ is obtained by the relation

which generates the sequence

where by $f_{c}^{k}(0)$ one understands $f_{c}(f_{c}^{k-1}(0))$.

A complex number $c\in IO\mathcal{M}$ set if the absolute value of $z_{n}$ in the sequence (2) remains bounded for all $n\in\mathbb{N}^{*}$, $|z_{n}|<2$, for all $n\geq 0$.

As proposed by Mandelbrot, the boundary of the Mandelbrot set represents a fractal curve (Fig. 1 (a)). For the history related to the origin of the $IO\mathcal{M}$ set compare [23].

Consider the set of all points $z_{0}$ which tend to $\infty$ through the iteration (1)

The set $A_{c}(\infty)$ depends on $c$, and his frontier represents the $IO\mathcal{J}_{c}$ set of $f_{c}$.

The filled Julia set for fixed $c$, of IO, related to $f_{c}$, $IO\mathcal{K}_{c}$ which, for computationally reasons is considered in this paper, is the set of all points $z_{0}\in\mathbb{C}$ for which the orbit (2) is bounded

The $IO\mathcal{J}_{c}$ set is contained in the $IO\mathcal{K}_{c}$ set and is the boundary of the $IO\mathcal{K}_{c}$ set

In this paper, beside some new properties related to the Mandelbrot set of FO, the following known properties of Mandelbrot set and Julia sets of IO will be analytically or numerically studied on their FO counterparts.

1. 1.

   The $IO\mathcal{K}_{c}$ sets for purely real $c$, and $IO\mathcal{M}$ set, are symmetric about the real axis (reflection symmetry). Julia sets of $f_{c}$ are symmetric about the origin.

2. 2.

   As known, it is conjectured that the Mandelbrot set is locally connected [7, 23], the full conjecture of this very technical and complicated property being still open. Mandelbrot had decided empirically that his isolated islands were actually connected to the mainland by very thin filaments [24]. In this paper we are interested in the connectivity property as a computationally property which by using, beside the escape-time algorithm (see Appendix A), a performing method (Distance Estimator Method), revel the filaments connecting the apparently isolated islands to the mainland of the Mandelbrot set or Julia sets, of IO or FO. Empirical small areas are considered.

3. 3.

   The $IO\mathcal{M}$ set is the set of all parameters $c$ for which $IO\mathcal{K}_{c}$ are connected sets;

4. 4.

   The $FO\mathcal{M}$ set is bounded;

5. 5.

   For $c$ situated outside of $IO\mathcal{M}$ one obtains Cantor sets (“dust” sets composed of infinitely disjoint points);

6. 6.

   The Julia sets can be connected, disconnected or totally disconnected (see e.g. [25, 26, 27]). To note that the $IO\mathcal{M}$ set represents the set of all points in the complex plane for which the alternated Julia sets [28] are disconnected (but not totally disconnected).

In this paper the filled Julia sets are considered.

Hereafter, by point in the complex plane, one understands the image in the complex plane of a complex number.

Drawing Mandelbrot set and Julia sets of IO and FO, bases 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 $\infty$ [5].

In Fig. 1 (a) is presented the $IO\mathcal{M}$ set, while in Figs. 1 (c)-(j) there are presented several $IO\mathcal{K}_{c}$ sets for different $c$ values. In Fig. 1 (c), there is presented an $IO\mathcal{J}_{c}$ set with no interior, with $c=0.359+\i 0.599$, while in Fig.1 (d) there are presented comparatively, both the $IO\mathcal{K}_{c}$ and $IO\mathcal{J}_{c}$ sets for the same value of parameter $c=0.276+\i 0.536$, obtained with the time-escape algorithm and IIM, [5], respectively.

In this section the fractional discretization of the Mandelbrot and Julia sets of FO, in the sense of Caputo-like are introduced.

To obtain the fractional discretization of the $FO\mathcal{M}$, consider the time scale $N_{a}=\{a,a+1,a+2,...\}$. For $q>0$ and $q\not\in\mathbb{N}$ the $q$-th Caputo-like discrete fractional difference of a function $u:N_{a}\rightarrow\mathbb{R}$ is defined as is defined [29, 30] as

for $t\in N_{a+n-q}$ and $n=[q]+1$.

$\Delta^{n}$ is the $n$-th order forward difference operator,

while $\Delta_{a}^{-q}$ represents the fractional sum of order $q$ of $u$, namely,

with the falling factorial $t^{(q)}$ in the following form:

The fractional operator $\Delta_{a}^{-q}$ maps functions defined on $\mathbb{N}_{a}$ to functions on $N_{a+q}$ (for time scales see, e.g., [31]).

For the case considered in this paper, $q\in(0,1)$, when $\Delta u(s)=u(s+1)-u(s)$, $n=1$, and starting point $a=0$, $q$-th Caputo’s fractional derivative, $\Delta^{q}$, becomes

Consider next, the FO autonomous IVP in the sense of Caputo’s derivative

with $f$ a continuous real valued map and $q\in(0,1)$. The numerical solution is

or, in the convenient form for numerical simulation [21]

The recursive iteration implies that (7) is equivalent to the IVP (5) [29] and, therefore, the study of the IVP (5) can be realized, both analytically and numerically, on (7).

Consider next, the following FO variant of (5), in the sense of Caputo’s fractional derivative (4)

with $f$ a complex continuous function of variable $z=x+\i y\in\mathbb{C}$ and $z_{0}\in\mathbb{C}$. Then, the numerical integral (7) of the IVP (8) becomes

If one consider the FO variant of the IVP (1), with scaled $c$ within a parametric complex domain, $z(0)=0$, one obtains the numerical integral defining the $FO\mathcal{M}$ map

To facilitate understanding the numerical implementation of (10), with $\Re(f_{c}(z))=x^{2}-y^{2}+c_{x}$, $\Im(f_{c}(z))=2xy+c_{y}$, one gets the following scalar form

where $x(0)=y(0)=0$.

If $z(0)$ is variable within a complex domain and $c$ is fixed, one obtains the integral defining the $FO\mathcal{K}_{c}$ sets

The scalar form of (12) is

Note that for the $FO\mathcal{K}_{c}$ sets, $z(0)$ (i.e. $x(0)$, and $y(0)$ in (12) and (13) are variable and scan a complex domain of the variable $z(0)$ (see variables $xx$ and $yy$ in Algorithm 2, Appendix A).

If $c\in$$FO\mathcal{M}$, the orbits generating by (10) are bounded¹¹1Boundedness property of $FO\mathcal{M}$ for $c\in\mathbb{C}$ is not analyzed here (The proof, for $c\in\mathbb{R}$, is presented in Proposition 5).. Then

with $|\overline{z(n)}|=|z(n)|$, so if $c\in$$FO\mathcal{M}$ then also $\overline{c}\in$$FO\mathcal{M}$.

Also, for $c$ purely real, following the same reasoning, because $\overline{c}=c$, like for $IO\mathcal{K}_{c}$ sets, one has the following property

Next, for $c\in\mathbb{R}$ one has

Using

one obtains for $c>0$

Thus, we get

Property 3 is verified numerically too as indicated by the light blue line AB where $c\in R$ and $c>0$, and $c\notin$ $FO\mathcal{M}$ set in Fig. 3 (a) for $q=1$ and also Fig. 5 (a) for $q=0.5$.

Furthermore, for $c\in\mathbb{R}$ and $c<0$, one obtains

Assuming

one obtains $z(2)\geq 0$.

From Propositions 3 and 4 one can deduce the boundedness of the $FO\mathcal{M}$ for real $c$.

If the IVP (8) is considered in the sense of $FO\mathcal{M}$ map (with scanning $c$ and $z(0)=0$), for $q\uparrow 1$,the numerical integral (21) becomes

while for $q\downarrow 0$

If the IVP (8) is considered in the sense of $FO\mathcal{K}_{c}$ sets (with scanning $z(0)$ and fixed $c$), for $q\uparrow 1$, the numerical integral (21) becomes

while for $q\downarrow 0$

$FO\mathcal{M}$ set and $FO\mathcal{K}_{c}$ sets are generated with the escape-time algorithm adapted to complex FO discrete equations and is presented in Appendix A.

In (10), numerically, $q$ can take the limit value 1 (see Remark 6)). Therefore, hereafter, the limit case $q\uparrow 1$ will be considered $q=1$.

Regarding the other limit case, $q\downarrow 0$, due to the fact $\Gamma(0)$ is not defined and the limit is considered on small intervals, in this paper for numerically reasons, the limit has been considered empirically, as $q=1e-15$, and will be denoted $0_{+}$.

For $q\downarrow 0$, linked to Remark 6, one get numerically the following important identity properties (congruent-like shapes) which, for now, are introduced as conjectures

$FO\mathcal{K}_{c}$ sets are generated for $c\in$$FO\mathcal{M}$, $c\notin$$FO\mathcal{M}$ and, for some cases, for $c$ close to $FO\mathcal{M}$ neighborhood. To verify the connectedness property, some empirical zoomed details are considered, where DEM [5] is used.

In Figs. 2 (a)-(d) four representative cases of $FO\mathcal{M}$ sets for $q=0_{+}$, $q=0.25$, $q=0.5$ and $q=1$, respectively, are comparatively presented.

• 1.

  Consider the $FO\mathcal{M}$ for $q=1$ (Figs. 3). As can be seen, there are several differences compared to the IO case. Thus, contrary to expectations, according to which $FO\mathcal{M}$ should be similar with $IO\mathcal{M}$ while $q$ approaches $1$ (FO cases are considered, generally, that they are generalizations of IO cases), the $FO\mathcal{M}$ set for $q=1$ is far from being similar to the $IO\mathcal{M}$ set. Also, the connectedness property of the $IO\mathcal{M}$ seems to be broken. For example, the zoom of the detail D in Fig. 3 (a), obtained with DEM method (Fig. 3 (b)), shows that the $FO\mathcal{M}$ seems to not have the connectedness property. On the other hand, again contrary to expectations, for $q=1$, $FO\mathcal{K}_{c}$ sets look similar to IO Julia sets, but translated with $-0.25$ along the real axis. See for example, the $FOK_{0}$ and $IOK_{0.25}$ (Fig 3 (j) and Fig. 1 (j)), or $FOK_{-0.25}$ and $IOK_{0}$ (Fig. 3 (i) and Fig. 1 (i)). Also, the central symmetry encountered at $IO\mathcal{K}_{c}$ sets is verified, but not with respect the origin, but with respect to a horizontally translated center. Like for the $IO\mathcal{M}$ set, points $c$ within $FO\mathcal{M}$ generate connected $FO\mathcal{K}_{c}$ (Figs. 3 (d), (f), (h-j)), while points outside $FO\mathcal{M}$ generate disconnected fractals (Fig. 3 (c)). Due to the inherent numerical errors, connectedness property cannot be precisely numerically stated for points close to $FO\mathcal{M}$ frontier (see e.g. Fig. 3 (e) or (g)).

• 2.

  Let the other extreme case, $q=0_{+}$ (Figs. 4). Again, contrary to expectations, in this case, when $q$ approaches $0$, the $FO\mathcal{M}$ is similar to $IO\mathcal{M}$ (Conjecture 1). Empirically verified connectedness property seems to verify (see detail D and his zoom in Fig. 4 (a) and (b)). The $FO\mathcal{K}_{c}$ sets look like some parts of a $\mathcal{M}$ set (probably parts of the $FO\mathcal{M}$) obtained with a magnifier-like. Another property is the fact that, for all values of $c$, within or outside the $FO\mathcal{M}$ set, the $FO\mathcal{K}_{c}$ sets still look connected sets. Also, except the points $c$ on the real axis, the $FO\mathcal{K}_{c}$ sets have not central symmetry. However, probably the most important property is the fact that for $c=0$, the $FO\mathcal{K}_{c}$ set is actually the $IO\mathcal{M}$ set (see the red rectangle in Fig. 4 (i) and Conjecture 2).

• 3.

  For the intermediate case $q=0.5$, see Figs. 5. Now, excepting the values of $c$ belonging to the real axis, where the $FO\mathcal{K}_{c}$ sets are symmetric with respect the real axis (see Figs. 5 (h), (g), (e), (d) and Proposition 2), the central symmetry and also connectedness of the $FO\mathcal{K}_{c}$ are lost (see the zoom of detail D in Fig. 5 (a), presented in Fig. 5 (b)). Also, the frontier of the $FO\mathcal{K}_{c}$ is not connected (see the zoomed detail of the region $D_{1}$, Figs. 5 (e) and (f)).

In this paper the FO Mandelbrot map and set and FO Julia sets in the Caputo’s sense are introduced. Some properties of the $FO\mathcal{M}$ map and $FO\mathcal{K}_{c}$ are analytically studied and some of them numerically verified. The $FO\mathcal{M}$ set and $FO\mathcal{K}_{c}$ sets are obtained with the escape-time algorithm adapted for FO discrete complex maps, while to verify computationally the connectedness properties for some cases, zoomed details are obtained with DEM adapted to FO discrete complex maps. Similarities, and especially differences, between the IO case and FO case are summarizing next.

• •

  Probably the most interesting property is the one according to which, for $q\downarrow 0$, the $FO\mathcal{K}_{c}$ set corresponding to $c=0$, looks similar to the Mandelbrot of IO set (Fig. 4, Conjecture 2);

• •

  For all $q$ values, The $FO\mathcal{M}$ set and the $FO\mathcal{K}_{c}$ sets have symmetry with respect the real axis (Proposition 1 and Proposition 2, respectively);

• •

  Excepting the case $q=1$, the $FO\mathcal{K}_{c}$ sets have not central symmetry;

• •

  For $q=0_{+}$, the underlying $FO\mathcal{K}_{c}$ corresponding to $c=0$, is not the filled circled, like in the IO case, but the $IO\mathcal{M}$ set (see Fig. 1 (i) and Fig. 4 (i));

• •

  For $q>0$, the $FO\mathcal{M}$ set loose connectedness property (see Paragraphs 1 and Paragraph 3, Section 5), while for $q=0_{+}$, this property seems verify (see detail D in Figs. 4 (a) and (b));

• •

  For $q=0_{+}$ and also for some $q>0$, the $FO\mathcal{K}_{c}$ sets are imbedded within rectangular domains $\mathcal{L}$ which, contrary to the $IO\mathcal{K}_{c}$ sets, are not centered at the origin, but translated along the real axis (see Paragraphs 1 and 3, Section 5);

• •

  The $FO\mathcal{M}$, with $q=1$, is translated horizontally with $-0.25$ (see Figs. 1 and Figs. 3);

• •

  For $q=1$, $FO\mathcal{K}_{c}$ sets presents a central symmetry with respect a translated center (see Fig. 3);

• •

  Contrary to the IO case, except the case $q=1$, for $q\in(0,1)$, the $FO\mathcal{K}_{c}$ sets have not central symmetry (see Paragraphs 2 and Paragraph 3, Section 5);

• •

  The FO discretization for the Mandelbrot set, for $q=1$, is not as expected, a generalization of the $IO\mathcal{M}$ set (Fig. 3).