Mandelbrot set and Julia sets of fractional order Marius-F. Danca STAR-UBB Institute

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Mandelbrot set and Julia sets of fractional order

Marius-F. Danca

STAR-UBB Institute,

Babes-Bolyai University,

400084, Cluj-Napoca, Romania

and

Romanian Institute of Science and Technology,

400487, Cluj-Napoca, Romania

danca@rist.ro

and

Michal Feˇckan

Faculty of Mathematics, Physics and Informatics,

Comenius University in Bratislava,

84215 Bratislava, Slovakia

and

Mathematical Institute, Slovak Academy of Sciences,

84104 Bratislava, Slovakia

michal.feckan@gmail.com

October 6, 2022

Abstract

In this paper the fractional-order Mandelbrot and Julia sets in the sense of q-th Caputo-

like discrete fractional diﬀerences, for q∈(0,1), are introduced and several properties are

analytically and numerically studied. Some intriguing properties of the fractional models are

revealed. Thus, for q↑1, contrary to expectations, it is not obtained the known shape of the

Mandelbrot of integer order, but for q↓0. Also, we conjecture that for q↓0, the fractional-

order Mandelbrot set is similar to the integer-order Mandelbrot set, while for q↓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.

keywords: Mandelbrot set of fractional order; Julia set of fractional order; Caputo-like

discrete fractional diﬀerence

arXiv:2210.02037v1 [nlin.CD] 5 Oct 2022

1 Introduction

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 C, the

Integer Order (IO) Mandelbrot set represents the set of complex numbers (parameters) cfor which

the quadratic map fc(z) = z2+c(Mandelbrot map) does not diverge to inﬁnity when it is iterated

with fcfrom 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 fcrepresent 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 cis variable in the parametric plane C, Julia sets

are obtained for ﬁxed cand the origin of iterations variable in C.

The Mandelbrot set is known as the set of all points cfor 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:

•IOM: Mandelbrot of IO;

•IOKc: Filled Julia set of IO;

•IOJc: Julia set of IO;

•F OM: Mandelbrot set of FO;

•F OKc: Filled Julia set of FO;

•DEM: Distance Estimator Method;

•IIM: Inverse Iteration Method

2 Mandelbrot set and Julia sets of integer order

In this section, few of the most important characteristics of IOMset, and IOKcsets, which will

be revealed analytically or numerically in the cases of Mandelbrot and Julia sets of FO, are brieﬂy

presented.

The iteration of fcis obtained by the relation

zn=z2

n−1+c, z0= 0, n ∈N∗={1,2, ...},(1)

which generates the sequence

z0= 0, z1=fc(0) = c, z2=f2

c(0) = c2+c, z3=f3

c(0) = (c2+c)2+c, ... (2)

where by fk

c(0) one understands fc(fk−1

c(0)).

A complex number c∈IOMset if the absolute value of znin the sequence (2) remains bounded

for all n∈N∗,|zn|<2, for all n≥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 IOMset compare [23].

Consider the set of all points z0which tend to ∞through the iteration (1)

Ac(∞) = {z0∈C:fk

c(z0)→ ∞,as k→ ∞}.

The set Ac(∞) depends on c, and his frontier represents the IOJcset of fc.

The ﬁlled Julia set for ﬁxed c, of IO, related to fc,IOKcwhich, for computationally reasons is

considered in this paper, is the set of all points z0∈Cfor which the orbit (2) is bounded

IOKc=C\Ac(∞) = {z0∈C:fk

c(z0) remains bounded for all k}.

The IOJcset is contained in the IOKcset and is the boundary of the IOKcset

∂IOKc=IOJc=∂Ac(∞).

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. The IOKcsets for purely real c, and IOMset, are symmetric about the real axis (reﬂection

symmetry). Julia sets of fcare symmetric about the origin.

2. As known, it is conjectured that the Mandelbrot set is locally connected [7, 23], the full conjec-

ture 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 ﬁl-

aments [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. The IOMset is the set of all parameters cfor which IOKcare connected sets;

4. The F OMset is bounded;

5. For csituated outside of IOMone obtains Cantor sets (“dust” sets composed of inﬁnitely

disjoint points);

6. The Julia sets can be connected, disconnected or totally disconnected (see e.g. [25, 26, 27]).

To note that the IOMset 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 fc, with starting value z0, only one of the following possibilities happens: either the obtained

orbit remains bounded by 2, or diverges to ∞[5].

In Fig. 1 (a) is presented the IOMset, while in Figs. 1 (c)-(j) there are presented several

IOKcsets for diﬀerent cvalues. In Fig. 1 (c), there is presented an IOJcset with no interior, with

c= 0.359 + ı0.599, while in Fig.1 (d) there are presented comparatively, both the IOKcand IOJc

sets for the same value of parameter c= 0.276 + ı0.536, obtained with the time-escape algorithm

and IIM, [5], respectively.

3 Fractional Order Mandelbrot map

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 F OM, consider the time scale Na={a, a +

1, a + 2, ...}. For q > 0 and q6∈ Nthe q-th Caputo-like discrete fractional diﬀerence of a function

u:Na→Ris deﬁned as is deﬁned [29, 30] as

∆q

au(t)=∆−(n−q)

a∆nu(t) = 1

Γ(n−q)

t−(n−q)

s=a

(t−s−1)(n−q−1)∆nu(s),(3)

for t∈Na+n−qand n= [q] + 1.

∆nis the n-th order forward diﬀerence operator,

∆nu(s) =

k=0 n

k(−1)n−ku(s+k),

while ∆−q

arepresents the fractional sum of order qof u, namely,

∆−q

au(t) = 1

Γ(q)

t−q

s=a

(t−s−1)(q−1)u(s), t ∈Na+q,

with the falling factorial t(q)in the following form:

t(q)=Γ(t+ 1)

Γ(t−q+ 1).

The fractional operator ∆−q

amaps functions deﬁned on Nato functions on Na+q(for time scales

see, e.g., [31]).

For the case considered in this paper, q∈(0,1), when ∆u(s) = u(s+ 1) −u(s), n= 1, and

starting point a= 0, q-th Caputo’s fractional derivative, ∆q, becomes

∆qu(t) = 1

Γ(1 −q)

t−(1−q)

s=a

(t−s−1)(−q)∆u(s).(4)

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

∆qu(t) = f(u(t+q−1)), t ∈N1−q, u(0) = u0,(5)

with fa continuous real valued map and q∈(0,1). The numerical solution is

u(t) = u0+1

Γ(q)

t−q

s=1−q

(t−s−1)(q−1)f(u(s+q−1)),(6)

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MandelbrotsetandJuliasetsoffractionalorderMarius-F.DancaSTAR-UBBInstitute,Babes-BolyaiUniversity,400084,Cluj-Napoca,RomaniaandRomanianInstituteofScienceandTechnology,400487,Cluj-Napoca,Romaniadanca@rist.roandMichalFeckanFacultyofMathematics,PhysicsandInformatics,ComeniusUniversityinBratislava,84215...

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Mandelbrot set and Julia sets of fractional order Marius-F. Danca STAR-UBB Institute

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