Applied Mathematics and Computation 217 (2010) 2883–2890

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

On a topological closeness of perturbed Julia sets

Ioannis Andreadis a, Theodoros E. Karakasidis b,⇑a International School of The Hague, Wijndaelerduin 1, 2554 BX The Hague, The Netherlandsb Department of Civil Engineering, University of Thessaly, GR-38334 Volos, Greece

a r t i c l e i n f o

Keywords:Julia setMandelbrot mapAdditive dynamic noiseDistance

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.08.024

⇑ Corresponding author.E-mail addresses: i.andreadis@ish-rijnlandslyceu

a b s t r a c t

In the present work we expand our previous work in [1] by introducing the Julia DeviationDistance and the Julia Deviation Plot in order to study the stability of the Julia sets of noise-perturbed Mandelbrot maps. We observe a power-law behaviour of the Julia Deviation Dis-tance of the Julia sets of a family of additive dynamic noise Mandelbrot maps from the Juliaset of the Mandelbrot map as a function of the noise level. Additionally, using the abovetools, we support the invariance of the Julia set of a noise-perturbed Mandelbrot mapunder different noise realizations.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

In this work, we provide a mathematical framework for studying the stability and the changes of the morphology of theJulia sets of the Mandelbrot map under small changes of the parameters. This framework also applies for the study of theJulia sets of noise-perturbed Mandelbrot maps. For that aim, we define a metric, the Julia Deviation Distance, among the Juliasets of the same Mandelbrot map associated with different values of the parameters and perturbed Mandelbrot maps, as in[1], which is based on the finite escape algorithm [2,3]. Hence, we provide a mathematical framework for the notion of sim-ilar shape of the Julia sets of perturbed Mandelbrot maps used in [4–6].

Furthermore we introduce a graphical tool the Julia Deviation Plot in order to investigate the way in which the Julia set isdeformed due to the changes of the parameters of the Mandelbrot map or to the perturbations of the Mandelbrot map as in[1] for the study of the Mandelbrot sets of perturbed Mandelbrot maps. In fact, using this graphical tool, one can localize theregions of the Julia set that are affected by the changes of the parameter or by the effects of the noise. One of the interestingfindings that comes out from this analysis is that the perturbations lead to creations of points belonging to the Julia set of aperturbed Mandelbrot map along with the points that lose their property of belonging to the Julia set of the originalMandelbrot map.

In addition, our analysis reveals the presence of a three-regime power-law behaviour of the Julia Deviation Distance of theJulia sets of a family of additive dynamic noise Mandelbrot maps from the Julia set of the Mandelbrot map as a function ofthe level of noise. Subsequently, using the new tools, we support the independence of the Julia set of a noise-perturbedMandelbrot map [4–6] by different realizations of the noise used to create them.

Furthermore it would be of interest to consider the application of the tools introduced in this paper, the Julia DeviationDistance and the Julia Deviation plot, to the study of the Julia sets of the noise perturbed generalized Mandelbrot and Juliasets [7–10], of the Noise-perturbed quaternionic Mandelbrot sets [11], and of the superior Julia sets [12]. These tools couldprovide further insight on the way the morphology of the generalized Mandelbrot and Julia sets introduced in the references[7–12] is changing due to the application of noise.

. All rights reserved.

m.nl (I. Andreadis), thkarak@uth.gr (T.E. Karakasidis).

2884 I. Andreadis, T.E. Karakasidis / Applied Mathematics and Computation 217 (2010) 2883–2890

Here, we consider a Gaussian noise with values in the interval [0,1]. In this paper, using different noise time series, weconstruct a series of different noise-perturbed Mandelbrot maps. As the resulting Julia Deviation Distances of the Julia sets ofthe perturbed Mandelbrot maps from the Julia set of the Mandelbrot map are all equal within two decimal places, we sup-port the independence of the Julia set of a noise-perturbed Mandelbrot map under different noise realizations.

2. On a Julia Deviation Distance and a Julia Deviation Plot of the Julia sets of a Mandelbrot map

We consider a Mandelbrot map, denoted by QC in this paper, which is also known as the complex logistic map [2,3] and isdefined as follows:

xnþ1 ¼ ðxnÞ2 � ðynÞ2 þ c1; ð1Þ

ynþ1 ¼ 2xnyn þ c2;

where c1, c2 2 R.Let us recall the definition of the Julia set [2,3] of the Mandelbrot map QC which is denoted in this paper by J(QC) . We fix a

value of the parameter C = (c1,c2), and consider variations of the initial condition (x0,y0) . The Julia set of the map QC , is theset of all the values of the initial condition (x0,y0) such that limn!1jQn

Cðx0; y0Þj <1.

Fig. 1. The Julia set of the Mandelbrot Map QC with (a) C = C1 = (�0.3904,�0.58769) and (b) C = C2 = (�0.3905,�0.58779).

Fig. 2. The Julia Deviation Plot of the Julia sets of the Mandelbrot Map QC with C = C1 = (�0.3904,�0.58769) and of C = C2 = (�0.3905,�0.58779).

I. Andreadis, T.E. Karakasidis / Applied Mathematics and Computation 217 (2010) 2883–2890 2885

In the following, we recall briefly the finite escape algorithm as explained in [3] for the numerical calculation of the Juliaset. We fix an interval of the space of initial conditions (x0,y0) as follows: �1.5 6 x0 6 1.5 and �1 6 y0 6 1. These are the val-ues that we also used in our previous publications [4–6]. Subsequently, we consider a lattice of initial conditions. Then, forthe value of the parameter C = (c1,c2), we consider the initial conditions of the lattice. Following 500 iterations, we calculatethe value of the distance r of the 500th iteration from the origin, i.e. r = x2 (500) + y2 (500). If r 6 10, then we keep the initialcondition (x0,y0) in a file, otherwise we ignore this point and, finally, we plot the resulting file, which gives us the corre-sponding Julia set of the Mandelbrot map.

In Fig. 1a and b we present the Julia sets of the maps QC, with C = C1 = (�0.3904,�0.58769) considered in [4] and with avalue of C = C2 = (�0.3905,�0.58779), a point close to C1 within a radius of 0.0001. It is clear that the Julia sets J(QC1) andJ(QC2) of the Mandelbrot map QC look similar and close to each other. In order to provide a mathematical quantificationof this observation, we introduce the Julia Deviation Distance between two Julia sets J(QC1) and J(QC2) of the same Mandelbrotmap based on the finite escape algorithm, followed the method introduced in [1] for defining the Mandelbrot deviation dis-tance among perturbed Mandelbrot maps. Although the steps we follow are similar with those presented in our previouspublication [1], we provide them in the Appendix A, for the completeness of the present paper. The calculated Julia DeviationDistance for the Julia sets presented in Fig. 1 is distJ(J(QC1), J(QC2)) = 0.0935.

However, the Julia Deviation Distance, although it provides a quantification of the difference between perturbed Julia sets,it does not provide us with information about the way that the deformation of the one Julia set to the other takes place. Thiswill be achieved with a graphical tool that we name Julia Deviation Plot, denoted as JDPQc1,2, followed the method introducedin [1] for defining the Mandelbrot Deviation Plot among perturbed Mandelbrot maps. For the completeness of the presentpaper, we provide its definition in Appendix B. In Fig. 2, we present the Julia Deviation Plot of the Julia sets of the MandelbrotMap QC with C = C1 = (�0.3904,�0.58769) and of C = C2 = (�0.3905,�0.58779).

3. On a topological stability of the Julia sets of perturbed Mandelbrot map

In this paragraph, we consider the Julia sets of an additive dynamic noise Mandelbrot map considered in [13], and definedas

xnþ1 ¼ ðxnÞ2 � ðynÞ2 þ c1 þ a1wn; ð2Þ

ynþ1 ¼ 2xnyn þ c2 þ a2wn;

where wn is a noise input and a1, a2 2 R, denote the strength of the additive dynamic noise. The special case a1 = a2 = a isdenoted as ADa

c .In this work, we will consider first the case of the additive dynamic noise Mandelbrot map AD0:01

c . In Fig. 3, we present itsJulia sets of the maps, with C = C1 = (�0.3904,�0.58769) (Fig. 3a) considered in [4] and C = C2 = (�0.3905,�0.58779) (Fig. 3b).

Fig. 3. The Julia set of the additive dynamic noise Mandelbrot Map AD0:01c with (a) C = C1 = (�0.3904,�0.58769) and (b) C = C2 = (�0.3905,�0.58779).

Table 1The Julia Deviation Distance of J(ADC) and J(QC) .

Parameter C1 = (�0.3904,�0.58769) C2 = (�0.3905,�0.58779)

distJ J AD0:01c

� �; JðQCÞ

� �0.07074 0.01

2886 I. Andreadis, T.E. Karakasidis / Applied Mathematics and Computation 217 (2010) 2883–2890

It is clear that the Julia sets J AD0:01c1

� �with J(QC1) and J AD0:01

c2

� �with J(QC2) look similar and close to each other. The corre-

sponding calculated Julia Deviation Distances as we introduced in paragraph 2, are summarized in Table 1.In order to provide an insight into the mechanism which affects the change of the morphology of the J AD0:01

c

� �and J(QC)

we produce in Fig. 4 the Julia Deviation Plot of the two sets J AD0:01c

� �and J(QC) . Here again one can localize the regions that

are affected by the noise.

4. The Julia Deviation Distance as a function of the level of noise

In this paragraph, we consider the Julia Deviation Distance of the Julia sets of a family of additive dynamic noise Mandelb-rot Maps ADa

c from the Julia set of the original Mandelbrot map Q C ;distJðJ ADac

� �; JðQCÞÞ, as a function of the level of noise a ,

Fig. 4. The Julia Deviation Plot of J AD0:01c

� �and J(QC) with C = (�0.3904,�0.58769).

I. Andreadis, T.E. Karakasidis / Applied Mathematics and Computation 217 (2010) 2883–2890 2887

while fixing the values of the parameter C. For this aim, we consider values for the level of noise a starting from 0.0001 up to1 for two different noise realizations.

The results appear in Fig. 5, where the distJðJ ADac

� �; JðQCÞÞ as a function of the level of the noise a while fixing the value of

the parameter C, with C = (�0.3904,�0.58769) is presented. We can see that for small values of noise level (10�4–10�3), the

0.01

0.1

1

0.0001 0.001 0.01 0.1 1

dist

J

noise levelFig. 5. The Julia Deviation Distances of additive dynamic noise Mandelbrot maps ADa

c as a function of the values of a, with C = (�0.3904,�0.58769) for twodifferent noise realizations represented with blue and red colour on the graph. (For interpretation of the references to color in this figure legend, the readeris referred to the web version of this article.)

Fig. 6. The Julia Deviation Plot of J ADac

� �and J(QC) with C = (�0.3904,�0.58769) for a = a*.

2888 I. Andreadis, T.E. Karakasidis / Applied Mathematics and Computation 217 (2010) 2883–2890

distJðJ ADac

� �; JðQ CÞÞ is very small since there are only small differences from the unperturbed Julia set ADa

c . For higher valuesof noise level in the region 10�3�10�2 the changes become more important; the distJðJ ADa

c

� �; JðQ CÞÞ increases as the noise

level increases. Finally for values of noise level in the region 10�2 �1 the distJðJ ADac

� �; JðQCÞÞ increases more significantly.

It is of interest to note that above a given value of noise level, denoted a* , the perturbed Julia set has no points, i.e. it iscompletely destroyed by the noise. This results in that for noise levels a P a* the Julia Deviation Distances distJðJ ADa

c

� �;

JðQ CÞÞ takes a constant value. This happens since the Julia set of the additive dynamic noise Mandelbrot map loses its Juliastructure and as a result, on the lattice points there are only the points of the Julia set of the unperturbed Mandelbrot mapwhich have the Julia property. This limit value of the Julia Deviation Distance is given by:

Table 3The Juli

Pertu

distJ

distJðJðADac Þ; JðQCÞÞ ¼

nðJðQCÞNM

; ð3Þ

for all the values of the level of noise a P a* . This can be seen in Fig. 6 where we present the Julia Deviation Plot of the twosets J ADa�

c

� �and J(QC) where only death points are present and it is the same set as those of the unperturbed Julia set of

Fig. 1a.Furthermore our analysis showed that we can fit a three region power-law behaviour of distJ as a function of noise level a,

i.e. of the form

distJ ¼ Aab; ð4Þ

Table 2The corresponding exponents for the power-law behaviour of the distJ as a function of noise level regions as well as thecorresponding noise realizations (the two values of A and b correspond to two different noise realizations).

Noise level regions ANoise realization Fig. 5 red colour /blue colour

bNoise realization Fig. 5 red colour /blue colour

10�4–10�3 0.056/0.053 0.002/�0.00310�3–10�2 0.128/0.118 0.125/0.113

10�2–a* 0.26/0.22 0.302/0.263

a Deviation Distances of J AD0:01c;i

� �from J(QC) with C = (�0.3904,�0.58769) for a1 = a2 = 0.01.

rbation AD0:01c;1 AD0:01

c;2 AD0:01c;3 AD0:01

c;4 AD0:01c;5

ðJ AD0:01c;i

� �; JðQC ÞÞ 0.07074 0.07304 0.07458 0.07435 0.07902

I. Andreadis, T.E. Karakasidis / Applied Mathematics and Computation 217 (2010) 2883–2890 2889

where A, b are constants depending on the noise realisation for the three intervals of noise level mentioned above. The cor-responding coefficients and exponents were calculated for these regions and for the two noise realizations and the corre-sponding results are summarized in Table 2.

5. On the noise invariance of the Julia set of a noise-perturbed Mandelbrot map

In this paragraph, we consider the question of the independence of a Julia set of a noise-perturbed Mandelbrot map con-sidered in [4–6] under different noise realizations, i.e. if there is any strong dependence on the set of random numbers em-ployed to simulate noise perturbation. In this direction, we present results for the additive dynamic noise Mandelbrot MapAD0:01

c introduced above.Firstly, we consider five different realizations of noise and we name the corresponding additive dynamic noise Mandelb-

rot maps AD0:01c;i with 1 6 i 6 5. For that, we consider first five different noise time series, denoted as Ti with 1 6 i 6 5, which

are Gaussian noises with values in the interval [0,1]. In Table 3, we present the corresponding Julia Deviation Distances fromthe Julia set of the unperturbed Mandelbrot map. Since the resulting distances are all equal within two decimal places, weconclude that the resulting Julia sets of the additive dynamic noise Mandelbrot map are independent of the noiserealizations.

6. Conclusions

In this paper, we expanded our results of [1] to the study of the morphology of the Julia sets either of the same Mandelbrotmap while varying its parameter values or while fixing the parameter values and introducing noise-perturbation of the Man-delbrot map. These tools give the possibility to quantify the difference of the Julia set of the Mandelbrot map from the Julia setof the noise-perturbed Mandelbrot map; as well as to localize the regions of the set that are affected by noise. A first step to-wards the study of the Julia Deviation Distance as a function of the level of noise was established revealing a power-law behav-iour. It is of interest that there are different regions and that the power-law exponent increases as the noise level increases.

Appendix A. The Julia Deviation Distance

For any four real numbers a, b, c and d such that a< b and c< d, we consider a lattice of points (x,y) with x an element of theinterval [a,b] and y an element of the interval [c,d]. Then, for any two natural numbers N, M, we consider N subintervals ofthe x-interval [a,b], defined via the formula:

aþ ði� 1Þ b�aN�1

� �, with 1 6 i 6 N and M subintervals of the y-interval [c,d], defined via the formula:

c þ ðj� 1Þ d�cM�1

� �, with 1 6 j 6M. Hence, we obtain NM points on the lattice with coordinates P aþ ði� 1Þ b�a

N�1

� �; c

�þðj� 1Þ d�c

M�1

� �Þ, with 1 6 i 6 N, 1 6 j 6M. Now, we define for the NM matrix of coordinates of the lattice point a one-dimen-

sional quantity, in order to define a time series like behaviour, by the following schema:P corresponds to the one-dimensional index k, defined as k = (i � 1)M + j with 1 6 i 6 N, 1 6 j 6M.For every point P of the lattice, and hence for any value of k, we define the Julia index, denoted as JQc,

JQcðkÞ ¼1; P 2 ðQ cÞ;0; P R JðQ cÞ:

�

This means that if the point P belongs to the Julia set of J(QC) then the value of the Julia index is 1, otherwise it is zero.Then, we define for the two Julia sets J(QC1) and J(QC2) of the unperturbed Mandelbrot map, Qc and for every point P of the

lattice, and hence for any value of k, the Julia Deviation Index, denoted as DJQc1,2 :

DJQc1;2ðkÞ ¼ jJQc1ðkÞ � JQc2

ðkÞj;

where jxj denotes the absolute value of a real number, x.Hence, for any point P of the lattice, the value of DJQc1,2 is 1 if the point P belongs to the Julia set of J(QC1) but not to the

Julia set of J(QC2) or vice versa. If the point P belongs to both Julia sets or does not belong to both Julia sets then the value ofDJQc1,2 is zero at that point.

Now, we define a metric distance between the two Julia sets J(QC1) and J(QC2) which we name Julia Deviation Distance andwe denote distJ as follows:

distJðJðQ C1Þ; JðQ C2ÞÞ ¼1

NM

XNM

k¼1

DJQc1;2ðkÞ;

It is clear that distJ satisfies the axioms of a metric [14] and hence it defines a topology on the space of all the Julia sets de-fined on the same lattice of points. Therefore, we could define that two Julia sets are close to each other when their JuliaDeviation Distance is a small number.

In general, the Julia Deviation Distance between the two Julia sets J(QC1) and J(QC2) satisfies the inequality 0 6 distJ(J(QC1),J(QC2)) < 1.

2890 I. Andreadis, T.E. Karakasidis / Applied Mathematics and Computation 217 (2010) 2883–2890

Appendix B. The Julia Deviation Plot

Let us consider two Julia sets J(QC1) and J(QC2) defined on the same lattice of points. We define firstly that a point P of thelattice possesses the Julia property if it belongs to the one or to the other Julia set. Hence, every point P of the lattice belongsto one of the following four disjoint sets according to the following:

(1) If it belongs to both Julia sets, it is considered as a stable point for the Julia property. Such points are coloured yellowon the corresponding plots. We denote by JS the set of all the stable points of the lattice.

(2) If it was a point of the Julia set J(QC1) but not of the Julia set J(QC2), it is considered as a death point for the Julia prop-erty. Such points are coloured red on the corresponding plots. We denote by JD the set of all the death points of thelattice.

(3) If it was not on the Julia set set J(QC1) but now it becomes a point of the Julia set J(QC2), it is considered as a birth pointfor the Julia property. Such points are coloured blue on the corresponding plots. We denote by JB the set of all the birthpoints of the lattice.

(4) If it does not belong to either of Julia sets, it is considered as a neutral point for the Julia property. Such points arecoloured white. We denote by JN the set of all the neutral points of the lattice.

Thus, we plot every point P of the lattice with four colours as follows:

JDPQc1;2ðPÞ ¼

yellowðstableÞ; P 2 JðQ c1Þ \ JðQ c2Þ;redðdeathÞ; P 2 JðQc1Þ \ JðQ c2Þc;blueðbirthÞ; P 2 JðQ c1Þc \ JðQ c2Þ;whiteðneutralÞ; P 2 JðQ c1Þc \ JðQc2Þc;

8>>><>>>:

where J(QC1)c denotes the complement of the set J(QC1); i.e. all of the points of the lattice which do not belong to J(QC1) .Hence it is clear that J(QC1) = JS [ JD and J(QC2) = JS [ JB. Using the above notations, the formula for the Julia Deviation Dis-

tance becomes:

distJðJðQC1Þ; JðQ C2ÞÞ ¼nðJDÞ þ nðJBÞ

NM:

In addition, the distribution of the death and birth points provides us insight to the way in which the Julia set of the Man-delbrot map is deformed due to the changes of the parameters.

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