The aim of this paper is to offer an integrated approach to study the additive and multiplicative noises with respect to perturbations in superior Julia sets. External and internal perturbations in superior Julia sets are analyzed under the mixed effect of additive and multiplicative noises

Mamta Rani

Rashi Agarwal

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General Noise-Perturbed Superior
Julia Sets
Mamta Rani
Department of Computer Applications,
Krishna Engineering College,
95, Loni Road, Mohan Nagar,
Ghaziabad 201007, UP, India
e-mail: mamtarsingh@rediffmail.com
Rashi Agarwal
Department of Information Technology,
Sharda University,
Knowledge Park II,
Greater Noida 201306, UP, India
e-mail: rasagarwal@gmail.com
The aim of this paper is to offer an integrated approach to study
the additive and multiplicative noises with respect to perturba-
tions in superior Julia sets. External and internal perturbations in
superior Julia sets are analyzed under the mixed effect of additive
and multiplicative noises. [DOI: 10.1115/1.4006785]
Keywords: superior Julia set, general noise, additive noise,
multiplicative noise, internal perturbation, external perturbation
1 Introduction
The classical work of Gaston Julia, in 1918, has lead to new
excitements in complex dynamics. Rani and Kumar [1,2] intro-
duced superior iterates in the study of complex nonlinear dynam-
ics and computed effective new Julia and Mandelbrot sets with
respect to superior orbits (see also Refs. [3,4]).
Argyris et al. [5–8], in their pioneering work, have studied the
effect of dynamic noise in complex dynamical systems. They dis-
cussed the structural characteristics of the Mandelbrot and Julia
sets after importing dynamic noise (additive and multiplicative
both) into the complex map z2nþ1¼ z2n þ c (cf. Refs. [6–8]). Fur-
ther, Wang et al. offered a nice study on the effects of additive
noises on a generalized Mandelbrot map [9]. Recently, Rani and
Agarwal [10] showed that under the effect of dynamical noise,
distortion in Julia sets in superior orbit and Julia sets in Picard
orbit is different. They also discussed the usefulness and other
issues of the superior orbit in discrete dynamics.
In contrast to studying the additive and multiplicative noises sep-
arately, Negi and Rani [11] proposed a general approach and for-
mulated a new method to combine the criterions regarding additive
and multiplicative noises. Indeed, they put forward the theory of
general noise, which integrates additive and multiplicative noises
and allows application of both the noises simultaneously. They
studied the mixed effect of additive and multiplicative noises on
the superior Mandelbrot map. Using the general noise, Wang, Jia,
and Zhenfeng [12] studied the mixed effect of additive and multi-
plicative noises on the generalized Mandelbrot map and studied the
structural characteristics and the fission-evolution law of the noise-
perturbed generalized Mandelbrot map. Also, Wang, Jia, and Sun
[13] presented a collective study of the effect of additive noise,
effect of multiplicative noise, and their mixed effected on zaþ c for
jaj> 1.
The main purpose of this paper is to study perturbations in Julia
sets, generated in superior orbit, under the effect of general noise.
We focus on the distortions in superior Julia sets, when additive
and multiplicative noises both are available simultaneously.
Indeed, we apply general noise on superior Julia sets and show the
pattern of perturbation. Finally, the results have been summarized
and precise conclusions have been drawn.
2 Preliminaries
Consider a continuous time dynamical system in the Euclidean
space Rn expressed by the equation s¼ f(x, l, t), where x, l [ Rn
are vectors and t [ R.
Negi and Rani [11] considered the notion of general noise in
the form of a disturbance, which influences the evolution of a dy-
namical system. Its general formulation in dynamical systems is
given by k¼ f(x, l, t, w), where x, l [ Rn are vectors and t [ R and
w is noise.
Definition 1: General Noise. The general noise, denoted by
GN, on the Mandelbrot map is given by
Xnþ1 ¼ kxa þ 1 kð Þxm (1)
Ynþ1 ¼ kya þ 1 kð Þym (2)
where (xa, ya) are additive noises and (xm, ym) are multiplicative
noises in a dynamical system, and 0  k  1.
The parameter k controls the contribution of both the noises in
GN. Thus, GN behaves as a multiplicative noise when k¼ 0, and
with k¼ 1 it reduces to an additive noise [11].
Superior iterations, an example of a two-step feedback
machine, are constructed by the formula zn¼ sn f(zn1)þ (1  sn)
zn1, where zn is a complex number and 0< sn  1 and {sn} is
convergent to a nonzero number.
Definition 2: Superior Orbit. The sequence {zn} constructed
above is called superior sequence of iterates, denoted by SO(f, z0,
sn) (see Refs. [1–4,10,11]).
The superior orbit SO(f, z0, sn) with sn¼ 1 reduces to the Picard
orbit (also called the function iteration in discrete dynamics). In
this paper, we perform experiments for sn¼ s. Now, we recall the
definition of superior Julia set, denoted by SJ, i.e., Julia set for a
function with respect to an SO.
Definition 3: Superior Julia Set. The set of points SK whose
orbits are bounded under superior iteration of a function Q(z) is
called the filled superior Julia set. Superior Julia set SJ of Q is the
boundary of filled superior Julia set SK [1].
An SJ set for Q(z) reduces to Julia set when the parameter
sn¼ 1 in SO. Now, we consider the polynomial Qc(z)¼ z2þ c. In
order to compute the SJ sets for Qc, a new escape criterion in SO
is required. The superior escape criterion for the polynomial
Qc(z)¼ z2þ c is max {jcj, 2=s}. For the details of the superior
escape criterion, one may refer to Rani and Kumar [1].
3 General Noise-Perturbed Superior Julia Sets
Argyris et al. [8] have studied the influence of dynamical
noise on the quadratic Julia sets Qc(z)¼ z2þ c, for c¼ (cx, cy)
¼ (0.3904, 0.58769). They showed that Julia sets suffer from
external perturbation due to intrusion of additive noise and get
internally perturbed due to multiplicative noise. For the sake of
comparison of the results obtained by intrusion of dynamical noise
and general noise, we also take the same parameters in our study
and observe the effect of general noise on SJ for Q(0.3904,0.58769).
Now, we fix up the parameter k¼ [1=2], so that both of the additive
and multiplicative noises may contribute equally in our study. Also,
for the symmetrical distortion in the objects, we take the parameters
xa¼ ya¼m and xm¼ ym¼ k (cf. Refs. [5,6]).
With all these assumptions, we made a program in Cþþ and
applied GN on quadratic SJ sets. This section is subdivided into
Contributed by the Design Engineering Division of ASME for publication in the
JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January
13, 2011; final manuscript received June 16, 2011; published online July 23, 2012.
Assoc. Editor: Eric A. Butcher.
Journal of Computational and Nonlinear Dynamics APRIL 2013, Vol. 8 / 024501-1
Copyright VC 2013 by ASME
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parts to study GN perturbed SJ sets. Indeed, in order to consider
the mixed effect of additive and multiplicative noises on SJ sets,
we divide our study into four parts.
3.1 Low Additive and Low Multiplicative Noises. Figure
1(a) shows SJ at s¼ 0.9, and when a small amount of additive and
multiplicative noises are applied, there is negligible distortion in
SJ (see Fig. 1(b)), which is obvious.
3.2 Low Additive and Higher Multiplicative Noises. At
low additive noise, it was found that with the increasing strength
of multiplicative noise, internal perturbation increases, which
leads to disconnected SJ and further, SJ vanishes (see Fig. 2).
3.3 Higher Additive and Low Multiplicative Noises. At
low multiplicative noise, we find that the increase in the strength
of additive noise, up to m¼ 1, reflects external perturbation, i.e.,
SJ tends to be somewhat circular and connectivity increases, see
Fig. 3.
Further, an increase in the strength of additive noise (i.e.,
m> 1) reflects internal perturbation in SJ. One can see distortion
in the shape of SJ and its reducing connectivity in Fig. 4.
3.4 Higher Additive and Higher Multiplicative Noises.
When both of the dynamic noises in GN are higher, for m¼ k  1,
SJ tends to take a disturbed elliptical shape (see Fig. 5). Further
Fig. 2 Effect of increasing strength of multiplicative noise on SJ at low additive noise m 5 0.01
at s 5 0.8 (row 1) and s 5 0.9 (row 2)
Fig. 3 Effect of increasing additive noise for m £ 1 on SJ at low multiplicative noise
k 5 0.01 at s 5 0.9 (row 1) and s 5 0.8 (row 2)
Fig. 1 (a) SJ at s 5 0.9 (b) SJ at low additive and low multiplica-
tive noise (s, m, k) 5 (0.9, 0.01, 0.01)
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increase in both of the dynamic noises, i.e. (m¼ k  1), causes in-
ternal perturbation in the shape of SJ as shown in Fig. 6.
4 Discussion and Conclusion
The results obtained in the paper are summarized as follows:
^ When both the noises in GN are low, there is negligible dis-
tortion in the shape of SJ, which is obvious.
^ At low additive and higher multiplicative noises, there is in-
ternal perturbation, which tends to destroy the connectivity
of SJ.
^ The distortion in SJ is due to dominance of multiplicative
noise. The result is the same as given by Argyris et al. [8].Fig. 5 Effect of higher additive noise and higher multiplicative
noise at (m, k, s) 5 (0.9, 0.9, 0.9)
Fig. 6 Effect of higher additive noise and higher multiplicative noise at m > 1 and k > 1
at s 5 0.9 (row 1) and s 5 0.6 (row 2)
Fig. 4 Effect of increasing additive noise for m > 1 on SJ at low multiplicative noise
k 5 0.01 at s 5 0.9 (row 1) and s 5 0.8 (row 2)
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^ At low multiplicative noise, for higher additive noise, a
mixed effect is observed. Initial increase in the additive
noise reflects external perturbation. However, its further
higher strengths lead to internal perturbation in SJ (cf. the
SJ for additive noise> 1).
^ Here, distortion in SJ is due to dominance of additive noise.
As per Argyris et al. [8] under the dominance of additive
noise, Julia sets get externally perturbed.
^ When both the noises in GN are higher, again a mixed effect of
them is observed. Initial increase in the strength of GN reflects
external perturbation. However, further higher strengths lead to
internal perturbation in SJ (cf. the SJ for GN> 1).
^ Distortion in all the above four cases is controlled by the pa-
rameter s. Distortion can be increased or decreased by increas-
ing or decreasing the value of the parameter in any case.
We conclude by asserting that:
In a general-noise perturbed SJ, if we keep the additive noise on
the higher side, then by decreasing and increasing the strength of
additive noise, we can achieve external and internal perturbations
respectively, irrespective of the strength of multiplicative noise.
Distortion in SJ can be decreased or increased by decreasing or
increasing the value of the controlling parameter s, respectively.
We close our study by observing that there is a good scope to
continue the present study for various choices of the parameter s
and possibly considering a sequence of parameters {sn} instead of s.
Acknowledgment
The authors thank to Professor S. L. Singh, Government P. G.
College, Rishikesh, India for his valuable suggestions in the
development of the manuscript. We are grateful to the referees for
their valuable comments and suggestions that helped in improve-
ment of the manuscript.
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Article ID 781976, 18 pages doi:10.1155/2009/781976.
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General Noise-Perturbed Superior Julia Sets

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General Noise-Perturbed Superior Julia Sets

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