The Generalized Julia Set Perturbed by Composing Additive and Multiplicative Noises

Abstract

This paper contrastively researches the structural characteristic and the fission-evolution law of four different kinds of generalized Julia set (generalized J set in short) with different parameter 𝑐, which includes the generalized J set without any perturbation, the generalized J set perturbed by additive noises, the generalized J set perturbed by multiplicative noise, and the generalized J set perturbed by composing additive and multiplicative noises, analyzes the effect of random perturbation to the generalized J set, and illuminates the stability of the generalized J set.

1. Introduction

In the area of fractals, deep investigations have been made on the Mandelbrot set (M set in short) of the complex mapping 𝑧←𝑧^𝛼+𝑐(𝛼=2) by using computer technologies. During last twenty years, people studied the generalized M set with 𝛼∈𝑅 and found that it has a regularity structure [1–8]. In recent years, Argyris et al. discussed the classification and affection of noise in complex dynamical system [9]; Argyris et al. studied the structural characteristic of M-J sets containing noise after importing additive noise and multiplicative noise into the complex map 𝑧_𝑛+1=𝑧²_𝑛+𝑐 [10–14]. The authors studied the structural characteristic and fission-evolution law of additive perturbed generalized M-J sets, analyzed the dynamic additive noise perturbed and multiplicative noise perturbed generalized M-J sets [15, 16].

Recently, Negi and Rani put forward a new noise criterion which integrates the dynamic additive noise and dynamic multiplicative noise, and discussed its effect on the usual and superior Mandelbrot maps [17]. Andreadis and Karakasidis proposed a definition for a probabilistic Mandelbrot map and studied the numerical stability of the Mandelbrot and the Julia set of a probabilistic Mandelbrot map [18]. The authors studied the generalized M set perturbed by composing noise of additive and multiplicative and analyzed the effect of random perturbation to the generalized M set [19].

In the present work, we investigate on the structural relationship of the generalized Julia sets with different parameter 𝑐 perturbed by random noise, study the structural characteristics of the generalized J sets perturbed by additive noise, multiplicative noise, and the composing noise of additive and multiplicative, and analyze the effect of random perturbation to the generalized M set.

2. Theory and Method

According to [19], we give the definition of the generalized J set with composing additive and multiplicative noise. The method in the paper is the same with that of [19].

Definition 2.1. Assume that 𝑓(𝑧_𝑛)=𝑧^𝛼_𝑛+𝑊+𝑐 or 𝑔(𝑧_𝑛)=𝛾(𝜆,𝑘₁,𝑘₂,𝐰_𝑛)𝑧^𝛼_𝑛+𝑐(|𝛼|>1) is a complex map in Riemann sphere 𝐶, 𝜑=𝜇𝑓+(1−𝜇)𝑔(𝜇∈[0,1]), 𝐹_𝜑 is a collection of points 𝑧 whose trajectories do not converge to infinite in C, that is, 𝐹_𝜑={𝑧∈𝐶∶{|𝜑^𝑘(𝑧)|}^∞_𝑘=1isbounded}. (2.1) Then this set is called the filled generalized J set with composing additive and multiplicative noise corresponding to 𝜑; the boundary of 𝐹_𝜑 is called the generalized J set of the complex map 𝜑, recorded as 𝐽_𝜑, that is, 𝐽_𝜑=𝜕𝐹_𝜑. (2.2)

3. Experiment and Result

Selecting escape-radius as 𝑅=30 and escape-time restriction as 𝑁=100, we plot the composing noise perturbed generalized J set from Equation (GN) using escape-time algorithm. When 𝛼>0, the black in the figure is the stable region while the white is the escape region; when 𝛼<0, the white in the figure is the stable region while the black is the escape region. As to the additive noise perturbed generalized J set, when |𝑚₁,𝑚₂|<0.1 or |𝑚₁,𝑚₂|>1, the change of the structure can be ignored; also as to the multiplicative noise perturbed generalized J set, when |𝑘₁,𝑘₂|<0.1 or |𝑘₁,𝑘₂|>1, the structure almost does not change. Therefore, we select 0.1≤|𝑚₁,𝑚₂,𝑘₁,𝑘₂|≤1 and 𝜇=0.5. According to the structure characteristics of the generalized J set, we start from the following aspects.

3.1. The Generalized J Set without Any Perturbation

When 𝛼 is odd, observing the relationship between the generalized J sets without any perturbation and parameter c (Figures 1 and 2), we can find the following properties:

(a) 𝑐=−0.5+0.5𝑖

(b) 𝑐=0.5+0.5𝑖

(c) 𝑐=0.5−0.5𝑖

(d) 𝑐=−0.5−0.5𝑖

Figure 1 

The generalized J sets without any perturbation when 𝛼=3.

(a) 𝑐=0.5+0.5𝑖

(b) 𝑐=−0.5+0.5𝑖

(c) 𝑐=0.5−0.5𝑖

(d) 𝑐=−0.5−0.5𝑖

Figure 2 

The generalized J sets without any perturbation when 𝛼=−5.

(1)the graph when 𝑐=𝑎+𝑏𝑖 and the graph when 𝑐=−𝑎+𝑏𝑖 have mirror symmetry about y-axis;(2)Clockwise rotating the graph when 𝑐=𝑎+𝑏𝑖 for 180 degrees we will get the graph when 𝑐=−𝑎−𝑏𝑖;(3)The graph when 𝑐=𝑎+𝑏𝑖 and the graph when 𝑐=𝑎−𝑏𝑖 have mirror symmetry about x-axis.

Observing the generalized J sets without any perturbation when 𝛼 is even (Figure 3), we can find that they have the following properties:

(a) 𝑐=0.5+0.5𝑖

(b) 𝑐=−0.5+0.5𝑖

(c) 𝑐=0.5−0.5𝑖

(d) 𝑐=−0.5−0.5𝑖

Figure 3 

The generalized J sets without any perturbation when 𝛼=−4.

(1) The graph when 𝑐=𝑎+𝑏𝑖 and the graph when 𝑐=𝑎−𝑏𝑖 have mirror symmetry about x-axis.

Observing the generalized J sets without any perturbation when 𝛼 is positive decimal and phase angle 𝜃∈[−𝜋,𝜋) (Figure 4), we can find that they have the following property:

(a) 𝑐=0.5+0.5𝑖

(b) 𝑐=−0.5+0.5𝑖

(c) 𝑐=0.5−0.5𝑖

(d) 𝑐=−0.5−0.5𝑖

Figure 4 

The generalized J sets without any perturbation when 𝛼=5.5, 𝜃∈[−𝜋,𝜋).

(1) the graph when 𝑐=𝑎+𝑏𝑖 and the graph when 𝑐=𝑎−𝑏𝑖 have mirror symmetry about x-axis.

Observing Figures 5 and 6, we can find that when 𝛼 is positive decimal and the value bound of phase angle is [−𝜋/2, 3𝜋/2) or [−3𝜋/2,𝜋/2), the generalized J sets have the following property:

(a) 𝑐=0.5+0.5𝑖

(b) 𝑐=−0.5+0.5𝑖

(c) 𝑐=0.5−0.5𝑖

(d) 𝑐=−0.5−0.5𝑖

Figure 5 

The generalized J sets without any perturbation when 𝛼=5.5, 𝜃∈[−𝜋/2,3𝜋/2).

(a) 𝑐=0.5+0.5𝑖

(b) 𝑐=−0.5+0.5𝑖

(c) 𝑐=0.5−0.5𝑖

(d) 𝑐=−0.5−0.5𝑖

Figure 6 

The generalized J sets without any perturbation when 𝛼=5.5, 𝜃∈[−3𝜋/2,𝜋/2).

(1) the graph when phase angle 𝜃∈[−𝜋/2, 3𝜋/2) with 𝑐=𝑎+𝑏𝑖 and the graph when phase angle 𝜃∈[−3𝜋/2,𝜋/2) with 𝑐=𝑎−𝑏𝑖 have mirror symmetry about x-axis.

In fact, when 𝛼 is decimal, the generalized J sets without any perturbation satisfy the following properties:

(1)when phase angle 𝜃∈[−𝜋,𝜋), the graph with 𝑐=𝑎+𝑏𝑖 and the graph with 𝑐=𝑎−𝑏𝑖 have mirror symmetry about x-axis;(2)the graph when phase angle 𝜃∈[−𝜋/2, 3𝜋/2) with 𝑐=𝑎+𝑏𝑖 and the graph when phase angle 𝜃∈[−3𝜋/2,𝜋/2) with 𝑐=𝑎−𝑏𝑖 have mirror symmetry about x-axis;(3)when phase angle 𝜃∈[0, 2𝜋), with different values of parameter c corresponding to different figures, there are basically no corresponding rules as above;(4)the proof of the above symmetry property of generalized J set can refer to [7, 15].

3.2. The Generalized J Set Perturbed by Additive Noise

Let the order of parameters be (𝑎,𝑏,𝑚₁,𝑚₂); s here a, b are the real part and imaginary part of parameter c, respectively; 𝑚₁,𝑚₂ are the intensity coefficients of the real part and imaginary part of additive perturbation 𝑊, respectively. Observing the additive perturbed generalized J sets when 𝛼 is odd (Figure 7), we can get the following properties:

(a) (−0.5,−0.5,0.1,0.3)

(b) (0.5,0.5,−0.1,−0.3)

(c) (−0.5,0.5,0.1,−0.3)

(d) (0.5,−0.5,−0.1,0.3)

Figure 7 

The generalized J sets perturbed by additive noise when 𝛼=3.

(1)the graph with 𝑐=𝑎+𝑏𝑖 and additive noise parameter (𝑚₁,𝑚₂) and the graph with 𝑐=−𝑎+𝑏𝑖 and additive noise parameter (−𝑚₁,𝑚₂) have mirror symmetry about y-axis; (2)the graph with 𝑐=𝑎+𝑏𝑖 and additive noise parameter (𝑚₁,𝑚₂) and the graph with 𝑐=𝑎−𝑏𝑖 and additive noise parameter (𝑚₁,−𝑚₂) have mirror symmetry about x-axis; (3) rotating the graph with 𝑐=𝑎+𝑏𝑖 and additive noise parameter (𝑚₁,𝑚₂) by 180 degrees we will get the graph with 𝑐=−𝑎−𝑏𝑖 and additive noise parameter(−𝑚₁,−𝑚₂).

Observing the additive perturbed generalized J sets when 𝛼 is even (Figure 8), we can get the following property:

(a) (0.5,0.5,−0.1,0.3)

(b) (0.5,−0.5,−0.1,−0.3)

(c) (−0.5,0.5,−0.1,−0.3)

(d) (−0.5,−0.5,−0.1,0.3)

Figure 8 

The generalized J sets perturbed by additive noise when 𝛼=4.

(1) the graph with 𝑐=𝑎+𝑏𝑖 and additive noise parameter (𝑚₁,𝑚₂) and the graph with 𝑐=𝑎−𝑏𝑖 and additive noise parameter (𝑚₁,−𝑚₂) have mirror symmetry about x-axis.

Next is the property of the generalized J set perturbed by additive noise when 𝛼 is decimal.

Observing the additive perturbed generalized J sets when 𝛼=5.5, 𝜃∈[−𝜋,𝜋) (Figure 9), we can get the following property:

(a) (0.5,0.5,0.1,0.2)

(b) (0.5,−0.5,0.1,−0.2)

(c) (−0.5,0.5,−0.1,−0.2)

(d) (−0.5,−0.5,−0.1,0.2)

Figure 9 

The generalized J sets perturbed by additive noise when 𝛼=5.5, 𝜃∈[−𝜋,𝜋).

the graph with 𝑐=𝑎+𝑏𝑖 and additive noise parameter (𝑚₁,𝑚₂) and the graph with 𝑐=𝑎−𝑏𝑖 and additive noise parameter (𝑚₁,−𝑚₂) have mirror symmetry about x-axis.

Observe the additive perturbed generalized J sets when 𝛼=5.5, 𝜃∈[−𝜋/2,3𝜋/2) and when 𝛼=5.5, 𝜃∈[−3𝜋/2,𝜋/2) (Figures 10 and 11), we can get the following property:

(a) (0.5,−0.5,0.1,−0.2)

(b) (−0.5,0.5,−0.1,−0.2)

(c) (0.5,0.5,0.1,0.2)

(d) (−0.5,−0.5,−0.1,0.2)

Figure 10 

The generalized J sets perturbed by additive noise with different parameters (𝑎,𝑏,𝑚1,𝑚2) when 𝛼=5.5, 𝜃∈[−𝜋/2,3𝜋/2).

(a) (0.5,0.5,0.1,0.2)

(b) (−0.5,−0.5,−0.1,0.2)

(c) (0.5,−0.5,0.1,−0.2)

(d) (−0.5,0.5,−0.1,−0.2)

Figure 11 

The generalized J sets perturbed by additive noise with different parameters (𝑎,𝑏,𝑚1,𝑚2) when 𝛼=5.5, 𝜃∈[−3𝜋/2,𝜋/2).

The graph when phase angle 𝜃∈[−𝜋/2, 3𝜋/2) with 𝑐=𝑎+𝑏𝑖 and additive noise parameter (𝑚₁,𝑚₂) and the graph when phase angle 𝜃∈[−3𝜋/2,𝜋/2) with 𝑐=𝑎−𝑏𝑖 and additive noise parameter (𝑚₁,−𝑚₂) have mirror symmetry about x-axis.

When 𝛼 is decimal and phase angle 𝜃∈[0, 2𝜋), different c values corresponding to different graphs, basically there are no obvious rules among the additive noise perturbed generalized J sets with different parameter c.

3.3. The Generalized J Set Perturbed by Multiplicative Noise

Let the order of the parameters be (𝑎,𝑏,𝑘₁,𝑘₂); here a, b are the real part and the imaginary part of parameter c,respectively; 𝑘₁, 𝑘₂ are the intensity coefficients of entrance perturbation noise 𝐰_𝑛 in the direction of x axis and y axis of multiplicative perturbation. Observing the generalized J sets perturbed by multiplicative noise when 𝛼 is odd (Figure 12), we can find the following rules:

(a) (0.5,0.5,0.1,0.1)

(b) (−0.5,0.5,0.1,0.1)

(c) (0.5,−0.5,0.1,0.1)

(d) (−0.5,−0.5,0.1,0.1)

Figure 12 

The generalized J sets perturbed by multiplicative noise when 𝛼=3.

(1)the graph when 𝑐=𝑎+𝑏𝑖 with multiplicative noise parameter (𝑘₁,𝑘₂) and the graph when 𝑐=−𝑎+𝑏𝑖 with the same multiplicative noise parameter (𝑘₁,𝑘₂) have mirror symmetry about y-axis;(2)the graph when 𝑐=𝑎+𝑏𝑖 with multiplicative noise parameter (𝑘₁,𝑘₂) and the graph when 𝑐=𝑎−𝑏𝑖 with the same multiplicative noise parameter (𝑘₁,𝑘₂) have mirror symmetry about x-axis;(3)Clockwise rotating the graph when 𝑐=𝑎+𝑏𝑖 with multiplicative noise parameter (𝑘₁,𝑘₂) for 180-degree, we will get the graph when 𝑐=−𝑎−𝑏𝑖 with the same multiplicative noise parameter (𝑘₁,𝑘₂).

Observing the generalized J sets perturbed by multiplicative noise when 𝛼 is even (Figure 13), we can find the following rule:

(a) (0.5,0.5,0.1,0.1)

(b) (0.5,−0.5,0.1,0.1)

(c) (−0.5,0.5,−0.1,−0.1)

(d) (−0.5,−0.5,−0.1,−0.1)

Figure 13 

The generalized J sets perturbed by multiplicative noise when 𝛼=4.

(1) The multiplicative noise perturbed graph with 𝑐=𝑎+𝑏𝑖 and the multiplicative noise perturbed graph with 𝑐=𝑎−𝑏𝑖 of corresponding parameters have mirror symmetry about x-axis.

Next is the property of the generalized J set perturbed by multiplicative noise when 𝛼 is decimal.

Observing the generalized J sets perturbed by multiplicative noise when 𝛼 is decimal and phase angle 𝜃∈[−𝜋,𝜋) (Figure 14), we can find the following rule:

(a) (0.5,0.5,0.1,0.1)

(b) (0.5,−0.5,0.1,0.1)

(c) (−0.5,0.5,0.1,0.1)

(d) (−0.5,−0.5,0.1,0.1)

Figure 14 

The generalized J sets perturbed by multiplicative noise when 𝛼=5.5, 𝜃∈[−𝜋,𝜋).

the multiplicative noise perturbed graph with 𝑐=𝑎+𝑏𝑖 and the multiplicative noise perturbed graph with 𝑐=𝑎−𝑏𝑖 of corresponding parameters have mirror symmetry about x-axis.

Observing the generalized J sets perturbed by multiplicative noise when 𝛼 is decimal and phase angle 𝜃∈[−𝜋/2, 3𝜋/2) and phase angle 𝜃∈[−3𝜋/2,𝜋/2) (Figures 15 and 16), we can find the following rule:

(a) (0.5,0.5,0.1,0.1)

(b) (−0.5,0.5,−0.1,0.1)

(c) (0.5,−0.5,0.1,−0.1)

(d) (−0.5,−0.5,−0.1,−0.1)

Figure 15 

The generalized J sets perturbed by multiplicative noise when 𝛼=5.5, 𝜃∈[−𝜋/2,3𝜋/2).

(a) (0.5,−0.5,0.1,0.1)

(b) (−0.5,−0.5,−0.1,0.1)

(c) (0.5,0.5,0.1,−0.1)

(d) (−0.5,0.5,−0.1,−0.1)

Figure 16 

The generalized J sets perturbed by multiplicative noise when 𝛼=5.5, 𝜃∈[−3𝜋/2,𝜋/2).

when 𝛼 is decimal, the graph when phase angle 𝜃∈[−𝜋/2, 3𝜋/2) with 𝑐=𝑎+𝑏𝑖 and the graph with corresponding multiplicative parameters when phase angle 𝜃∈[−3𝜋/2,𝜋/2) with 𝑐=𝑎−𝑏𝑖 have mirror symmetry about x-axis.

When 𝛼 is decimal and phase angle 𝜃∈[0, 2𝜋), as to the multiplicative noise perturbed generalized J sets, different c values correspond to different graphs; basically the changes of the graphs are only related to the values of c, but have little connection with the intensity coefficients (𝑘₁,𝑘₂) of the multiplicative noise; the general outside shape is determined by the value of c, while the mini changes in detail are related to multiplicative parameters.

3.4. The Generalized J Set Perturbed by the Composing Noise of Additive and Multiplicative

Let the order of parameters be (𝜇,𝑚₁,𝑚₂,𝑘₁,𝑘₂); here 𝜇 is the proportion coefficient in the composing noise of Equation (GN); 𝑚₁,𝑚₂ are the intensity coefficients of the real part and imaginary part of additive perturbation 𝑊, respectively; 𝑘₁, 𝑘₂ are the intensity coefficients of entrance perturbation noise 𝐰_𝑛 in the direction of x axis and y axis of multiplicative perturbation.

(1) When 𝛼 is even

Observe that the generalized J sets perturbed by the composing noise when 𝛼 is even (Figures 17 and 18) have the following property:

(a) (0.5,0.1,0.1,0.1,0.1)

(b) (0.5,0.1,0.1,0.1,−0.1)

(c) (0.5,0.1,−0.1,0.1,0.1)

(d) (0.5,0.1,−0.1,0.1,−0.1)

Figure 17 

The generalized J sets perturbed by the composing noise when 𝛼=4, here Figures (a) and (b) are the graphs with 𝑐=0.5+0.5𝑖, Figures (c) and (d) are the graphs with 𝑐=0.5−0.5𝑖.

(a) (0.5,0.1,−0.1,−0.1,0.1)

(b) (0.5,0.1,−0.1,−0.1,−0.1)

(c) (0.5,0.1,0.1,−0.1,0.1)

(d) (0.5,0.1,0.1,−0.1,−0.1)

Figure 18 

The generalized J sets perturbed by the composing noise when 𝛼=4, here Figures (a) and (b) are the graphs with 𝑐=−0.5+0.5𝑖, Figures (c) and (d) are the graphs with 𝑐=−0.5−0.5𝑖.

the graph with 𝑐=𝑎+𝑏𝑖 and the composing noise parameters (𝑚₁,𝑚₂,𝑘₁,𝑘₂) and the graph with 𝑐=𝑎−𝑏𝑖 and composing noise parameters (𝑚₁,−𝑚₂,𝑘₁,𝑘₂) have mirror symmetry about x-axis.

Property 1. Assume that 𝑓(𝑧_𝑛)=𝑧^𝛼_𝑛+𝑊+𝑐 or 𝑔(𝑧_𝑛)=𝛾(𝜆,𝑘₁,𝑘₂,𝐰_𝑛)𝑧^𝛼_𝑛+𝑐(|𝛼|>1) is a complex map in Riemann sphere 𝐶; 𝜑=𝜇𝑓+(1−𝜇)𝑔(𝜇∈[0,1]) is the composed mapping of 𝑓 and 𝑔; constructing the generalized J sets perturbed by the composing noise of additive and multiplicative noise with it, when 𝛼is integer, there is 𝜑^𝑘(𝑧,𝑊,𝑐,𝛾)=𝜑^𝑘(𝑧,𝑊,𝑐,𝛾)(𝑘=1,2,…,𝑁). (3.1)

Proof. Use mathematical induction: because 𝑓(𝑧,𝑊,𝑐)=𝑧^𝛼+𝑊+𝑐,𝑓(𝑧,𝑊,𝑐)=(𝑧)^𝛼+𝑊+𝑐=𝑧^𝛼+𝑊+𝑐, (3.2) then 𝑓(𝑧,𝑊,𝑐)=𝑓(𝑧,𝑊,𝑐). (3.3)
Suppose
𝑓^𝑘−1(𝑧,𝑊,𝑐)=𝑓^𝑘−1(𝑧,𝑊,𝑐), (3.4) then there is 𝑓^𝑘(𝑧,𝑊,𝑐)=𝑓^𝑘−1(𝑓(𝑧,𝑊,𝑐),𝑊,𝑐)=𝑓^𝑘−1(𝑓(𝑧,𝑊,=𝑐),𝑊,𝑐)𝑓^𝑘−1(𝑓(𝑧,𝑊,𝑐),𝑊,𝑐)=𝑓^𝑘(𝑧,𝑊,𝑐). (3.5)
Again Use mathematical induction: for
𝑔¹(𝑧,𝑐,𝛾)=𝛾𝑧^𝛼+𝑐,𝑔¹(𝑧,𝑐,𝛾)=𝛾(𝑧)^𝛼+𝑐. (3.6)𝛾 is inlet perturbations of 𝐹 on the x-axis and y-axis, where 𝛾_𝑥=𝜆₁+𝑘₁𝐰_𝑛 and 𝛾_𝑦=𝜆₂+𝑘₂𝐰_𝑛. So 𝛾_𝑥=𝛾_𝑥, 𝛾_𝑦=𝛾_𝑦. Hence 𝑔(𝑧,𝑐,𝛾)=𝑔(𝑧,𝑐,𝛾). (3.7)
Suppose
𝑔^𝑘−1(𝑧,𝑐,𝛾)=𝑔^𝑘−1(𝑧,𝑐,𝛾), (3.8) then, 𝑔^𝑘(𝑧,𝑐,𝛾)=𝑔^𝑘−1(𝑔(𝑧,𝑐,𝛾),𝑐,𝛾)=𝑔^𝑘−1(𝑔(𝑧,=𝑐,𝛾),𝑐,𝛾)𝑔^𝑘−1(𝑔(𝑧,𝑐,𝛾),𝑐,𝛾)=𝑔^𝑘(𝑧,𝜑𝑐,𝛾),_𝑘(𝑧,𝑊,𝑐,𝛾)=𝜇𝑓^𝑘(𝑧,𝑊,𝑐)+(1−𝜇)𝑔^𝑘(𝑧,𝑐,𝛾)=𝜇𝑓^𝑘(𝑧,𝑊,𝑐)+(1−𝜇)𝑔^𝑘(𝑧,=𝑐,𝛾)𝜑^𝑘(𝑧,𝑊,𝑐,𝛾). (3.9)
The proposition is tenable. Property 1 indicates that when 𝛼 is integer, as to the composing noise perturbed generalized J sets, the graph with parameter 𝑐 and composing noise parameters (𝑚₁,𝑚₂,𝑘₁,𝑘₂) and the graph with parameter 𝑐 and composing noise parameters (𝑚₁,−𝑚₂,𝑘₁,𝑘₂) have mirror symmetry about x-axis.
(2) When 𝛼 is odd
Observe that the generalized J sets perturbed by the composing noise when 𝛼 is odd (Figures 19, 20, 21, and 22) have the following properties
(1)the graph with 𝑐=𝑎+𝑏𝑖 and the composing noise parameters (𝑚₁,𝑚₂,𝑘₁,𝑘₂) and the graph with 𝑐=−𝑎+𝑏𝑖 and the composing noise parameters (−𝑚₁,𝑚₂,𝑘₁,𝑘₂) have mirror symmetry about y-axis;(2)the graph with 𝑐=𝑎+𝑏𝑖 and the composing noise parameters (𝑚₁,𝑚₂,𝑘₁,𝑘₂) and the graph with 𝑐=𝑎−𝑏𝑖 and the composing noise parameters (𝑚₁,−𝑚₂,𝑘₁,𝑘₂) have mirror symmetry about x-axis (see proof of Property 1);(3) rotating the graph with 𝑐=𝑎+𝑏𝑖 and the composing noise parameters (𝑚₁,𝑚₂,𝑘₁,𝑘₂) for 180-degree will get the graph with 𝑐=−𝑎−𝑏𝑖 and composing noise parameters (−𝑚₁,−𝑚₂,𝑘₁,𝑘₂).

(a) (0.5,