# Fractal Dimension Analysis of the Julia Sets of Controlled Brusselator Model

Authors: Yuqian Deng; Xiuxiong Liu; Yongping Zhang
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

Fractal theory is a branch of nonlinear scientific research, and its research object is the irregular geometric form in nature. On account of the complexity of the fractal set, the traditional Euclidean dimension is no longer applicable and the measurement method of fractal dimension is required. In the numerous fractal dimension definitions, box-counting dimension is taken to characterize the complexity of Julia set since the calculation of box-counting dimension is relatively achievable. In this paper, the Julia set of Brusselator model which is a class of reaction diffusion equations from the viewpoint of fractal dynamics is discussed, and the control of the Julia set is researched by feedback control method, optimal control method, and gradient control method, respectively. Meanwhile, we calculate the box-counting dimension of the Julia set of controlled Brusselator model in each control method, which is used to describe the complexity of the controlled Julia set and the system. Ultimately we demonstrate the effectiveness of each control method.


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## Body

## 1. Introduction

In order to recognize the essence of some extremely sophisticated phenomena, researchers attempt to figure out the regularity and unity which exist behind these phenomena so that they can control and predict them better. In the early 20th century, the fundamental theory of chaos and fractal was proposed. The theory explains the unity of determinacy and randomness and the unity of order and disorder. It is considered to be the third major revolution of science after the theory of relativity and quantum mechanics [1, 2].Fractal theory, first proposed in the 1970s, comes from the study of nonlinear science. Its primary research object is the geometric form of nature and nonlinear system, which is complex but has some kind of self similarity and regularity. In 1977, Mandelbrot, a professor of mathematics of the Harvard University, published the landmark workFractal: Form, Chance and Dimension[3]. It marked the fractal geometry that had become an independent discipline. Subsequently, he published another workThe Fractal Geometry of Nature[4], which implied that fractal theory had been basically formed.Nowadays, with the emergence of some new mathematical tools and methods, especially the combination of the study of fractal theory and computer, the theory has been developed rapidly. In addition, researchers not only constantly establish and improve the theory of fractals, but also apply it in various fields, such as the diffusion processes and chemical kinetics in crowded media, the protein structure and complex vascular branches in biomedicine, dynamical system and hydromechanics in physics, and landforms evolution and earthquake monitoring [5–16]. Even in social and economic activities, the theory of fractals also has numerous applications [17–19].Considering the complexity of fractal sets, traditional Euclidean geometry dimension cannot accurately depict their geometric forms. Mathematicians propose many definitions of noninteger dimension and use different names to distinguish them. For example, Hausdorff dimension which was proposed by the German mathematician Hausdorff in 1919 has a rigorous mathematical definition. It is established on the basis of Hausdorff measure and can define most fractal sets, so it is easier to deal with in mathematics [20]. Moreover, box-counting dimension is one of the most widely used dimensions. Its popularity is largely due to its relative ease of mathematical calculation and empirical estimation. Besides, other fractal dimensions, such as similar dimension, capacity dimension, and Lyapunov dimension, also have their own applications in the corresponding fields [21–24].Brusselator model [25–27] is a kind of reaction diffusion equations which describe the change of chemical elements in the process of chemical reaction [28]. It is significant in the study of the chaos and fractal behavior of nonlinear differential equations. Researchers have studied Brusselator model from different aspects and proven some properties of it [29–33]. These studies of the model have contributed much to the development of nonlinear mathematics. Nowadays, with the development of research, people apply some of the property of the model in all kinds of disciplines and social production activities and have achieved fruitful results.Notably, in the nonlinear system, the Julia set of the system is an important nonlinear feature. According to the objective requirement, we often need restrict the size of the nonlinear attractive domain. And sometimes it is required that the system possess different or similar behavior and performance in compliance with the actual requirements of technical problems. As a result, how to effectively control the Julia set is particularly critical.Based on the Julia set of Brusselator model, feedback control method, optimal control method, and gradient control method [34] are taken to control the Julia set of the model. And the box-counting dimension of the Julia set of controlled Brusselator model is calculated in each control method to describe the complexity of the Julia set of the system.


## 2. Basic Theory

In 1918, Julia Gaston, a famous French mathematician, discovered an important fractal set in fractal theory, when he studied the iteration of complex functions, which was named Julia set. He noticed that functions on the complex plane as simple asfz=z2+c, with a complex constant c, can give rise to fractals of an exotic appearance. The precise definition of Julia set is given below [35].Takef:C→C to be a polynomial of degree n≥2 with complex coefficients, fz=a0+a1z+⋯+anzn. Write fk for the k-fold composition f∘⋯∘f, so that fk(w) is the kth iterate ff⋯fw of w. If fw=w we call w a fixed point of f, and if fpw=w for some integer p≥1 we call w a periodic point of f; the least such p is called the period of w. We call w,f(w),…,fp(w) a period p orbit. Let w be a periodic point of period p, with fp′w=λ, where the prime denotes complex differentiation. The point w is called superattractive, if λ=0; attractive, if 0≤λ<1; neutral, if λ=1; and repelling, if λ>1.Definition 1.
Letf:C→C be a polynomial of degree n>1; Julia set of f is defined to be the closure of repelling period points of f.In fractal theory, fractal dimension is one of the most elemental concepts. At present, there are many definitions of fractal dimensions, including Hausdorff dimension, box-counting dimension, similarity dimension. In all kinds of definitions of fractal dimensions, Hausdorff dimension is the basis of the fractal theory. It can even be considered the theoretical basis of the fractal geometry. However, Hausdorff dimension is just suitable for the theoretical analysis of fractal theory, and there is only a small class of fairly solid mathematical regular fractal graphics that can be calculated for their Hausdorff dimension. It is hard to calculate the fractal dimension which is proposed in the practical applications. Therefore, people propose the concept of box-counting dimension. Its popularity is largely due to its relative ease of mathematical calculation and empirical estimation. In fact, in practical applications, the dimension is generally referred to as box-counting dimension. The precise definition of box-counting dimension is as follows.Definition 2 (see [35]).
LetF be any nonempty bounded subset of Rn and let Nδ(F) be the smallest number of sets of diameter at most δ which can cover F. Thelower andupper box-counting dimensions of F ((1), (2)), respectively, are defined as(1)dim_BF=limδ→0-⁡log⁡NδF-log⁡δ,(2)dim¯BF=limδ→0⁡-log⁡NδF-log⁡δ.If these are equal we refer to the common value as thebox-counting dimension of F (3): (3)dimBF=limδ→0log⁡NδF-log⁡δ.


## 3. The Control of the Julia Set of Brusselator Model

Brusselator model is one of the most fundamental models in nonlinear systems, and the dynamic equations are as follows:(4)xn+1=A-B+1xn+xn2yn,yn+1=Bxn-xn2yn,where x, y denote concentration of reactant in the process of chemical reaction. A,B>0 denote initial concentration of reactant.Brusselator equations were first discovered by A. Turing in 1952 [36], and then I. Pigogine and Leefver did some systematic studies on it. They pointed out that the Brusselator equations were the most elementary and essential mathematic model which describes the oscillation of biochemistry. They proved that when B>1+A2, the equations have stable and unique limit cycle. When B≤1+A2, there is no limit cycle [37]. It can be known that the initial concentration of reactant has an important influence on the system. In fractal theory, Julia set is a set of initial points of the system that satisfy certain conditions. With the same thought, we define the Julia set of Brusselator model. Let Fx,y=A-B+1xn+xn2yn,Bxn-xn2yn.Definition 3.
SetK={x,y∈R2∣Fn(x,y)n≥1is  bounded} is called the filled Julia set of Brusselator model. The boundary of the filled Julia set is defined to be the Julia set of Brusselator model; that is, J(F)=∂K.The research results demonstrate that, coupled with a piecewise constant value control function, we can control the size of the limit cycle of (4) [36]. From the definition of Julia set, we found that there is a close relation between the boundedness of iterative orbits and the structure of the Julia set. So if we want to control the Julia set of Brusselator model, how to control the boundedness of the system iterative is critical. We consider especially the stability of the fixed point of Brusselator model, and by designing controllers unstable fixed points are turned into stable fixed points to control the boundedness of iterative orbits effectively. Then the control of the Julia set can be realized.As was mentioned above, considering the stability of the fixed points of system (5), we try to find controllers to make the fixed point of the system stable. Let the controlled Brusselator model be(5)xn+1=a+1-b-γxn+γxn2yn+un,yn+1=yn+bxn-γxn2yn+vn,where un and vn denote the designed controllers.


### 3.1. Feedback Control Method

Takeun=kxn-x∗, vn=kyn-y∗, with the control parameter k, and the controlled system is(6)xn+1=a+1-b-γxn+γxn2yn+kxn-x∗,yn+1=yn+bxn-γxn2yn+kyn-y∗.Theorem 4.
LetA1=1-b-γ+2γx∗y∗+k, B1=1-γx∗2+k, C1=γx∗2, and D1=b-2γx∗y∗, where x∗,y∗ denotes the fixed point of the controlled system (6). If A1+B1±A1-B12+4C1D1<2, the fixed point of the system is attractive.Proof.
WriteB~1x,y=a+1-b-γx+γx2y+kx-x∗,y+bx-γx2y+ky-y∗. The Jacobi matrix of the controlled system (6) is(7)DB~1=1-b-γ+2γxnyn+kγxn2b-2γxnyn1-γxn2+kand its eigenmatrix is (8)DB~1-λE=1-b-γ+2γxnyn+k-λγxn2b-2γxnyn1-γxn2+k-λ.
And the characteristic equation is(9)λ2-2-b-γ+2γxnyn-γxn2+2kλ+1-b-γ+2γxnyn+k1-γxn2+k-γxn2b-2γxnyn=0.
When the modulus of the eigenvaluesλ11, λ12 of the Jacobi matrix at the fixed point is less than 1, the fixed point is attractive:(10)λ11,12=A1+B1±A1-B12+4C1D12<1;that is,(11)A1+B1±A1-B12+4C1D1<2.
By Theorem4, we know that Brusselator model can be controlled by selecting the value of k which satisfies the condition, and then the control of the Julia set can be realized.
For example, takea=0.07, b=0.02, and γ=0.01 in system (6) and the initial Julia set is shown in Figure 1; then we get x∗=7,y∗=2/7. By Theorem 4, the range of the value of k is -1.9895<k<0.4695.
Six simulation diagrams are chosen corresponding to the values ofk from 0.06 to 0.45 in Figure 2, and we find that the trend of the change of the Julia sets is obvious. In Figure 3 the box-counting dimensions of the controlled Julia sets are computed in this control method.
In the same control, six simulation diagrams are chosen corresponding to the values ofk from −1 to −0.22 in Figure 4 to illustrate the change of the Julia sets in feedback control method. In Figure 5 the box-counting dimensions of the controlled Julia sets are computed in this control method.
In feedback control method, the contraction of the left and the lower parts is faster than the right and the upper parts when the interval of control parameterk is between 0.06 and 0.45. In addition, the complexity of the boundary of the Julia set significantly decreases and the lower part rapidly contracts. When the interval of control parameter k is between −1 and −0.22, the complexity of the boundary of the Julia set significantly decreases, and the Julia set tends to be centrally symmetric. In general, with the absolute value of k increasing, the Julia set contracts to the center gradually.
From the perspective of the change of box-counting dimensions, with the absolute values ofk increasing, the change generally shows a monotonic decreasing trend. Particularly when the control parameter k is in the interval from 0.16 to 0.37 and −0.9 to −0.4, the monotonic change of box-counting dimensions is obvious, which indicates the effectiveness of this control method on the Julia set of Brusselator model.Figure 1
The original Julia set of the system.Figure 2
The change of the Julia sets of the controlled system whena=0.07,b=0.02, and γ=0.01. (a) k=0.06; (b) k=0.14; (c) k=0.22; (d) k=0.30; (e) k=0.38; (f) k=0.45.
(a) (b) (c) (d) (e) (f)Figure 3
The change of box-counting dimensions of the Julia sets of the controlled system whenk is from 0.06 to 0.45.Figure 4
The change of the Julia sets of the controlled system whena=0.07,b=0.02, and γ=0.01. (a) k=-0.22; (b) k=-0.38; (c) k=-0.54; (d) k=-0.70; (e) k=-0.86; (f) k=-1.
(a) (b) (c) (d) (e) (f)Figure 5
The change of box-counting dimensions of the Julia sets of the controlled system whenk is from −1 to −0.22.


### 3.2. Optimal Control Method

Takeun=kfxn,yn-xn,vn=kgxn,yn-yn, with fxn,yn=a+1-b-γxn+γxn2yn,gxn,yn=yn+bxn-γxn2yn, where k is the control parameter. Then the controlled system is(12)xn+1=a+1-b-γxn+γxn2yn+kfxn,yn-xn,yn+1=yn+bxn-γxn2yn+kgxn,yn-yn.Theorem 5.
LetA2=1-b-γ+2γx∗y∗k+1-k, B2=1-γx∗2k+1-k, C2=γx∗2k+1, and D2=b-2γx∗y∗k+1, where x∗,y∗ denotes the fixed point of the controlled system (12). If A2+B2±A2-B22+4C2D2<2, the fixed point of the system is attractive.Proof.
WriteB~2x,y=a+1-b-γx+γx2y+kfx,y-x,y+bx-γx2y+kgx,y-y. The Jacobi matrix of the controlled system (12) is(13)DB~2=1-b-γ+2γxnynk+1-kγxn2k+1b-2γxnynk+11-γxn2k+1-kand its eigenmatrix is(14)DB~2-λE=1-b-γ+2γxnynk+1-k-λγxn2k+1b-2γxnynk+11-γxn2k+1-k-λ.
And the characteristic equation is(15)λ2-2-b-γ+2γxnyn-γxn2k+1-2kλ+1-b-γ+2γxnynk+1-k1-γxn2k+1-k-γxn2b-2γxnynk+12=0.
When the modulus of the eigenvalueλ21, λ22 of the Jacobi matrix at the fixed point is less than 1, the fixed point is attractive:(16)λ21,22=A2+B2±A2-B22+4C2D22<1;that is,(17)A2+B2±A2-B22+4C2D2<2.
By Theorem5, we know that Brusselator model can be controlled by selecting the value of k which satisfies the condition, and then the control of the Julia set can be realized.
For example, we take the values of system parametersa=0.07,b=0.02,andγ=0.01, which are the same as Section 3.1. Then the range of the value of k is -1<k<190.659.
Six simulation diagrams are chosen corresponding to the values ofk from 0.64 to 2.6 in Figure 6 to illustrate the change of the Julia sets in optimal control method. In Figure 7 the box-counting dimensions of the controlled Julia sets are computed in this control method.
In optimal control method, the effective controlled interval ofk is from −1 to 190.6. But when the interval of k is from 0.64 to 2.6, this method has the best controlled effectiveness. In this interval, with the absolute values of k increasing, the Julia sets gradually contract to the center with nearly the same speed, and no significant change in the overall shape of the Julia sets occurs.
From the perspective of the change of box-counting dimensions, box-counting dimensions of the Julia sets generally show a monotonic decreasing trend with the absolute values ofk increasing. For the reason that the complexity of the Julia set can be depicted by box-counting dimension, when the complexity of the Julia set decreases with the absolute value of k increasing, it indicates that this control method has great control effectiveness.Figure 6
The change of the Julia sets of the controlled system whena=0.07, b=0.02, and γ=0.01. (a) k=0.64; (b) k=1.04; (c) k=1.44; (d) k=1.84; (e) k=2.24; (f) k=2.6.
(a) (b) (c) (d) (e) (f)Figure 7
The change of box-counting dimensions of the Julia sets of the controlled system whenk is from 0.64 to 2.6.


### 3.3. Gradient Control Method

Takeun=kxn2-x∗2,vn=kyn2-y∗2, with the control parameter k. Therefore the controlled system is(18)xn+1=a+1-b-γxn+γxn2yn+kxn2-x∗2,yn+1=yn+bxn-γxn2yn+kyn2-y∗2.Theorem 6.
LetA3=1-b-γ+2γx∗y∗+2kx∗,B3=1-γx∗2+2ky∗,C3=γx∗2,andD3=b-2γx∗y∗, where x∗,y∗ denotes the fixed point of the controlled system (18). If A3+B3±A3-B32+4C3D3<2, the fixed point of the system is attractive.Proof.
WriteB~3x,y=a+1-b-γx+γx2y+kx2-x∗2,y+bx-γx2y+ky2-y∗2. The Jacobi matrix of the controlled system (18) is(19)DB~3=1-b-γ+2γxnyn+2kxnγxn2b-2γxnyn1-γxn2+2kynand its eigenmatrix is(20)DB~3-λE=1-b-γ+2γxnyn+2kxn-λγxn2b-2γxnyn1-γxn2+2kyn-λ.
And the characteristic equation is(21)λ2-2-b-γ+2γxnyn+2kxn-γxn2+2kynλ+1-b-γ+2γxnyn+2kxn1-γxn2+2kyn-γxn2b-2γxnyn=0.
When the modulus of the eigenvalueλ31,λ32 of the Jacobi matrix at the fixed point is less than 1, the fixed point is attractive:(22)λ31,32=A3+B3±A3-B32+4C3D32<1;that is, (23)A3+B3±A3-B32+4C3D3<2.
By Theorem6, we know that the Brusselator model can be controlled by selecting the value of k which satisfies the condition, and then the control of the Julia set can be realized.
For example, we take the values of system parametersa=0.07,b=0.02,andγ=0.01, which are the same as Section 3.1. Then the range of the value of k is -2.6420<k<0.8561.
Six simulation diagrams are chosen corresponding to the values ofk from −0.0005 to −0.02 in Figure 8 to illustrate the change of the Julia sets in gradient control method. In Figure 9 the box-counting dimensions of the controlled Julia sets are computed in this control method.
In the same control, six simulation diagrams are chosen corresponding to the values ofk from 0.0005 to 0.02 in Figure 10 to illustrate the change of the Julia sets in gradient control method. In Figure 11 the box-counting dimensions of the controlled Julia sets are computed in this control method.
In gradient control method, we consider two groups of interval of the parameterk. It is worth noticing that the contraction of the left and the lower parts of the Julia set are faster than the right and the upper parts when the interval of control parameter k is from −0.0005 to −0.02, while the right and the upper parts are faster than the left and the lower parts when the interval of control parameter k is from 0.0005 to 0.02. In general, with the absolute values of k increasing, the Julia sets contract to the center gradually, and the complexity of the boundary of the Julia sets decreases.
From the perspective of the change of box-counting dimensions, with the absolute values ofk increasing, the box-counting dimensions of the Julia sets generally show a monotonic decreasing trend. Particularly when the control parameter k is in the range from −0.016 to −0.0075 and from 0.0075 to 0.016, the change of box-counting dimensions is obviously monotonic, which indicates the effectiveness of this method on the Julia set of the Brusselator model.Figure 8
The change of the Julia sets of the controlled system whena=0.07,b=0.02, and γ=0.01. (a) k=-0.0005; (b) k=-0.004; (c) k=-0.008; (d) k=-0.012; (e) k=-0.016; (f) k=-0.020.
(a) (b) (c) (d) (e) (f)Figure 9
The change of box-counting dimensions of the Julia sets of the controlled system whenk is from −0.02 to −0.0005.Figure 10
The change of the Julia sets of the controlled system whena=0.07,b=0.02, and γ=0.01. (a) k=0.0005; (b) k=0.0045; (c) k=0.0085; (d) k=0.0125; (e) k=0.0165; (f) k=0.0200.
(a) (b) (c) (d) (e) (f)Figure 11
The change of box-counting dimensions of the Julia sets of the controlled system whenk is from 0.0005 to 0.02.


## 3.1. Feedback Control Method

Takeun=kxn-x∗, vn=kyn-y∗, with the control parameter k, and the controlled system is(6)xn+1=a+1-b-γxn+γxn2yn+kxn-x∗,yn+1=yn+bxn-γxn2yn+kyn-y∗.Theorem 4.
LetA1=1-b-γ+2γx∗y∗+k, B1=1-γx∗2+k, C1=γx∗2, and D1=b-2γx∗y∗, where x∗,y∗ denotes the fixed point of the controlled system (6). If A1+B1±A1-B12+4C1D1<2, the fixed point of the system is attractive.Proof.
WriteB~1x,y=a+1-b-γx+γx2y+kx-x∗,y+bx-γx2y+ky-y∗. The Jacobi matrix of the controlled system (6) is(7)DB~1=1-b-γ+2γxnyn+kγxn2b-2γxnyn1-γxn2+kand its eigenmatrix is (8)DB~1-λE=1-b-γ+2γxnyn+k-λγxn2b-2γxnyn1-γxn2+k-λ.
And the characteristic equation is(9)λ2-2-b-γ+2γxnyn-γxn2+2kλ+1-b-γ+2γxnyn+k1-γxn2+k-γxn2b-2γxnyn=0.
When the modulus of the eigenvaluesλ11, λ12 of the Jacobi matrix at the fixed point is less than 1, the fixed point is attractive:(10)λ11,12=A1+B1±A1-B12+4C1D12<1;that is,(11)A1+B1±A1-B12+4C1D1<2.
By Theorem4, we know that Brusselator model can be controlled by selecting the value of k which satisfies the condition, and then the control of the Julia set can be realized.
For example, takea=0.07, b=0.02, and γ=0.01 in system (6) and the initial Julia set is shown in Figure 1; then we get x∗=7,y∗=2/7. By Theorem 4, the range of the value of k is -1.9895<k<0.4695.
Six simulation diagrams are chosen corresponding to the values ofk from 0.06 to 0.45 in Figure 2, and we find that the trend of the change of the Julia sets is obvious. In Figure 3 the box-counting dimensions of the controlled Julia sets are computed in this control method.
In the same control, six simulation diagrams are chosen corresponding to the values ofk from −1 to −0.22 in Figure 4 to illustrate the change of the Julia sets in feedback control method. In Figure 5 the box-counting dimensions of the controlled Julia sets are computed in this control method.
In feedback control method, the contraction of the left and the lower parts is faster than the right and the upper parts when the interval of control parameterk is between 0.06 and 0.45. In addition, the complexity of the boundary of the Julia set significantly decreases and the lower part rapidly contracts. When the interval of control parameter k is between −1 and −0.22, the complexity of the boundary of the Julia set significantly decreases, and the Julia set tends to be centrally symmetric. In general, with the absolute value of k increasing, the Julia set contracts to the center gradually.
From the perspective of the change of box-counting dimensions, with the absolute values ofk increasing, the change generally shows a monotonic decreasing trend. Particularly when the control parameter k is in the interval from 0.16 to 0.37 and −0.9 to −0.4, the monotonic change of box-counting dimensions is obvious, which indicates the effectiveness of this control method on the Julia set of Brusselator model.Figure 1
The original Julia set of the system.Figure 2
The change of the Julia sets of the controlled system whena=0.07,b=0.02, and γ=0.01. (a) k=0.06; (b) k=0.14; (c) k=0.22; (d) k=0.30; (e) k=0.38; (f) k=0.45.
(a) (b) (c) (d) (e) (f)Figure 3
The change of box-counting dimensions of the Julia sets of the controlled system whenk is from 0.06 to 0.45.Figure 4
The change of the Julia sets of the controlled system whena=0.07,b=0.02, and γ=0.01. (a) k=-0.22; (b) k=-0.38; (c) k=-0.54; (d) k=-0.70; (e) k=-0.86; (f) k=-1.
(a) (b) (c) (d) (e) (f)Figure 5
The change of box-counting dimensions of the Julia sets of the controlled system whenk is from −1 to −0.22.


## 3.2. Optimal Control Method

Takeun=kfxn,yn-xn,vn=kgxn,yn-yn, with fxn,yn=a+1-b-γxn+γxn2yn,gxn,yn=yn+bxn-γxn2yn, where k is the control parameter. Then the controlled system is(12)xn+1=a+1-b-γxn+γxn2yn+kfxn,yn-xn,yn+1=yn+bxn-γxn2yn+kgxn,yn-yn.Theorem 5.
LetA2=1-b-γ+2γx∗y∗k+1-k, B2=1-γx∗2k+1-k, C2=γx∗2k+1, and D2=b-2γx∗y∗k+1, where x∗,y∗ denotes the fixed point of the controlled system (12). If A2+B2±A2-B22+4C2D2<2, the fixed point of the system is attractive.Proof.
WriteB~2x,y=a+1-b-γx+γx2y+kfx,y-x,y+bx-γx2y+kgx,y-y. The Jacobi matrix of the controlled system (12) is(13)DB~2=1-b-γ+2γxnynk+1-kγxn2k+1b-2γxnynk+11-γxn2k+1-kand its eigenmatrix is(14)DB~2-λE=1-b-γ+2γxnynk+1-k-λγxn2k+1b-2γxnynk+11-γxn2k+1-k-λ.
And the characteristic equation is(15)λ2-2-b-γ+2γxnyn-γxn2k+1-2kλ+1-b-γ+2γxnynk+1-k1-γxn2k+1-k-γxn2b-2γxnynk+12=0.
When the modulus of the eigenvalueλ21, λ22 of the Jacobi matrix at the fixed point is less than 1, the fixed point is attractive:(16)λ21,22=A2+B2±A2-B22+4C2D22<1;that is,(17)A2+B2±A2-B22+4C2D2<2.
By Theorem5, we know that Brusselator model can be controlled by selecting the value of k which satisfies the condition, and then the control of the Julia set can be realized.
For example, we take the values of system parametersa=0.07,b=0.02,andγ=0.01, which are the same as Section 3.1. Then the range of the value of k is -1<k<190.659.
Six simulation diagrams are chosen corresponding to the values ofk from 0.64 to 2.6 in Figure 6 to illustrate the change of the Julia sets in optimal control method. In Figure 7 the box-counting dimensions of the controlled Julia sets are computed in this control method.
In optimal control method, the effective controlled interval ofk is from −1 to 190.6. But when the interval of k is from 0.64 to 2.6, this method has the best controlled effectiveness. In this interval, with the absolute values of k increasing, the Julia sets gradually contract to the center with nearly the same speed, and no significant change in the overall shape of the Julia sets occurs.
From the perspective of the change of box-counting dimensions, box-counting dimensions of the Julia sets generally show a monotonic decreasing trend with the absolute values ofk increasing. For the reason that the complexity of the Julia set can be depicted by box-counting dimension, when the complexity of the Julia set decreases with the absolute value of k increasing, it indicates that this control method has great control effectiveness.Figure 6
The change of the Julia sets of the controlled system whena=0.07, b=0.02, and γ=0.01. (a) k=0.64; (b) k=1.04; (c) k=1.44; (d) k=1.84; (e) k=2.24; (f) k=2.6.
(a) (b) (c) (d) (e) (f)Figure 7
The change of box-counting dimensions of the Julia sets of the controlled system whenk is from 0.64 to 2.6.


## 3.3. Gradient Control Method

Takeun=kxn2-x∗2,vn=kyn2-y∗2, with the control parameter k. Therefore the controlled system is(18)xn+1=a+1-b-γxn+γxn2yn+kxn2-x∗2,yn+1=yn+bxn-γxn2yn+kyn2-y∗2.Theorem 6.
LetA3=1-b-γ+2γx∗y∗+2kx∗,B3=1-γx∗2+2ky∗,C3=γx∗2,andD3=b-2γx∗y∗, where x∗,y∗ denotes the fixed point of the controlled system (18). If A3+B3±A3-B32+4C3D3<2, the fixed point of the system is attractive.Proof.
WriteB~3x,y=a+1-b-γx+γx2y+kx2-x∗2,y+bx-γx2y+ky2-y∗2. The Jacobi matrix of the controlled system (18) is(19)DB~3=1-b-γ+2γxnyn+2kxnγxn2b-2γxnyn1-γxn2+2kynand its eigenmatrix is(20)DB~3-λE=1-b-γ+2γxnyn+2kxn-λγxn2b-2γxnyn1-γxn2+2kyn-λ.
And the characteristic equation is(21)λ2-2-b-γ+2γxnyn+2kxn-γxn2+2kynλ+1-b-γ+2γxnyn+2kxn1-γxn2+2kyn-γxn2b-2γxnyn=0.
When the modulus of the eigenvalueλ31,λ32 of the Jacobi matrix at the fixed point is less than 1, the fixed point is attractive:(22)λ31,32=A3+B3±A3-B32+4C3D32<1;that is, (23)A3+B3±A3-B32+4C3D3<2.
By Theorem6, we know that the Brusselator model can be controlled by selecting the value of k which satisfies the condition, and then the control of the Julia set can be realized.
For example, we take the values of system parametersa=0.07,b=0.02,andγ=0.01, which are the same as Section 3.1. Then the range of the value of k is -2.6420<k<0.8561.
Six simulation diagrams are chosen corresponding to the values ofk from −0.0005 to −0.02 in Figure 8 to illustrate the change of the Julia sets in gradient control method. In Figure 9 the box-counting dimensions of the controlled Julia sets are computed in this control method.
In the same control, six simulation diagrams are chosen corresponding to the values ofk from 0.0005 to 0.02 in Figure 10 to illustrate the change of the Julia sets in gradient control method. In Figure 11 the box-counting dimensions of the controlled Julia sets are computed in this control method.
In gradient control method, we consider two groups of interval of the parameterk. It is worth noticing that the contraction of the left and the lower parts of the Julia set are faster than the right and the upper parts when the interval of control parameter k is from −0.0005 to −0.02, while the right and the upper parts are faster than the left and the lower parts when the interval of control parameter k is from 0.0005 to 0.02. In general, with the absolute values of k increasing, the Julia sets contract to the center gradually, and the complexity of the boundary of the Julia sets decreases.
From the perspective of the change of box-counting dimensions, with the absolute values ofk increasing, the box-counting dimensions of the Julia sets generally show a monotonic decreasing trend. Particularly when the control parameter k is in the range from −0.016 to −0.0075 and from 0.0075 to 0.016, the change of box-counting dimensions is obviously monotonic, which indicates the effectiveness of this method on the Julia set of the Brusselator model.Figure 8
The change of the Julia sets of the controlled system whena=0.07,b=0.02, and γ=0.01. (a) k=-0.0005; (b) k=-0.004; (c) k=-0.008; (d) k=-0.012; (e) k=-0.016; (f) k=-0.020.
(a) (b) (c) (d) (e) (f)Figure 9
The change of box-counting dimensions of the Julia sets of the controlled system whenk is from −0.02 to −0.0005.Figure 10
The change of the Julia sets of the controlled system whena=0.07,b=0.02, and γ=0.01. (a) k=0.0005; (b) k=0.0045; (c) k=0.0085; (d) k=0.0125; (e) k=0.0165; (f) k=0.0200.
(a) (b) (c) (d) (e) (f)Figure 11
The change of box-counting dimensions of the Julia sets of the controlled system whenk is from 0.0005 to 0.02.


## 4. Conclusion

Fractal theory is a hot topic in the research of nonlinear science. Describing the change of chemical elements in the chemical reaction process, Brusselator model, an important class of reaction diffusion equations, is significant in the study of chaotic and fractal behaviors of nonlinear differential equations. In technological applications, it is often required that the behavior and performance of the system can be controlled effectively.In this paper, feedback control method, optimal control method, and gradient control method are taken to control the Julia set of Brusselator model, respectively, and the box-counting dimensions of the Julia set are calculated. In each control method, when the absolute value of control parameterk increases discretely, the box-counting dimension of Julia set decreases and the Julia set contracts to the center gradually. The decrease of box-counting dimension is nearly monotonic, which indicates that the complexity of the Julia set of the system is declined gradually. Thus when the control parameter k is selected, the Julia set of Brusselator model could be controlled. Most importantly, the three control methods have consistency of conclusion and effectiveness.



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