research article fractal dimension analysis of the...
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Research ArticleFractal Dimension Analysis of the Julia Sets of ControlledBrusselator Model
Yuqian Deng Xiuxiong Liu and Yongping Zhang
School of Mathematics and Statistics Shandong University at Weihai Weihai 264209 China
Correspondence should be addressed to Yongping Zhang ypzhangsdueducn
Received 21 September 2016 Accepted 7 November 2016
Academic Editor Cengiz Cinar
Copyright copy 2016 Yuqian Deng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Fractal theory is a branch of nonlinear scientific research and its research object is the irregular geometric form in nature Onaccount of the complexity of the fractal set the traditional Euclidean dimension is no longer applicable and the measurementmethod of fractal dimension is required In the numerous fractal dimension definitions box-counting dimension is taken tocharacterize the complexity of Julia set since the calculation of box-counting dimension is relatively achievable In this paper theJulia set of Brusselator model which is a class of reaction diffusion equations from the viewpoint of fractal dynamics is discussedand the control of the Julia set is researched by feedback control method optimal control method and gradient control methodrespectively Meanwhile we calculate the box-counting dimension of the Julia set of controlled Brusselator model in each controlmethod which is used to describe the complexity of the controlled Julia set and the system Ultimately we demonstrate theeffectiveness of each control method
1 Introduction
In order to recognize the essence of some extremely sophis-ticated phenomena researchers attempt to figure out theregularity and unity which exist behind these phenomena sothat they can control and predict them better In the early20th century the fundamental theory of chaos and fractalwas proposed The theory explains the unity of determinacyand randomness and the unity of order and disorder It isconsidered to be the third major revolution of science afterthe theory of relativity and quantum mechanics [1 2]
Fractal theory first proposed in the 1970s comes fromthe study of nonlinear science Its primary research object isthe geometric form of nature and nonlinear system which iscomplex but has some kind of self similarity and regularity In1977 Mandelbrot a professor of mathematics of the HarvardUniversity published the landmark work Fractal FormChance and Dimension [3] It marked the fractal geometrythat had become an independent discipline Subsequently hepublished another work The Fractal Geometry of Nature [4]which implied that fractal theory had been basically formed
Nowadays with the emergence of some new mathemati-cal tools andmethods especially the combination of the study
of fractal theory and computer the theory has been developedrapidly In addition researchers not only constantly establishand improve the theory of fractals but also apply it in variousfields such as the diffusion processes and chemical kinetics incrowded media the protein structure and complex vascularbranches in biomedicine dynamical system and hydrome-chanics in physics and landforms evolution and earthquakemonitoring [5ndash16] Even in social and economic activi-ties the theory of fractals also has numerous applications[17ndash19]
Considering the complexity of fractal sets traditionalEuclidean geometry dimension cannot accurately depict theirgeometric forms Mathematicians propose many definitionsof noninteger dimension and use different names to distin-guish them For example Hausdorff dimension which wasproposed by the German mathematician Hausdorff in 1919has a rigorous mathematical definition It is established onthe basis of Hausdorff measure and can define most fractalsets so it is easier to deal with inmathematics [20]Moreoverbox-counting dimension is one of the most widely useddimensions Its popularity is largely due to its relative ease ofmathematical calculation and empirical estimation Besidesother fractal dimensions such as similar dimension capacity
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 8234108 13 pageshttpdxdoiorg10115520168234108
2 Discrete Dynamics in Nature and Society
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Figure 1 The original Julia set of the system
dimension and Lyapunov dimension also have their ownapplications in the corresponding fields [21ndash24]
Brusselator model [25ndash27] is a kind of reaction diffusionequations which describe the change of chemical elementsin the process of chemical reaction [28] It is significant inthe study of the chaos and fractal behavior of nonlineardifferential equations Researchers have studied Brusselatormodel from different aspects and proven some properties ofit [29ndash33]These studies of the model have contributed muchto the development of nonlinear mathematics Nowadayswith the development of research people apply some of theproperty of the model in all kinds of disciplines and socialproduction activities and have achieved fruitful results
Notably in the nonlinear system the Julia set of thesystem is an important nonlinear feature According to theobjective requirement we often need restrict the size of thenonlinear attractive domain And sometimes it is requiredthat the system possess different or similar behavior andperformance in compliance with the actual requirements oftechnical problems As a result how to effectively control theJulia set is particularly critical
Based on the Julia set of Brusselatormodel feedback con-trol method optimal control method and gradient controlmethod [34] are taken to control the Julia set of the modelAnd the box-counting dimension of the Julia set of controlledBrusselator model is calculated in each control method todescribe the complexity of the Julia set of the system
2 Basic Theory
In 1918 Julia Gaston a famous French mathematician dis-covered an important fractal set in fractal theory when hestudied the iteration of complex functions which was namedJulia set He noticed that functions on the complex plane assimple as 119891(119911) = 1199112 + 119888 with a complex constant 119888 can giverise to fractals of an exotic appearanceThe precise definitionof Julia set is given below [35]
Take 119891 119862 rarr 119862 to be a polynomial of degree 119899 ge 2 withcomplex coefficients 119891(119911) = 1198860 + 1198861119911 + sdot sdot sdot + 119886119899119911119899 Write 119891119896for the 119896-fold composition 119891 ∘ sdot sdot sdot ∘ 119891 so that 119891119896(119908) is the 119896thiterate 119891(119891(sdot sdot sdot (119891(119908)))) of 119908 If 119891(119908) = 119908 we call 119908 a fixed
point of 119891 and if 119891119901(119908) = 119908 for some integer 119901 ge 1 we call119908 a periodic point of 119891 the least such 119901 is called the periodof 119908 We call 119908119891(119908) 119891119901(119908) a period 119901 orbit Let 119908 bea periodic point of period 119901 with (119891119901)1015840(119908) = 120582 where theprime denotes complex differentiation The point 119908 is calledsuperattractive if 120582 = 0 attractive if 0 le |120582| lt 1 neutral if|120582| = 1 and repelling if |120582| gt 1Definition 1 Let 119891 119862 rarr 119862 be a polynomial of degree 119899 gt 1Julia set of 119891 is defined to be the closure of repelling periodpoints of 119891
In fractal theory fractal dimension is one of the mostelemental concepts At present there are many definitionsof fractal dimensions including Hausdorff dimension box-counting dimension similarity dimension In all kinds ofdefinitions of fractal dimensions Hausdorff dimension is thebasis of the fractal theory It can even be considered thetheoretical basis of the fractal geometry However Hausdorffdimension is just suitable for the theoretical analysis offractal theory and there is only a small class of fairly solidmathematical regular fractal graphics that can be calculatedfor their Hausdorff dimension It is hard to calculate thefractal dimension which is proposed in the practical applica-tionsTherefore people propose the concept of box-countingdimension Its popularity is largely due to its relative ease ofmathematical calculation and empirical estimation In factin practical applications the dimension is generally referredto as box-counting dimension The precise definition of box-counting dimension is as follows
Definition 2 (see [35]) Let 119865 be any nonempty boundedsubset of 119877119899 and let 119873120575(119865) be the smallest number of setsof diameter at most 120575 which can cover 119865 The lower andupper box-counting dimensions of 119865 ((1) (2)) respectively aredefined as
dim119861119865 = lim120575rarr0
log119873120575 (119865)minus log 120575 (1)
dim119861119865 = lim120575rarr0
log119873120575 (119865)minus log 120575 (2)
Discrete Dynamics in Nature and Society 3
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Figure 2 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 006 (b) 119896 = 014 (c) 119896 = 022(d) 119896 = 030 (e) 119896 = 038 (f) 119896 = 045
If these are equal we refer to the common value as thebox-counting dimension of 119865 (3)
dim119861119865 = lim120575rarr0
log119873120575 (119865)minus log 120575 (3)
3 The Control of the Julia Set ofBrusselator Model
Brusselator model is one of the most fundamental models innonlinear systems and the dynamic equations are as follows
4 Discrete Dynamics in Nature and Society
005Control parameter k
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Figure 3 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from 006 to 045
119909119899+1 = 119860 minus (119861 + 1) 119909119899 + 1199092119899119910119899119910119899+1 = 119861119909119899 minus 1199092119899119910119899
(4)
where 119909 119910 denote concentration of reactant in the process ofchemical reaction 119860 119861 gt 0 denote initial concentration ofreactant
Brusselator equations were first discovered by A Turingin 1952 [36] and then I Pigogine and Leefver did somesystematic studies on itThey pointed out that the Brusselatorequations were the most elementary and essential mathe-matic model which describes the oscillation of biochemistryThey proved that when 119861 gt 1 + 1198602 the equations have stableand unique limit cycle When 119861 le 1 + 1198602 there is no limitcycle [37] It can be known that the initial concentration ofreactant has an important influence on the system In fractaltheory Julia set is a set of initial points of the system thatsatisfy certain conditions With the same thought we definethe Julia set of Brusselator model Let 119865(119909 119910) = (119860 minus (119861 +1)119909119899 + 1199092119899119910119899 119861119909119899 minus 1199092119899119910119899)Definition 3 Set 119870 = (119909 119910) isin 1198772 | 119865119899(119909 119910)119899ge1 is bound-ed is called the filled Julia set of Brusselator model Theboundary of the filled Julia set is defined to be the Julia setof Brusselator model that is 119869(119865) = 120597119870
The research results demonstrate that coupled with apiecewise constant value control function we can controlthe size of the limit cycle of (4) [36] From the definition ofJulia set we found that there is a close relation between theboundedness of iterative orbits and the structure of the Juliaset So if we want to control the Julia set of Brusselator modelhow to control the boundedness of the system iterative iscritical We consider especially the stability of the fixed pointof Brusselator model and by designing controllers unstablefixed points are turned into stable fixed points to control theboundedness of iterative orbits effectivelyThen the control ofthe Julia set can be realized
As was mentioned above considering the stability ofthe fixed points of system (5) we try to find controllers tomake the fixed point of the system stable Let the controlledBrusselator model be
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119906119899119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + V119899 (5)
where 119906119899 and V119899 denote the designed controllers
31 Feedback Control Method Take 119906119899 = 119896(119909119899 minus 119909lowast) V119899 =119896(119910119899 minus 119910lowast) with the control parameter 119896 and the controlledsystem is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119896 (119909119899 minus 119909lowast) 119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (119910119899 minus 119910lowast) (6)
Theorem 4 Let 1198601 = 1 minus 119887 minus 120574 + 2120574119909lowast119910lowast + 119896 1198611 = 1 minus120574119909lowast2 + 119896 1198621 = 120574119909lowast2 and 1198631 = 119887 minus 2120574119909lowast119910lowast where (119909lowast 119910lowast)denotes the fixed point of the controlled system (6) If |(1198601 +1198611) plusmnradic(1198601 minus 1198611)2 + 411986211198631| lt 2 the fixed point of the systemis attractive
Proof Write 1(119909 119910) = (119886+(1minus119887minus120574)119909+1205741199092119910+119896(119909minus119909lowast) 119910+119887119909 minus 1205741199092119910 + 119896(119910 minus 119910lowast)) The Jacobi matrix of the controlledsystem (6) is
1198631 = [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896 1205741199092119899119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 119896] (7)
and its eigenmatrix is
1198631 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 119896 minus 120582] (8)
Discrete Dynamics in Nature and Society 5
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Figure 4 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus022 (b) 119896 = minus038 (c)119896 = minus054 (d) 119896 = minus070 (e) 119896 = minus086 (f) 119896 = minus1
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899 + 2119896) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896) (1 minus 1205741199092119899 + 119896)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(9)
When the modulus of the eigenvalues 12058211 12058212 of theJacobi matrix at the fixed point is less than 1 the fixed pointis attractive
100381610038161003816100381612058211121003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198601 + 1198611) plusmn radic(1198601 minus 1198611)2 + 4119862111986312
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (10)
6 Discrete Dynamics in Nature and Society
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minus1 minus09 minus08 minus07 minus06 minus05 minus04 minus03 minus02
Figure 5 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus1 to minus022
that is
10038161003816100381610038161003816100381610038161003816(1198601 + 1198611) plusmn radic(1198601 minus 1198611)2 + 41198621119863110038161003816100381610038161003816100381610038161003816 lt 2 (11)
By Theorem 4 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example take 119886 = 007 119887 = 002 and 120574 = 001 insystem (6) and the initial Julia set is shown in Figure 1 thenwe get 119909lowast = 7 119910lowast = 27 By Theorem 4 the range of thevalue of 119896 is minus19895 lt 119896 lt 04695
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 006 to 045 in Figure 2 and we find that thetrend of the change of the Julia sets is obvious In Figure 3the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from minus1 to minus022 in Figure 4to illustrate the change of the Julia sets in feedback controlmethod In Figure 5 the box-counting dimensions of thecontrolled Julia sets are computed in this control method
In feedback control method the contraction of the leftand the lower parts is faster than the right and the upperparts when the interval of control parameter 119896 is between006 and 045 In addition the complexity of the boundaryof the Julia set significantly decreases and the lower partrapidly contracts When the interval of control parameter 119896is between minus1 and minus022 the complexity of the boundary ofthe Julia set significantly decreases and the Julia set tends tobe centrally symmetric In general with the absolute value of119896 increasing the Julia set contracts to the center gradually
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing the
change generally shows a monotonic decreasing trend Par-ticularly when the control parameter 119896 is in the intervalfrom 016 to 037 and minus09 to minus04 the monotonic changeof box-counting dimensions is obvious which indicates theeffectiveness of this control method on the Julia set ofBrusselator model
32 Optimal Control Method Take 119906119899 = 119896(119891(119909119899 119910119899) minus119909119899) V119899 = 119896(119892(119909119899 119910119899) minus 119910119899) with 119891(119909119899 119910119899) = 119886 + (1 minus 119887 minus120574)119909119899 + 1205741199092119899119910119899 119892(119909119899 119910119899) = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 where 119896 is thecontrol parameter Then the controlled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899+ 119896 (119891 (119909119899 119910119899) minus 119909119899)
119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (119892 (119909119899 119910119899) minus 119910119899) (12)
Theorem 5 Let1198602 = (1minus119887minus120574+2120574119909lowast119910lowast)(119896+1)minus119896 1198612 = (1minus120574119909lowast2)(119896+1)minus1198961198622 = 120574119909lowast2(119896+1) and1198632 = (119887minus2120574119909lowast119910lowast)(119896+1)where (119909lowast 119910lowast) denotes the fixed point of the controlled system(12) If |(1198602+1198612)plusmnradic(1198602 minus 1198612)2 + 411986221198632| lt 2 the fixed pointof the system is attractive
Proof Write 2(119909 119910) = (119886 + (1 minus 119887minus 120574)119909+ 1205741199092119910+119896(119891(119909 119910) minus119909) 119910 + 119887119909 minus 1205741199092119910 + 119896(119892(119909 119910) minus 119910)) The Jacobi matrix of thecontrolled system (12) is
1198632= [(1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896 1205741199092119899 (119896 + 1)
(119887 minus 2120574119909119899119910119899) (119896 + 1) (1 minus 1205741199092119899) (119896 + 1) minus 119896](13)
and its eigenmatrix is
1198632 minus 120582119864 = [(1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896 minus 120582 1205741199092119899 (119896 + 1)(119887 minus 2120574119909119899119910119899) (119896 + 1) (1 minus 1205741199092119899) (119896 + 1) minus 119896 minus 120582] (14)
Discrete Dynamics in Nature and Society 7
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Figure 6 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 064 (b) 119896 = 104 (c) 119896 = 144(d) 119896 = 184 (e) 119896 = 224 (f) 119896 = 26
8 Discrete Dynamics in Nature and Society
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Figure 7 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from 064 to 26
And the characteristic equation is
1205822 minus ((2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899) (119896 + 1) minus 2119896) 120582+ ((1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896)sdot ((1 minus 1205741199092119899) (119896 + 1) minus 119896) minus 1205741199092119899 (119887 minus 2120574119909119899119910119899)sdot (119896 + 1)2 = 0
(15)
When themodulus of the eigenvalue 12058221 12058222 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058221221003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 4119862211986322
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (16)
that is10038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 41198622119863210038161003816100381610038161003816100381610038161003816 lt 2 (17)
By Theorem 5 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus1 lt 119896 lt190659
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 064 to 26 in Figure 6 to illustrate thechange of the Julia sets in optimal controlmethod In Figure 7the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In optimal control method the effective controlled inter-val of 119896 is from minus1 to 1906 But when the interval of 119896 is from064 to 26 this method has the best controlled effectivenessIn this interval with the absolute values of 119896 increasing the
Julia sets gradually contract to the center with nearly the samespeed and no significant change in the overall shape of theJulia sets occurs
From the perspective of the change of box-countingdimensions box-counting dimensions of the Julia sets gen-erally show a monotonic decreasing trend with the absolutevalues of 119896 increasing For the reason that the complexityof the Julia set can be depicted by box-counting dimensionwhen the complexity of the Julia set decreases with theabsolute value of 119896 increasing it indicates that this controlmethod has great control effectiveness
33 Gradient Control Method Take 119906119899 = 119896(1199092119899 minus 119909lowast2) V119899 =119896(1199102119899 minus 119910lowast2) with the control parameter 119896 Therefore thecontrolled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119896 (1199092119899 minus 119909lowast2) 119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (1199102119899 minus 119910lowast2) (18)
Theorem 6 Let 1198603 = 1 minus 119887 minus 120574 + 2120574119909lowast119910lowast + 2119896119909lowast 1198613 =1 minus 120574119909lowast2 + 2119896119910lowast 1198623 = 120574119909lowast2 and 1198633 = 119887 minus 2120574119909lowast119910lowast where(119909lowast 119910lowast) denotes the fixed point of the controlled system (18) If|(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 411986231198633| lt 2 the fixed point of thesystem is attractive
Proof Write 3(119909 119910) = (119886 + (1 minus 119887 minus 120574)119909 + 1205741199092119910 + 119896(1199092 minus119909lowast2) 119910 + 119887119909 minus 1205741199092119910 + 119896(1199102 minus 119910lowast2)) The Jacobi matrix of thecontrolled system (18) is
1198633 = [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 1205741199092119899119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899] (19)
Discrete Dynamics in Nature and Society 9
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Figure 8 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus00005 (b) 119896 = minus0004 (c)119896 = minus0008 (d) 119896 = minus0012 (e) 119896 = minus0016 (f) 119896 = minus0020
10 Discrete Dynamics in Nature and Society
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Figure 9 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus002 to minus00005
and its eigenmatrix is
1198633 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899 minus 120582] (20)
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 1205741199092119899 + 2119896119910119899) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899) (1 minus 1205741199092119899 + 2119896119910119899)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(21)
When themodulus of the eigenvalue 12058231 12058232 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058231321003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 4119862311986332
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (22)
that is10038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 41198623119863310038161003816100381610038161003816100381610038161003816 lt 2 (23)
By Theorem 6 we know that the Brusselator model canbe controlled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus26420 lt 119896 lt08561
Six simulation diagrams are chosen corresponding to thevalues of 119896 from minus00005 to minus002 in Figure 8 to illustratethe change of the Julia sets in gradient control method In
Figure 9 the box-counting dimensions of the controlled Juliasets are computed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from 00005 to 002 inFigure 10 to illustrate the change of the Julia sets in gradientcontrol method In Figure 11 the box-counting dimensionsof the controlled Julia sets are computed in this controlmethod
In gradient control method we consider two groups ofinterval of the parameter 119896 It is worth noticing that thecontraction of the left and the lower parts of the Julia set arefaster than the right and the upper parts when the interval ofcontrol parameter 119896 is from minus00005 to minus002 while the rightand the upper parts are faster than the left and the lower partswhen the interval of control parameter 119896 is from 00005 to002 In general with the absolute values of 119896 increasing theJulia sets contract to the center gradually and the complexityof the boundary of the Julia sets decreases
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing thebox-counting dimensions of the Julia sets generally show amonotonic decreasing trend Particularly when the controlparameter 119896 is in the range from minus0016 to minus00075 and from00075 to 0016 the change of box-counting dimensions isobviouslymonotonic which indicates the effectiveness of thismethod on the Julia set of the Brusselator model
4 Conclusion
Fractal theory is a hot topic in the research of nonlinearscience Describing the change of chemical elements in thechemical reaction process Brusselator model an importantclass of reaction diffusion equations is significant in thestudy of chaotic and fractal behaviors of nonlinear differentialequations In technological applications it is often requiredthat the behavior and performance of the system can becontrolled effectively
Discrete Dynamics in Nature and Society 11
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 10 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 00005 (b) 119896 = 00045 (c)119896 = 00085 (d) 119896 = 00125 (e) 119896 = 00165 (f) 119896 = 00200
12 Discrete Dynamics in Nature and Society
135
136
137
138
139
14
141
142
143
144
1450
0002
0004
0006
0008
001
0012
0014
0016
0018
002
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 11 The change of box-counting dimensions of the Julia setsof the controlled system when 119896 is from 00005 to 002
In this paper feedback control method optimal controlmethod and gradient control method are taken to controlthe Julia set of Brusselator model respectively and thebox-counting dimensions of the Julia set are calculated Ineach control method when the absolute value of controlparameter 119896 increases discretely the box-counting dimensionof Julia set decreases and the Julia set contracts to the centergradually The decrease of box-counting dimension is nearlymonotonic which indicates that the complexity of the Juliaset of the system is declined graduallyThus when the controlparameter 119896 is selected the Julia set of Brusselator modelcould be controlled Most importantly the three controlmethods have consistency of conclusion and effectiveness
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61403231 and 11501328) and ChinaPostdoctoral Science Foundation (no 2016M592188)
References
[1] X Wang Fractal Mechanism of Generalized M-J Sets DalianUniversity of Technology Press Dalian China 2002
[2] G Chen and X Dong ldquoFrom chaos to order methodologiesperspectives and applicationsrdquo Methodologies Perspectives ampApplications World Scientific vol 31 no 2 pp 113ndash122 1998
[3] B B Mandelbrot Fractal Form Chance and Dimension W HFreeman San Francisco Calif USA 1977
[4] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Co San Francisco Calif USA 1982
[5] L Pitulice D Craciun E Vilaseca et al ldquoFractal dimension ofthe trajectory of a single particle diffusing in crowded mediardquoRomanian Journal of Physics vol 61 no 7-8 pp 1276ndash1286 2016
[6] L Pitulice E Vilaseca I Pastor et al ldquoMonte Carlo simulationsof enzymatic reactions in crowdedmedia Effect of the enzyme-obstacle relative sizerdquo Mathematical Biosciences vol 251 no 1pp 72ndash82 2014
[7] M Lewis and D C Rees ldquoFractal surfaces of proteinsrdquo Sciencevol 230 no 4730 pp 1163ndash1165 1985
[8] A Isvoran L Pitulice C T Craescu and A Chiriac ldquoFractalaspects of calcium binding protein structuresrdquo Chaos Solitonsamp Fractals vol 35 no 5 pp 960ndash966 2008
[9] P J Bentley ldquoFractal proteinsrdquo Genetic Programming amp Evolv-able Machines vol 5 no 1 pp 71ndash101 2004
[10] M Zamir ldquoFractal dimensions and multifractility in vascularbranchingrdquo Journal of Theoretical Biology vol 212 no 2 pp183ndash190 2001
[11] R L Devaney and J P Eckmann ldquoAn introduction to chaoticdynamical systemsrdquo Mathematical Gazette vol 74 no 468 p72 1990
[12] Z Huang ldquoFractal characteristics of turbulencerdquo Advances inMechanics vol 30 no 4 pp 581ndash596 2000
[13] M S Movahed and E Hermanis ldquoFractal analysis of riverflow fluctuationsrdquo Physica A Statistical Mechanics and ItsApplications vol 387 no 4 pp 915ndash932 2008
[14] N Yonaiguchi Y Ida M Hayakawa and S Masuda ldquoFractalanalysis for VHF electromagnetic noises and the identificationof preseismic signature of an earthquakerdquo Journal of Atmo-spheric and Solar-Terrestrial Physics vol 69 no 15 pp 1825ndash1832 2007
[15] N Ai R Chen and H Li ldquoTo the fractal geomorphologyrdquoGeography and Territorial Research vol 15 no 1 pp 92ndash961999
[16] M Li L Zhu and H Long ldquoAnalysis of several problems in theapplication of fractal in geomorphologyrdquo Earthquake Researchvol 25 no 2 pp 155ndash162 2002
[17] J Shu D Tan and J Wu ldquoFractal structure of Chinese stockmarketrdquo Journal of Southwest Jiao Tong University vol 38 no 2pp 212ndash215 2003
[18] Y Lin ldquoFractal and its application in securitymarketrdquo EconomicBusiness vol 8 pp 44ndash46 2001
[19] Z Peng C Li and H Li ldquoResearch on enterprises fractalmanagement model in knowledge economy era rdquo BusinessResearch vol 255 no 10 pp 33ndash35 2002
[20] W Zeng The Dimension Calculation of Several Special Self-Similar Sets Central China Normal University Wuhan China2007
[21] K J FalconerTheGeometry of Fractal Sets vol 85 ofCambridgeTracts inMathematics CambridgeUniversity Press CambridgeUK 1986
[22] Preiss andDavid ldquoNondifferentiable functionsrdquo Transactions ofthe American Mathematical Society vol 17 no 3 pp 301ndash3251916
[23] B B Mandelbrot Les Objets Fractals Flammarion 1984[24] X Gaoan Fractal Dimension Seismological Press Beijing
China 1989[25] T Erneux and E L Reiss ldquoBrussellator isolasrdquo SIAM Journal on
Applied Mathematics vol 43 no 6 pp 1240ndash1246 1983[26] G Nicolis ldquoPatterns of spatio-temporal organization in chem-
ical and biochemical kineticsrdquo American Mathematical Societyvol 8 pp 33ndash58 1974
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Discrete Dynamics in Nature and Society
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus60minus80
Figure 1 The original Julia set of the system
dimension and Lyapunov dimension also have their ownapplications in the corresponding fields [21ndash24]
Brusselator model [25ndash27] is a kind of reaction diffusionequations which describe the change of chemical elementsin the process of chemical reaction [28] It is significant inthe study of the chaos and fractal behavior of nonlineardifferential equations Researchers have studied Brusselatormodel from different aspects and proven some properties ofit [29ndash33]These studies of the model have contributed muchto the development of nonlinear mathematics Nowadayswith the development of research people apply some of theproperty of the model in all kinds of disciplines and socialproduction activities and have achieved fruitful results
Notably in the nonlinear system the Julia set of thesystem is an important nonlinear feature According to theobjective requirement we often need restrict the size of thenonlinear attractive domain And sometimes it is requiredthat the system possess different or similar behavior andperformance in compliance with the actual requirements oftechnical problems As a result how to effectively control theJulia set is particularly critical
Based on the Julia set of Brusselatormodel feedback con-trol method optimal control method and gradient controlmethod [34] are taken to control the Julia set of the modelAnd the box-counting dimension of the Julia set of controlledBrusselator model is calculated in each control method todescribe the complexity of the Julia set of the system
2 Basic Theory
In 1918 Julia Gaston a famous French mathematician dis-covered an important fractal set in fractal theory when hestudied the iteration of complex functions which was namedJulia set He noticed that functions on the complex plane assimple as 119891(119911) = 1199112 + 119888 with a complex constant 119888 can giverise to fractals of an exotic appearanceThe precise definitionof Julia set is given below [35]
Take 119891 119862 rarr 119862 to be a polynomial of degree 119899 ge 2 withcomplex coefficients 119891(119911) = 1198860 + 1198861119911 + sdot sdot sdot + 119886119899119911119899 Write 119891119896for the 119896-fold composition 119891 ∘ sdot sdot sdot ∘ 119891 so that 119891119896(119908) is the 119896thiterate 119891(119891(sdot sdot sdot (119891(119908)))) of 119908 If 119891(119908) = 119908 we call 119908 a fixed
point of 119891 and if 119891119901(119908) = 119908 for some integer 119901 ge 1 we call119908 a periodic point of 119891 the least such 119901 is called the periodof 119908 We call 119908119891(119908) 119891119901(119908) a period 119901 orbit Let 119908 bea periodic point of period 119901 with (119891119901)1015840(119908) = 120582 where theprime denotes complex differentiation The point 119908 is calledsuperattractive if 120582 = 0 attractive if 0 le |120582| lt 1 neutral if|120582| = 1 and repelling if |120582| gt 1Definition 1 Let 119891 119862 rarr 119862 be a polynomial of degree 119899 gt 1Julia set of 119891 is defined to be the closure of repelling periodpoints of 119891
In fractal theory fractal dimension is one of the mostelemental concepts At present there are many definitionsof fractal dimensions including Hausdorff dimension box-counting dimension similarity dimension In all kinds ofdefinitions of fractal dimensions Hausdorff dimension is thebasis of the fractal theory It can even be considered thetheoretical basis of the fractal geometry However Hausdorffdimension is just suitable for the theoretical analysis offractal theory and there is only a small class of fairly solidmathematical regular fractal graphics that can be calculatedfor their Hausdorff dimension It is hard to calculate thefractal dimension which is proposed in the practical applica-tionsTherefore people propose the concept of box-countingdimension Its popularity is largely due to its relative ease ofmathematical calculation and empirical estimation In factin practical applications the dimension is generally referredto as box-counting dimension The precise definition of box-counting dimension is as follows
Definition 2 (see [35]) Let 119865 be any nonempty boundedsubset of 119877119899 and let 119873120575(119865) be the smallest number of setsof diameter at most 120575 which can cover 119865 The lower andupper box-counting dimensions of 119865 ((1) (2)) respectively aredefined as
dim119861119865 = lim120575rarr0
log119873120575 (119865)minus log 120575 (1)
dim119861119865 = lim120575rarr0
log119873120575 (119865)minus log 120575 (2)
Discrete Dynamics in Nature and Society 3
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(d)
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus60minus80
(e)
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus60minus80
(f)
Figure 2 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 006 (b) 119896 = 014 (c) 119896 = 022(d) 119896 = 030 (e) 119896 = 038 (f) 119896 = 045
If these are equal we refer to the common value as thebox-counting dimension of 119865 (3)
dim119861119865 = lim120575rarr0
log119873120575 (119865)minus log 120575 (3)
3 The Control of the Julia Set ofBrusselator Model
Brusselator model is one of the most fundamental models innonlinear systems and the dynamic equations are as follows
4 Discrete Dynamics in Nature and Society
005Control parameter k
139
14
141
142
143
144
145
146
Box-
coun
ting
dim
ensio
n
04504035030250201501
Figure 3 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from 006 to 045
119909119899+1 = 119860 minus (119861 + 1) 119909119899 + 1199092119899119910119899119910119899+1 = 119861119909119899 minus 1199092119899119910119899
(4)
where 119909 119910 denote concentration of reactant in the process ofchemical reaction 119860 119861 gt 0 denote initial concentration ofreactant
Brusselator equations were first discovered by A Turingin 1952 [36] and then I Pigogine and Leefver did somesystematic studies on itThey pointed out that the Brusselatorequations were the most elementary and essential mathe-matic model which describes the oscillation of biochemistryThey proved that when 119861 gt 1 + 1198602 the equations have stableand unique limit cycle When 119861 le 1 + 1198602 there is no limitcycle [37] It can be known that the initial concentration ofreactant has an important influence on the system In fractaltheory Julia set is a set of initial points of the system thatsatisfy certain conditions With the same thought we definethe Julia set of Brusselator model Let 119865(119909 119910) = (119860 minus (119861 +1)119909119899 + 1199092119899119910119899 119861119909119899 minus 1199092119899119910119899)Definition 3 Set 119870 = (119909 119910) isin 1198772 | 119865119899(119909 119910)119899ge1 is bound-ed is called the filled Julia set of Brusselator model Theboundary of the filled Julia set is defined to be the Julia setof Brusselator model that is 119869(119865) = 120597119870
The research results demonstrate that coupled with apiecewise constant value control function we can controlthe size of the limit cycle of (4) [36] From the definition ofJulia set we found that there is a close relation between theboundedness of iterative orbits and the structure of the Juliaset So if we want to control the Julia set of Brusselator modelhow to control the boundedness of the system iterative iscritical We consider especially the stability of the fixed pointof Brusselator model and by designing controllers unstablefixed points are turned into stable fixed points to control theboundedness of iterative orbits effectivelyThen the control ofthe Julia set can be realized
As was mentioned above considering the stability ofthe fixed points of system (5) we try to find controllers tomake the fixed point of the system stable Let the controlledBrusselator model be
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119906119899119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + V119899 (5)
where 119906119899 and V119899 denote the designed controllers
31 Feedback Control Method Take 119906119899 = 119896(119909119899 minus 119909lowast) V119899 =119896(119910119899 minus 119910lowast) with the control parameter 119896 and the controlledsystem is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119896 (119909119899 minus 119909lowast) 119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (119910119899 minus 119910lowast) (6)
Theorem 4 Let 1198601 = 1 minus 119887 minus 120574 + 2120574119909lowast119910lowast + 119896 1198611 = 1 minus120574119909lowast2 + 119896 1198621 = 120574119909lowast2 and 1198631 = 119887 minus 2120574119909lowast119910lowast where (119909lowast 119910lowast)denotes the fixed point of the controlled system (6) If |(1198601 +1198611) plusmnradic(1198601 minus 1198611)2 + 411986211198631| lt 2 the fixed point of the systemis attractive
Proof Write 1(119909 119910) = (119886+(1minus119887minus120574)119909+1205741199092119910+119896(119909minus119909lowast) 119910+119887119909 minus 1205741199092119910 + 119896(119910 minus 119910lowast)) The Jacobi matrix of the controlledsystem (6) is
1198631 = [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896 1205741199092119899119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 119896] (7)
and its eigenmatrix is
1198631 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 119896 minus 120582] (8)
Discrete Dynamics in Nature and Society 5
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(f)
Figure 4 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus022 (b) 119896 = minus038 (c)119896 = minus054 (d) 119896 = minus070 (e) 119896 = minus086 (f) 119896 = minus1
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899 + 2119896) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896) (1 minus 1205741199092119899 + 119896)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(9)
When the modulus of the eigenvalues 12058211 12058212 of theJacobi matrix at the fixed point is less than 1 the fixed pointis attractive
100381610038161003816100381612058211121003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198601 + 1198611) plusmn radic(1198601 minus 1198611)2 + 4119862111986312
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (10)
6 Discrete Dynamics in Nature and Society
13
135
14
145
Control parameter k
Box-
coun
ting
dim
ensio
n
minus1 minus09 minus08 minus07 minus06 minus05 minus04 minus03 minus02
Figure 5 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus1 to minus022
that is
10038161003816100381610038161003816100381610038161003816(1198601 + 1198611) plusmn radic(1198601 minus 1198611)2 + 41198621119863110038161003816100381610038161003816100381610038161003816 lt 2 (11)
By Theorem 4 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example take 119886 = 007 119887 = 002 and 120574 = 001 insystem (6) and the initial Julia set is shown in Figure 1 thenwe get 119909lowast = 7 119910lowast = 27 By Theorem 4 the range of thevalue of 119896 is minus19895 lt 119896 lt 04695
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 006 to 045 in Figure 2 and we find that thetrend of the change of the Julia sets is obvious In Figure 3the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from minus1 to minus022 in Figure 4to illustrate the change of the Julia sets in feedback controlmethod In Figure 5 the box-counting dimensions of thecontrolled Julia sets are computed in this control method
In feedback control method the contraction of the leftand the lower parts is faster than the right and the upperparts when the interval of control parameter 119896 is between006 and 045 In addition the complexity of the boundaryof the Julia set significantly decreases and the lower partrapidly contracts When the interval of control parameter 119896is between minus1 and minus022 the complexity of the boundary ofthe Julia set significantly decreases and the Julia set tends tobe centrally symmetric In general with the absolute value of119896 increasing the Julia set contracts to the center gradually
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing the
change generally shows a monotonic decreasing trend Par-ticularly when the control parameter 119896 is in the intervalfrom 016 to 037 and minus09 to minus04 the monotonic changeof box-counting dimensions is obvious which indicates theeffectiveness of this control method on the Julia set ofBrusselator model
32 Optimal Control Method Take 119906119899 = 119896(119891(119909119899 119910119899) minus119909119899) V119899 = 119896(119892(119909119899 119910119899) minus 119910119899) with 119891(119909119899 119910119899) = 119886 + (1 minus 119887 minus120574)119909119899 + 1205741199092119899119910119899 119892(119909119899 119910119899) = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 where 119896 is thecontrol parameter Then the controlled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899+ 119896 (119891 (119909119899 119910119899) minus 119909119899)
119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (119892 (119909119899 119910119899) minus 119910119899) (12)
Theorem 5 Let1198602 = (1minus119887minus120574+2120574119909lowast119910lowast)(119896+1)minus119896 1198612 = (1minus120574119909lowast2)(119896+1)minus1198961198622 = 120574119909lowast2(119896+1) and1198632 = (119887minus2120574119909lowast119910lowast)(119896+1)where (119909lowast 119910lowast) denotes the fixed point of the controlled system(12) If |(1198602+1198612)plusmnradic(1198602 minus 1198612)2 + 411986221198632| lt 2 the fixed pointof the system is attractive
Proof Write 2(119909 119910) = (119886 + (1 minus 119887minus 120574)119909+ 1205741199092119910+119896(119891(119909 119910) minus119909) 119910 + 119887119909 minus 1205741199092119910 + 119896(119892(119909 119910) minus 119910)) The Jacobi matrix of thecontrolled system (12) is
1198632= [(1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896 1205741199092119899 (119896 + 1)
(119887 minus 2120574119909119899119910119899) (119896 + 1) (1 minus 1205741199092119899) (119896 + 1) minus 119896](13)
and its eigenmatrix is
1198632 minus 120582119864 = [(1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896 minus 120582 1205741199092119899 (119896 + 1)(119887 minus 2120574119909119899119910119899) (119896 + 1) (1 minus 1205741199092119899) (119896 + 1) minus 119896 minus 120582] (14)
Discrete Dynamics in Nature and Society 7
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 6 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 064 (b) 119896 = 104 (c) 119896 = 144(d) 119896 = 184 (e) 119896 = 224 (f) 119896 = 26
8 Discrete Dynamics in Nature and Society
13606 08 1 12 14 16 18 2 22 24 26
137
138
139
14
141
142
143
144
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 7 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from 064 to 26
And the characteristic equation is
1205822 minus ((2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899) (119896 + 1) minus 2119896) 120582+ ((1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896)sdot ((1 minus 1205741199092119899) (119896 + 1) minus 119896) minus 1205741199092119899 (119887 minus 2120574119909119899119910119899)sdot (119896 + 1)2 = 0
(15)
When themodulus of the eigenvalue 12058221 12058222 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058221221003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 4119862211986322
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (16)
that is10038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 41198622119863210038161003816100381610038161003816100381610038161003816 lt 2 (17)
By Theorem 5 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus1 lt 119896 lt190659
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 064 to 26 in Figure 6 to illustrate thechange of the Julia sets in optimal controlmethod In Figure 7the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In optimal control method the effective controlled inter-val of 119896 is from minus1 to 1906 But when the interval of 119896 is from064 to 26 this method has the best controlled effectivenessIn this interval with the absolute values of 119896 increasing the
Julia sets gradually contract to the center with nearly the samespeed and no significant change in the overall shape of theJulia sets occurs
From the perspective of the change of box-countingdimensions box-counting dimensions of the Julia sets gen-erally show a monotonic decreasing trend with the absolutevalues of 119896 increasing For the reason that the complexityof the Julia set can be depicted by box-counting dimensionwhen the complexity of the Julia set decreases with theabsolute value of 119896 increasing it indicates that this controlmethod has great control effectiveness
33 Gradient Control Method Take 119906119899 = 119896(1199092119899 minus 119909lowast2) V119899 =119896(1199102119899 minus 119910lowast2) with the control parameter 119896 Therefore thecontrolled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119896 (1199092119899 minus 119909lowast2) 119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (1199102119899 minus 119910lowast2) (18)
Theorem 6 Let 1198603 = 1 minus 119887 minus 120574 + 2120574119909lowast119910lowast + 2119896119909lowast 1198613 =1 minus 120574119909lowast2 + 2119896119910lowast 1198623 = 120574119909lowast2 and 1198633 = 119887 minus 2120574119909lowast119910lowast where(119909lowast 119910lowast) denotes the fixed point of the controlled system (18) If|(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 411986231198633| lt 2 the fixed point of thesystem is attractive
Proof Write 3(119909 119910) = (119886 + (1 minus 119887 minus 120574)119909 + 1205741199092119910 + 119896(1199092 minus119909lowast2) 119910 + 119887119909 minus 1205741199092119910 + 119896(1199102 minus 119910lowast2)) The Jacobi matrix of thecontrolled system (18) is
1198633 = [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 1205741199092119899119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899] (19)
Discrete Dynamics in Nature and Society 9
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 8 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus00005 (b) 119896 = minus0004 (c)119896 = minus0008 (d) 119896 = minus0012 (e) 119896 = minus0016 (f) 119896 = minus0020
10 Discrete Dynamics in Nature and Society
136
137
138
139
14
141
142
143
144
145
minus002
minus0018
minus0016
minus0014
minus0012
minus001
minus0008
minus0006
minus0004
minus0002 0
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 9 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus002 to minus00005
and its eigenmatrix is
1198633 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899 minus 120582] (20)
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 1205741199092119899 + 2119896119910119899) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899) (1 minus 1205741199092119899 + 2119896119910119899)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(21)
When themodulus of the eigenvalue 12058231 12058232 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058231321003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 4119862311986332
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (22)
that is10038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 41198623119863310038161003816100381610038161003816100381610038161003816 lt 2 (23)
By Theorem 6 we know that the Brusselator model canbe controlled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus26420 lt 119896 lt08561
Six simulation diagrams are chosen corresponding to thevalues of 119896 from minus00005 to minus002 in Figure 8 to illustratethe change of the Julia sets in gradient control method In
Figure 9 the box-counting dimensions of the controlled Juliasets are computed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from 00005 to 002 inFigure 10 to illustrate the change of the Julia sets in gradientcontrol method In Figure 11 the box-counting dimensionsof the controlled Julia sets are computed in this controlmethod
In gradient control method we consider two groups ofinterval of the parameter 119896 It is worth noticing that thecontraction of the left and the lower parts of the Julia set arefaster than the right and the upper parts when the interval ofcontrol parameter 119896 is from minus00005 to minus002 while the rightand the upper parts are faster than the left and the lower partswhen the interval of control parameter 119896 is from 00005 to002 In general with the absolute values of 119896 increasing theJulia sets contract to the center gradually and the complexityof the boundary of the Julia sets decreases
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing thebox-counting dimensions of the Julia sets generally show amonotonic decreasing trend Particularly when the controlparameter 119896 is in the range from minus0016 to minus00075 and from00075 to 0016 the change of box-counting dimensions isobviouslymonotonic which indicates the effectiveness of thismethod on the Julia set of the Brusselator model
4 Conclusion
Fractal theory is a hot topic in the research of nonlinearscience Describing the change of chemical elements in thechemical reaction process Brusselator model an importantclass of reaction diffusion equations is significant in thestudy of chaotic and fractal behaviors of nonlinear differentialequations In technological applications it is often requiredthat the behavior and performance of the system can becontrolled effectively
Discrete Dynamics in Nature and Society 11
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 10 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 00005 (b) 119896 = 00045 (c)119896 = 00085 (d) 119896 = 00125 (e) 119896 = 00165 (f) 119896 = 00200
12 Discrete Dynamics in Nature and Society
135
136
137
138
139
14
141
142
143
144
1450
0002
0004
0006
0008
001
0012
0014
0016
0018
002
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 11 The change of box-counting dimensions of the Julia setsof the controlled system when 119896 is from 00005 to 002
In this paper feedback control method optimal controlmethod and gradient control method are taken to controlthe Julia set of Brusselator model respectively and thebox-counting dimensions of the Julia set are calculated Ineach control method when the absolute value of controlparameter 119896 increases discretely the box-counting dimensionof Julia set decreases and the Julia set contracts to the centergradually The decrease of box-counting dimension is nearlymonotonic which indicates that the complexity of the Juliaset of the system is declined graduallyThus when the controlparameter 119896 is selected the Julia set of Brusselator modelcould be controlled Most importantly the three controlmethods have consistency of conclusion and effectiveness
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61403231 and 11501328) and ChinaPostdoctoral Science Foundation (no 2016M592188)
References
[1] X Wang Fractal Mechanism of Generalized M-J Sets DalianUniversity of Technology Press Dalian China 2002
[2] G Chen and X Dong ldquoFrom chaos to order methodologiesperspectives and applicationsrdquo Methodologies Perspectives ampApplications World Scientific vol 31 no 2 pp 113ndash122 1998
[3] B B Mandelbrot Fractal Form Chance and Dimension W HFreeman San Francisco Calif USA 1977
[4] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Co San Francisco Calif USA 1982
[5] L Pitulice D Craciun E Vilaseca et al ldquoFractal dimension ofthe trajectory of a single particle diffusing in crowded mediardquoRomanian Journal of Physics vol 61 no 7-8 pp 1276ndash1286 2016
[6] L Pitulice E Vilaseca I Pastor et al ldquoMonte Carlo simulationsof enzymatic reactions in crowdedmedia Effect of the enzyme-obstacle relative sizerdquo Mathematical Biosciences vol 251 no 1pp 72ndash82 2014
[7] M Lewis and D C Rees ldquoFractal surfaces of proteinsrdquo Sciencevol 230 no 4730 pp 1163ndash1165 1985
[8] A Isvoran L Pitulice C T Craescu and A Chiriac ldquoFractalaspects of calcium binding protein structuresrdquo Chaos Solitonsamp Fractals vol 35 no 5 pp 960ndash966 2008
[9] P J Bentley ldquoFractal proteinsrdquo Genetic Programming amp Evolv-able Machines vol 5 no 1 pp 71ndash101 2004
[10] M Zamir ldquoFractal dimensions and multifractility in vascularbranchingrdquo Journal of Theoretical Biology vol 212 no 2 pp183ndash190 2001
[11] R L Devaney and J P Eckmann ldquoAn introduction to chaoticdynamical systemsrdquo Mathematical Gazette vol 74 no 468 p72 1990
[12] Z Huang ldquoFractal characteristics of turbulencerdquo Advances inMechanics vol 30 no 4 pp 581ndash596 2000
[13] M S Movahed and E Hermanis ldquoFractal analysis of riverflow fluctuationsrdquo Physica A Statistical Mechanics and ItsApplications vol 387 no 4 pp 915ndash932 2008
[14] N Yonaiguchi Y Ida M Hayakawa and S Masuda ldquoFractalanalysis for VHF electromagnetic noises and the identificationof preseismic signature of an earthquakerdquo Journal of Atmo-spheric and Solar-Terrestrial Physics vol 69 no 15 pp 1825ndash1832 2007
[15] N Ai R Chen and H Li ldquoTo the fractal geomorphologyrdquoGeography and Territorial Research vol 15 no 1 pp 92ndash961999
[16] M Li L Zhu and H Long ldquoAnalysis of several problems in theapplication of fractal in geomorphologyrdquo Earthquake Researchvol 25 no 2 pp 155ndash162 2002
[17] J Shu D Tan and J Wu ldquoFractal structure of Chinese stockmarketrdquo Journal of Southwest Jiao Tong University vol 38 no 2pp 212ndash215 2003
[18] Y Lin ldquoFractal and its application in securitymarketrdquo EconomicBusiness vol 8 pp 44ndash46 2001
[19] Z Peng C Li and H Li ldquoResearch on enterprises fractalmanagement model in knowledge economy era rdquo BusinessResearch vol 255 no 10 pp 33ndash35 2002
[20] W Zeng The Dimension Calculation of Several Special Self-Similar Sets Central China Normal University Wuhan China2007
[21] K J FalconerTheGeometry of Fractal Sets vol 85 ofCambridgeTracts inMathematics CambridgeUniversity Press CambridgeUK 1986
[22] Preiss andDavid ldquoNondifferentiable functionsrdquo Transactions ofthe American Mathematical Society vol 17 no 3 pp 301ndash3251916
[23] B B Mandelbrot Les Objets Fractals Flammarion 1984[24] X Gaoan Fractal Dimension Seismological Press Beijing
China 1989[25] T Erneux and E L Reiss ldquoBrussellator isolasrdquo SIAM Journal on
Applied Mathematics vol 43 no 6 pp 1240ndash1246 1983[26] G Nicolis ldquoPatterns of spatio-temporal organization in chem-
ical and biochemical kineticsrdquo American Mathematical Societyvol 8 pp 33ndash58 1974
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 3
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(d)
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus60minus80
(e)
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus60minus80
(f)
Figure 2 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 006 (b) 119896 = 014 (c) 119896 = 022(d) 119896 = 030 (e) 119896 = 038 (f) 119896 = 045
If these are equal we refer to the common value as thebox-counting dimension of 119865 (3)
dim119861119865 = lim120575rarr0
log119873120575 (119865)minus log 120575 (3)
3 The Control of the Julia Set ofBrusselator Model
Brusselator model is one of the most fundamental models innonlinear systems and the dynamic equations are as follows
4 Discrete Dynamics in Nature and Society
005Control parameter k
139
14
141
142
143
144
145
146
Box-
coun
ting
dim
ensio
n
04504035030250201501
Figure 3 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from 006 to 045
119909119899+1 = 119860 minus (119861 + 1) 119909119899 + 1199092119899119910119899119910119899+1 = 119861119909119899 minus 1199092119899119910119899
(4)
where 119909 119910 denote concentration of reactant in the process ofchemical reaction 119860 119861 gt 0 denote initial concentration ofreactant
Brusselator equations were first discovered by A Turingin 1952 [36] and then I Pigogine and Leefver did somesystematic studies on itThey pointed out that the Brusselatorequations were the most elementary and essential mathe-matic model which describes the oscillation of biochemistryThey proved that when 119861 gt 1 + 1198602 the equations have stableand unique limit cycle When 119861 le 1 + 1198602 there is no limitcycle [37] It can be known that the initial concentration ofreactant has an important influence on the system In fractaltheory Julia set is a set of initial points of the system thatsatisfy certain conditions With the same thought we definethe Julia set of Brusselator model Let 119865(119909 119910) = (119860 minus (119861 +1)119909119899 + 1199092119899119910119899 119861119909119899 minus 1199092119899119910119899)Definition 3 Set 119870 = (119909 119910) isin 1198772 | 119865119899(119909 119910)119899ge1 is bound-ed is called the filled Julia set of Brusselator model Theboundary of the filled Julia set is defined to be the Julia setof Brusselator model that is 119869(119865) = 120597119870
The research results demonstrate that coupled with apiecewise constant value control function we can controlthe size of the limit cycle of (4) [36] From the definition ofJulia set we found that there is a close relation between theboundedness of iterative orbits and the structure of the Juliaset So if we want to control the Julia set of Brusselator modelhow to control the boundedness of the system iterative iscritical We consider especially the stability of the fixed pointof Brusselator model and by designing controllers unstablefixed points are turned into stable fixed points to control theboundedness of iterative orbits effectivelyThen the control ofthe Julia set can be realized
As was mentioned above considering the stability ofthe fixed points of system (5) we try to find controllers tomake the fixed point of the system stable Let the controlledBrusselator model be
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119906119899119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + V119899 (5)
where 119906119899 and V119899 denote the designed controllers
31 Feedback Control Method Take 119906119899 = 119896(119909119899 minus 119909lowast) V119899 =119896(119910119899 minus 119910lowast) with the control parameter 119896 and the controlledsystem is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119896 (119909119899 minus 119909lowast) 119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (119910119899 minus 119910lowast) (6)
Theorem 4 Let 1198601 = 1 minus 119887 minus 120574 + 2120574119909lowast119910lowast + 119896 1198611 = 1 minus120574119909lowast2 + 119896 1198621 = 120574119909lowast2 and 1198631 = 119887 minus 2120574119909lowast119910lowast where (119909lowast 119910lowast)denotes the fixed point of the controlled system (6) If |(1198601 +1198611) plusmnradic(1198601 minus 1198611)2 + 411986211198631| lt 2 the fixed point of the systemis attractive
Proof Write 1(119909 119910) = (119886+(1minus119887minus120574)119909+1205741199092119910+119896(119909minus119909lowast) 119910+119887119909 minus 1205741199092119910 + 119896(119910 minus 119910lowast)) The Jacobi matrix of the controlledsystem (6) is
1198631 = [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896 1205741199092119899119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 119896] (7)
and its eigenmatrix is
1198631 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 119896 minus 120582] (8)
Discrete Dynamics in Nature and Society 5
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(f)
Figure 4 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus022 (b) 119896 = minus038 (c)119896 = minus054 (d) 119896 = minus070 (e) 119896 = minus086 (f) 119896 = minus1
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899 + 2119896) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896) (1 minus 1205741199092119899 + 119896)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(9)
When the modulus of the eigenvalues 12058211 12058212 of theJacobi matrix at the fixed point is less than 1 the fixed pointis attractive
100381610038161003816100381612058211121003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198601 + 1198611) plusmn radic(1198601 minus 1198611)2 + 4119862111986312
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (10)
6 Discrete Dynamics in Nature and Society
13
135
14
145
Control parameter k
Box-
coun
ting
dim
ensio
n
minus1 minus09 minus08 minus07 minus06 minus05 minus04 minus03 minus02
Figure 5 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus1 to minus022
that is
10038161003816100381610038161003816100381610038161003816(1198601 + 1198611) plusmn radic(1198601 minus 1198611)2 + 41198621119863110038161003816100381610038161003816100381610038161003816 lt 2 (11)
By Theorem 4 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example take 119886 = 007 119887 = 002 and 120574 = 001 insystem (6) and the initial Julia set is shown in Figure 1 thenwe get 119909lowast = 7 119910lowast = 27 By Theorem 4 the range of thevalue of 119896 is minus19895 lt 119896 lt 04695
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 006 to 045 in Figure 2 and we find that thetrend of the change of the Julia sets is obvious In Figure 3the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from minus1 to minus022 in Figure 4to illustrate the change of the Julia sets in feedback controlmethod In Figure 5 the box-counting dimensions of thecontrolled Julia sets are computed in this control method
In feedback control method the contraction of the leftand the lower parts is faster than the right and the upperparts when the interval of control parameter 119896 is between006 and 045 In addition the complexity of the boundaryof the Julia set significantly decreases and the lower partrapidly contracts When the interval of control parameter 119896is between minus1 and minus022 the complexity of the boundary ofthe Julia set significantly decreases and the Julia set tends tobe centrally symmetric In general with the absolute value of119896 increasing the Julia set contracts to the center gradually
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing the
change generally shows a monotonic decreasing trend Par-ticularly when the control parameter 119896 is in the intervalfrom 016 to 037 and minus09 to minus04 the monotonic changeof box-counting dimensions is obvious which indicates theeffectiveness of this control method on the Julia set ofBrusselator model
32 Optimal Control Method Take 119906119899 = 119896(119891(119909119899 119910119899) minus119909119899) V119899 = 119896(119892(119909119899 119910119899) minus 119910119899) with 119891(119909119899 119910119899) = 119886 + (1 minus 119887 minus120574)119909119899 + 1205741199092119899119910119899 119892(119909119899 119910119899) = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 where 119896 is thecontrol parameter Then the controlled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899+ 119896 (119891 (119909119899 119910119899) minus 119909119899)
119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (119892 (119909119899 119910119899) minus 119910119899) (12)
Theorem 5 Let1198602 = (1minus119887minus120574+2120574119909lowast119910lowast)(119896+1)minus119896 1198612 = (1minus120574119909lowast2)(119896+1)minus1198961198622 = 120574119909lowast2(119896+1) and1198632 = (119887minus2120574119909lowast119910lowast)(119896+1)where (119909lowast 119910lowast) denotes the fixed point of the controlled system(12) If |(1198602+1198612)plusmnradic(1198602 minus 1198612)2 + 411986221198632| lt 2 the fixed pointof the system is attractive
Proof Write 2(119909 119910) = (119886 + (1 minus 119887minus 120574)119909+ 1205741199092119910+119896(119891(119909 119910) minus119909) 119910 + 119887119909 minus 1205741199092119910 + 119896(119892(119909 119910) minus 119910)) The Jacobi matrix of thecontrolled system (12) is
1198632= [(1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896 1205741199092119899 (119896 + 1)
(119887 minus 2120574119909119899119910119899) (119896 + 1) (1 minus 1205741199092119899) (119896 + 1) minus 119896](13)
and its eigenmatrix is
1198632 minus 120582119864 = [(1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896 minus 120582 1205741199092119899 (119896 + 1)(119887 minus 2120574119909119899119910119899) (119896 + 1) (1 minus 1205741199092119899) (119896 + 1) minus 119896 minus 120582] (14)
Discrete Dynamics in Nature and Society 7
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 6 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 064 (b) 119896 = 104 (c) 119896 = 144(d) 119896 = 184 (e) 119896 = 224 (f) 119896 = 26
8 Discrete Dynamics in Nature and Society
13606 08 1 12 14 16 18 2 22 24 26
137
138
139
14
141
142
143
144
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 7 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from 064 to 26
And the characteristic equation is
1205822 minus ((2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899) (119896 + 1) minus 2119896) 120582+ ((1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896)sdot ((1 minus 1205741199092119899) (119896 + 1) minus 119896) minus 1205741199092119899 (119887 minus 2120574119909119899119910119899)sdot (119896 + 1)2 = 0
(15)
When themodulus of the eigenvalue 12058221 12058222 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058221221003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 4119862211986322
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (16)
that is10038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 41198622119863210038161003816100381610038161003816100381610038161003816 lt 2 (17)
By Theorem 5 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus1 lt 119896 lt190659
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 064 to 26 in Figure 6 to illustrate thechange of the Julia sets in optimal controlmethod In Figure 7the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In optimal control method the effective controlled inter-val of 119896 is from minus1 to 1906 But when the interval of 119896 is from064 to 26 this method has the best controlled effectivenessIn this interval with the absolute values of 119896 increasing the
Julia sets gradually contract to the center with nearly the samespeed and no significant change in the overall shape of theJulia sets occurs
From the perspective of the change of box-countingdimensions box-counting dimensions of the Julia sets gen-erally show a monotonic decreasing trend with the absolutevalues of 119896 increasing For the reason that the complexityof the Julia set can be depicted by box-counting dimensionwhen the complexity of the Julia set decreases with theabsolute value of 119896 increasing it indicates that this controlmethod has great control effectiveness
33 Gradient Control Method Take 119906119899 = 119896(1199092119899 minus 119909lowast2) V119899 =119896(1199102119899 minus 119910lowast2) with the control parameter 119896 Therefore thecontrolled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119896 (1199092119899 minus 119909lowast2) 119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (1199102119899 minus 119910lowast2) (18)
Theorem 6 Let 1198603 = 1 minus 119887 minus 120574 + 2120574119909lowast119910lowast + 2119896119909lowast 1198613 =1 minus 120574119909lowast2 + 2119896119910lowast 1198623 = 120574119909lowast2 and 1198633 = 119887 minus 2120574119909lowast119910lowast where(119909lowast 119910lowast) denotes the fixed point of the controlled system (18) If|(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 411986231198633| lt 2 the fixed point of thesystem is attractive
Proof Write 3(119909 119910) = (119886 + (1 minus 119887 minus 120574)119909 + 1205741199092119910 + 119896(1199092 minus119909lowast2) 119910 + 119887119909 minus 1205741199092119910 + 119896(1199102 minus 119910lowast2)) The Jacobi matrix of thecontrolled system (18) is
1198633 = [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 1205741199092119899119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899] (19)
Discrete Dynamics in Nature and Society 9
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 8 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus00005 (b) 119896 = minus0004 (c)119896 = minus0008 (d) 119896 = minus0012 (e) 119896 = minus0016 (f) 119896 = minus0020
10 Discrete Dynamics in Nature and Society
136
137
138
139
14
141
142
143
144
145
minus002
minus0018
minus0016
minus0014
minus0012
minus001
minus0008
minus0006
minus0004
minus0002 0
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 9 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus002 to minus00005
and its eigenmatrix is
1198633 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899 minus 120582] (20)
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 1205741199092119899 + 2119896119910119899) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899) (1 minus 1205741199092119899 + 2119896119910119899)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(21)
When themodulus of the eigenvalue 12058231 12058232 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058231321003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 4119862311986332
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (22)
that is10038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 41198623119863310038161003816100381610038161003816100381610038161003816 lt 2 (23)
By Theorem 6 we know that the Brusselator model canbe controlled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus26420 lt 119896 lt08561
Six simulation diagrams are chosen corresponding to thevalues of 119896 from minus00005 to minus002 in Figure 8 to illustratethe change of the Julia sets in gradient control method In
Figure 9 the box-counting dimensions of the controlled Juliasets are computed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from 00005 to 002 inFigure 10 to illustrate the change of the Julia sets in gradientcontrol method In Figure 11 the box-counting dimensionsof the controlled Julia sets are computed in this controlmethod
In gradient control method we consider two groups ofinterval of the parameter 119896 It is worth noticing that thecontraction of the left and the lower parts of the Julia set arefaster than the right and the upper parts when the interval ofcontrol parameter 119896 is from minus00005 to minus002 while the rightand the upper parts are faster than the left and the lower partswhen the interval of control parameter 119896 is from 00005 to002 In general with the absolute values of 119896 increasing theJulia sets contract to the center gradually and the complexityof the boundary of the Julia sets decreases
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing thebox-counting dimensions of the Julia sets generally show amonotonic decreasing trend Particularly when the controlparameter 119896 is in the range from minus0016 to minus00075 and from00075 to 0016 the change of box-counting dimensions isobviouslymonotonic which indicates the effectiveness of thismethod on the Julia set of the Brusselator model
4 Conclusion
Fractal theory is a hot topic in the research of nonlinearscience Describing the change of chemical elements in thechemical reaction process Brusselator model an importantclass of reaction diffusion equations is significant in thestudy of chaotic and fractal behaviors of nonlinear differentialequations In technological applications it is often requiredthat the behavior and performance of the system can becontrolled effectively
Discrete Dynamics in Nature and Society 11
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 10 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 00005 (b) 119896 = 00045 (c)119896 = 00085 (d) 119896 = 00125 (e) 119896 = 00165 (f) 119896 = 00200
12 Discrete Dynamics in Nature and Society
135
136
137
138
139
14
141
142
143
144
1450
0002
0004
0006
0008
001
0012
0014
0016
0018
002
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 11 The change of box-counting dimensions of the Julia setsof the controlled system when 119896 is from 00005 to 002
In this paper feedback control method optimal controlmethod and gradient control method are taken to controlthe Julia set of Brusselator model respectively and thebox-counting dimensions of the Julia set are calculated Ineach control method when the absolute value of controlparameter 119896 increases discretely the box-counting dimensionof Julia set decreases and the Julia set contracts to the centergradually The decrease of box-counting dimension is nearlymonotonic which indicates that the complexity of the Juliaset of the system is declined graduallyThus when the controlparameter 119896 is selected the Julia set of Brusselator modelcould be controlled Most importantly the three controlmethods have consistency of conclusion and effectiveness
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61403231 and 11501328) and ChinaPostdoctoral Science Foundation (no 2016M592188)
References
[1] X Wang Fractal Mechanism of Generalized M-J Sets DalianUniversity of Technology Press Dalian China 2002
[2] G Chen and X Dong ldquoFrom chaos to order methodologiesperspectives and applicationsrdquo Methodologies Perspectives ampApplications World Scientific vol 31 no 2 pp 113ndash122 1998
[3] B B Mandelbrot Fractal Form Chance and Dimension W HFreeman San Francisco Calif USA 1977
[4] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Co San Francisco Calif USA 1982
[5] L Pitulice D Craciun E Vilaseca et al ldquoFractal dimension ofthe trajectory of a single particle diffusing in crowded mediardquoRomanian Journal of Physics vol 61 no 7-8 pp 1276ndash1286 2016
[6] L Pitulice E Vilaseca I Pastor et al ldquoMonte Carlo simulationsof enzymatic reactions in crowdedmedia Effect of the enzyme-obstacle relative sizerdquo Mathematical Biosciences vol 251 no 1pp 72ndash82 2014
[7] M Lewis and D C Rees ldquoFractal surfaces of proteinsrdquo Sciencevol 230 no 4730 pp 1163ndash1165 1985
[8] A Isvoran L Pitulice C T Craescu and A Chiriac ldquoFractalaspects of calcium binding protein structuresrdquo Chaos Solitonsamp Fractals vol 35 no 5 pp 960ndash966 2008
[9] P J Bentley ldquoFractal proteinsrdquo Genetic Programming amp Evolv-able Machines vol 5 no 1 pp 71ndash101 2004
[10] M Zamir ldquoFractal dimensions and multifractility in vascularbranchingrdquo Journal of Theoretical Biology vol 212 no 2 pp183ndash190 2001
[11] R L Devaney and J P Eckmann ldquoAn introduction to chaoticdynamical systemsrdquo Mathematical Gazette vol 74 no 468 p72 1990
[12] Z Huang ldquoFractal characteristics of turbulencerdquo Advances inMechanics vol 30 no 4 pp 581ndash596 2000
[13] M S Movahed and E Hermanis ldquoFractal analysis of riverflow fluctuationsrdquo Physica A Statistical Mechanics and ItsApplications vol 387 no 4 pp 915ndash932 2008
[14] N Yonaiguchi Y Ida M Hayakawa and S Masuda ldquoFractalanalysis for VHF electromagnetic noises and the identificationof preseismic signature of an earthquakerdquo Journal of Atmo-spheric and Solar-Terrestrial Physics vol 69 no 15 pp 1825ndash1832 2007
[15] N Ai R Chen and H Li ldquoTo the fractal geomorphologyrdquoGeography and Territorial Research vol 15 no 1 pp 92ndash961999
[16] M Li L Zhu and H Long ldquoAnalysis of several problems in theapplication of fractal in geomorphologyrdquo Earthquake Researchvol 25 no 2 pp 155ndash162 2002
[17] J Shu D Tan and J Wu ldquoFractal structure of Chinese stockmarketrdquo Journal of Southwest Jiao Tong University vol 38 no 2pp 212ndash215 2003
[18] Y Lin ldquoFractal and its application in securitymarketrdquo EconomicBusiness vol 8 pp 44ndash46 2001
[19] Z Peng C Li and H Li ldquoResearch on enterprises fractalmanagement model in knowledge economy era rdquo BusinessResearch vol 255 no 10 pp 33ndash35 2002
[20] W Zeng The Dimension Calculation of Several Special Self-Similar Sets Central China Normal University Wuhan China2007
[21] K J FalconerTheGeometry of Fractal Sets vol 85 ofCambridgeTracts inMathematics CambridgeUniversity Press CambridgeUK 1986
[22] Preiss andDavid ldquoNondifferentiable functionsrdquo Transactions ofthe American Mathematical Society vol 17 no 3 pp 301ndash3251916
[23] B B Mandelbrot Les Objets Fractals Flammarion 1984[24] X Gaoan Fractal Dimension Seismological Press Beijing
China 1989[25] T Erneux and E L Reiss ldquoBrussellator isolasrdquo SIAM Journal on
Applied Mathematics vol 43 no 6 pp 1240ndash1246 1983[26] G Nicolis ldquoPatterns of spatio-temporal organization in chem-
ical and biochemical kineticsrdquo American Mathematical Societyvol 8 pp 33ndash58 1974
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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4 Discrete Dynamics in Nature and Society
005Control parameter k
139
14
141
142
143
144
145
146
Box-
coun
ting
dim
ensio
n
04504035030250201501
Figure 3 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from 006 to 045
119909119899+1 = 119860 minus (119861 + 1) 119909119899 + 1199092119899119910119899119910119899+1 = 119861119909119899 minus 1199092119899119910119899
(4)
where 119909 119910 denote concentration of reactant in the process ofchemical reaction 119860 119861 gt 0 denote initial concentration ofreactant
Brusselator equations were first discovered by A Turingin 1952 [36] and then I Pigogine and Leefver did somesystematic studies on itThey pointed out that the Brusselatorequations were the most elementary and essential mathe-matic model which describes the oscillation of biochemistryThey proved that when 119861 gt 1 + 1198602 the equations have stableand unique limit cycle When 119861 le 1 + 1198602 there is no limitcycle [37] It can be known that the initial concentration ofreactant has an important influence on the system In fractaltheory Julia set is a set of initial points of the system thatsatisfy certain conditions With the same thought we definethe Julia set of Brusselator model Let 119865(119909 119910) = (119860 minus (119861 +1)119909119899 + 1199092119899119910119899 119861119909119899 minus 1199092119899119910119899)Definition 3 Set 119870 = (119909 119910) isin 1198772 | 119865119899(119909 119910)119899ge1 is bound-ed is called the filled Julia set of Brusselator model Theboundary of the filled Julia set is defined to be the Julia setof Brusselator model that is 119869(119865) = 120597119870
The research results demonstrate that coupled with apiecewise constant value control function we can controlthe size of the limit cycle of (4) [36] From the definition ofJulia set we found that there is a close relation between theboundedness of iterative orbits and the structure of the Juliaset So if we want to control the Julia set of Brusselator modelhow to control the boundedness of the system iterative iscritical We consider especially the stability of the fixed pointof Brusselator model and by designing controllers unstablefixed points are turned into stable fixed points to control theboundedness of iterative orbits effectivelyThen the control ofthe Julia set can be realized
As was mentioned above considering the stability ofthe fixed points of system (5) we try to find controllers tomake the fixed point of the system stable Let the controlledBrusselator model be
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119906119899119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + V119899 (5)
where 119906119899 and V119899 denote the designed controllers
31 Feedback Control Method Take 119906119899 = 119896(119909119899 minus 119909lowast) V119899 =119896(119910119899 minus 119910lowast) with the control parameter 119896 and the controlledsystem is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119896 (119909119899 minus 119909lowast) 119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (119910119899 minus 119910lowast) (6)
Theorem 4 Let 1198601 = 1 minus 119887 minus 120574 + 2120574119909lowast119910lowast + 119896 1198611 = 1 minus120574119909lowast2 + 119896 1198621 = 120574119909lowast2 and 1198631 = 119887 minus 2120574119909lowast119910lowast where (119909lowast 119910lowast)denotes the fixed point of the controlled system (6) If |(1198601 +1198611) plusmnradic(1198601 minus 1198611)2 + 411986211198631| lt 2 the fixed point of the systemis attractive
Proof Write 1(119909 119910) = (119886+(1minus119887minus120574)119909+1205741199092119910+119896(119909minus119909lowast) 119910+119887119909 minus 1205741199092119910 + 119896(119910 minus 119910lowast)) The Jacobi matrix of the controlledsystem (6) is
1198631 = [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896 1205741199092119899119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 119896] (7)
and its eigenmatrix is
1198631 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 119896 minus 120582] (8)
Discrete Dynamics in Nature and Society 5
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(f)
Figure 4 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus022 (b) 119896 = minus038 (c)119896 = minus054 (d) 119896 = minus070 (e) 119896 = minus086 (f) 119896 = minus1
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899 + 2119896) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896) (1 minus 1205741199092119899 + 119896)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(9)
When the modulus of the eigenvalues 12058211 12058212 of theJacobi matrix at the fixed point is less than 1 the fixed pointis attractive
100381610038161003816100381612058211121003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198601 + 1198611) plusmn radic(1198601 minus 1198611)2 + 4119862111986312
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (10)
6 Discrete Dynamics in Nature and Society
13
135
14
145
Control parameter k
Box-
coun
ting
dim
ensio
n
minus1 minus09 minus08 minus07 minus06 minus05 minus04 minus03 minus02
Figure 5 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus1 to minus022
that is
10038161003816100381610038161003816100381610038161003816(1198601 + 1198611) plusmn radic(1198601 minus 1198611)2 + 41198621119863110038161003816100381610038161003816100381610038161003816 lt 2 (11)
By Theorem 4 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example take 119886 = 007 119887 = 002 and 120574 = 001 insystem (6) and the initial Julia set is shown in Figure 1 thenwe get 119909lowast = 7 119910lowast = 27 By Theorem 4 the range of thevalue of 119896 is minus19895 lt 119896 lt 04695
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 006 to 045 in Figure 2 and we find that thetrend of the change of the Julia sets is obvious In Figure 3the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from minus1 to minus022 in Figure 4to illustrate the change of the Julia sets in feedback controlmethod In Figure 5 the box-counting dimensions of thecontrolled Julia sets are computed in this control method
In feedback control method the contraction of the leftand the lower parts is faster than the right and the upperparts when the interval of control parameter 119896 is between006 and 045 In addition the complexity of the boundaryof the Julia set significantly decreases and the lower partrapidly contracts When the interval of control parameter 119896is between minus1 and minus022 the complexity of the boundary ofthe Julia set significantly decreases and the Julia set tends tobe centrally symmetric In general with the absolute value of119896 increasing the Julia set contracts to the center gradually
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing the
change generally shows a monotonic decreasing trend Par-ticularly when the control parameter 119896 is in the intervalfrom 016 to 037 and minus09 to minus04 the monotonic changeof box-counting dimensions is obvious which indicates theeffectiveness of this control method on the Julia set ofBrusselator model
32 Optimal Control Method Take 119906119899 = 119896(119891(119909119899 119910119899) minus119909119899) V119899 = 119896(119892(119909119899 119910119899) minus 119910119899) with 119891(119909119899 119910119899) = 119886 + (1 minus 119887 minus120574)119909119899 + 1205741199092119899119910119899 119892(119909119899 119910119899) = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 where 119896 is thecontrol parameter Then the controlled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899+ 119896 (119891 (119909119899 119910119899) minus 119909119899)
119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (119892 (119909119899 119910119899) minus 119910119899) (12)
Theorem 5 Let1198602 = (1minus119887minus120574+2120574119909lowast119910lowast)(119896+1)minus119896 1198612 = (1minus120574119909lowast2)(119896+1)minus1198961198622 = 120574119909lowast2(119896+1) and1198632 = (119887minus2120574119909lowast119910lowast)(119896+1)where (119909lowast 119910lowast) denotes the fixed point of the controlled system(12) If |(1198602+1198612)plusmnradic(1198602 minus 1198612)2 + 411986221198632| lt 2 the fixed pointof the system is attractive
Proof Write 2(119909 119910) = (119886 + (1 minus 119887minus 120574)119909+ 1205741199092119910+119896(119891(119909 119910) minus119909) 119910 + 119887119909 minus 1205741199092119910 + 119896(119892(119909 119910) minus 119910)) The Jacobi matrix of thecontrolled system (12) is
1198632= [(1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896 1205741199092119899 (119896 + 1)
(119887 minus 2120574119909119899119910119899) (119896 + 1) (1 minus 1205741199092119899) (119896 + 1) minus 119896](13)
and its eigenmatrix is
1198632 minus 120582119864 = [(1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896 minus 120582 1205741199092119899 (119896 + 1)(119887 minus 2120574119909119899119910119899) (119896 + 1) (1 minus 1205741199092119899) (119896 + 1) minus 119896 minus 120582] (14)
Discrete Dynamics in Nature and Society 7
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 6 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 064 (b) 119896 = 104 (c) 119896 = 144(d) 119896 = 184 (e) 119896 = 224 (f) 119896 = 26
8 Discrete Dynamics in Nature and Society
13606 08 1 12 14 16 18 2 22 24 26
137
138
139
14
141
142
143
144
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 7 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from 064 to 26
And the characteristic equation is
1205822 minus ((2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899) (119896 + 1) minus 2119896) 120582+ ((1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896)sdot ((1 minus 1205741199092119899) (119896 + 1) minus 119896) minus 1205741199092119899 (119887 minus 2120574119909119899119910119899)sdot (119896 + 1)2 = 0
(15)
When themodulus of the eigenvalue 12058221 12058222 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058221221003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 4119862211986322
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (16)
that is10038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 41198622119863210038161003816100381610038161003816100381610038161003816 lt 2 (17)
By Theorem 5 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus1 lt 119896 lt190659
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 064 to 26 in Figure 6 to illustrate thechange of the Julia sets in optimal controlmethod In Figure 7the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In optimal control method the effective controlled inter-val of 119896 is from minus1 to 1906 But when the interval of 119896 is from064 to 26 this method has the best controlled effectivenessIn this interval with the absolute values of 119896 increasing the
Julia sets gradually contract to the center with nearly the samespeed and no significant change in the overall shape of theJulia sets occurs
From the perspective of the change of box-countingdimensions box-counting dimensions of the Julia sets gen-erally show a monotonic decreasing trend with the absolutevalues of 119896 increasing For the reason that the complexityof the Julia set can be depicted by box-counting dimensionwhen the complexity of the Julia set decreases with theabsolute value of 119896 increasing it indicates that this controlmethod has great control effectiveness
33 Gradient Control Method Take 119906119899 = 119896(1199092119899 minus 119909lowast2) V119899 =119896(1199102119899 minus 119910lowast2) with the control parameter 119896 Therefore thecontrolled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119896 (1199092119899 minus 119909lowast2) 119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (1199102119899 minus 119910lowast2) (18)
Theorem 6 Let 1198603 = 1 minus 119887 minus 120574 + 2120574119909lowast119910lowast + 2119896119909lowast 1198613 =1 minus 120574119909lowast2 + 2119896119910lowast 1198623 = 120574119909lowast2 and 1198633 = 119887 minus 2120574119909lowast119910lowast where(119909lowast 119910lowast) denotes the fixed point of the controlled system (18) If|(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 411986231198633| lt 2 the fixed point of thesystem is attractive
Proof Write 3(119909 119910) = (119886 + (1 minus 119887 minus 120574)119909 + 1205741199092119910 + 119896(1199092 minus119909lowast2) 119910 + 119887119909 minus 1205741199092119910 + 119896(1199102 minus 119910lowast2)) The Jacobi matrix of thecontrolled system (18) is
1198633 = [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 1205741199092119899119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899] (19)
Discrete Dynamics in Nature and Society 9
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 8 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus00005 (b) 119896 = minus0004 (c)119896 = minus0008 (d) 119896 = minus0012 (e) 119896 = minus0016 (f) 119896 = minus0020
10 Discrete Dynamics in Nature and Society
136
137
138
139
14
141
142
143
144
145
minus002
minus0018
minus0016
minus0014
minus0012
minus001
minus0008
minus0006
minus0004
minus0002 0
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 9 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus002 to minus00005
and its eigenmatrix is
1198633 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899 minus 120582] (20)
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 1205741199092119899 + 2119896119910119899) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899) (1 minus 1205741199092119899 + 2119896119910119899)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(21)
When themodulus of the eigenvalue 12058231 12058232 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058231321003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 4119862311986332
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (22)
that is10038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 41198623119863310038161003816100381610038161003816100381610038161003816 lt 2 (23)
By Theorem 6 we know that the Brusselator model canbe controlled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus26420 lt 119896 lt08561
Six simulation diagrams are chosen corresponding to thevalues of 119896 from minus00005 to minus002 in Figure 8 to illustratethe change of the Julia sets in gradient control method In
Figure 9 the box-counting dimensions of the controlled Juliasets are computed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from 00005 to 002 inFigure 10 to illustrate the change of the Julia sets in gradientcontrol method In Figure 11 the box-counting dimensionsof the controlled Julia sets are computed in this controlmethod
In gradient control method we consider two groups ofinterval of the parameter 119896 It is worth noticing that thecontraction of the left and the lower parts of the Julia set arefaster than the right and the upper parts when the interval ofcontrol parameter 119896 is from minus00005 to minus002 while the rightand the upper parts are faster than the left and the lower partswhen the interval of control parameter 119896 is from 00005 to002 In general with the absolute values of 119896 increasing theJulia sets contract to the center gradually and the complexityof the boundary of the Julia sets decreases
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing thebox-counting dimensions of the Julia sets generally show amonotonic decreasing trend Particularly when the controlparameter 119896 is in the range from minus0016 to minus00075 and from00075 to 0016 the change of box-counting dimensions isobviouslymonotonic which indicates the effectiveness of thismethod on the Julia set of the Brusselator model
4 Conclusion
Fractal theory is a hot topic in the research of nonlinearscience Describing the change of chemical elements in thechemical reaction process Brusselator model an importantclass of reaction diffusion equations is significant in thestudy of chaotic and fractal behaviors of nonlinear differentialequations In technological applications it is often requiredthat the behavior and performance of the system can becontrolled effectively
Discrete Dynamics in Nature and Society 11
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 10 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 00005 (b) 119896 = 00045 (c)119896 = 00085 (d) 119896 = 00125 (e) 119896 = 00165 (f) 119896 = 00200
12 Discrete Dynamics in Nature and Society
135
136
137
138
139
14
141
142
143
144
1450
0002
0004
0006
0008
001
0012
0014
0016
0018
002
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 11 The change of box-counting dimensions of the Julia setsof the controlled system when 119896 is from 00005 to 002
In this paper feedback control method optimal controlmethod and gradient control method are taken to controlthe Julia set of Brusselator model respectively and thebox-counting dimensions of the Julia set are calculated Ineach control method when the absolute value of controlparameter 119896 increases discretely the box-counting dimensionof Julia set decreases and the Julia set contracts to the centergradually The decrease of box-counting dimension is nearlymonotonic which indicates that the complexity of the Juliaset of the system is declined graduallyThus when the controlparameter 119896 is selected the Julia set of Brusselator modelcould be controlled Most importantly the three controlmethods have consistency of conclusion and effectiveness
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61403231 and 11501328) and ChinaPostdoctoral Science Foundation (no 2016M592188)
References
[1] X Wang Fractal Mechanism of Generalized M-J Sets DalianUniversity of Technology Press Dalian China 2002
[2] G Chen and X Dong ldquoFrom chaos to order methodologiesperspectives and applicationsrdquo Methodologies Perspectives ampApplications World Scientific vol 31 no 2 pp 113ndash122 1998
[3] B B Mandelbrot Fractal Form Chance and Dimension W HFreeman San Francisco Calif USA 1977
[4] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Co San Francisco Calif USA 1982
[5] L Pitulice D Craciun E Vilaseca et al ldquoFractal dimension ofthe trajectory of a single particle diffusing in crowded mediardquoRomanian Journal of Physics vol 61 no 7-8 pp 1276ndash1286 2016
[6] L Pitulice E Vilaseca I Pastor et al ldquoMonte Carlo simulationsof enzymatic reactions in crowdedmedia Effect of the enzyme-obstacle relative sizerdquo Mathematical Biosciences vol 251 no 1pp 72ndash82 2014
[7] M Lewis and D C Rees ldquoFractal surfaces of proteinsrdquo Sciencevol 230 no 4730 pp 1163ndash1165 1985
[8] A Isvoran L Pitulice C T Craescu and A Chiriac ldquoFractalaspects of calcium binding protein structuresrdquo Chaos Solitonsamp Fractals vol 35 no 5 pp 960ndash966 2008
[9] P J Bentley ldquoFractal proteinsrdquo Genetic Programming amp Evolv-able Machines vol 5 no 1 pp 71ndash101 2004
[10] M Zamir ldquoFractal dimensions and multifractility in vascularbranchingrdquo Journal of Theoretical Biology vol 212 no 2 pp183ndash190 2001
[11] R L Devaney and J P Eckmann ldquoAn introduction to chaoticdynamical systemsrdquo Mathematical Gazette vol 74 no 468 p72 1990
[12] Z Huang ldquoFractal characteristics of turbulencerdquo Advances inMechanics vol 30 no 4 pp 581ndash596 2000
[13] M S Movahed and E Hermanis ldquoFractal analysis of riverflow fluctuationsrdquo Physica A Statistical Mechanics and ItsApplications vol 387 no 4 pp 915ndash932 2008
[14] N Yonaiguchi Y Ida M Hayakawa and S Masuda ldquoFractalanalysis for VHF electromagnetic noises and the identificationof preseismic signature of an earthquakerdquo Journal of Atmo-spheric and Solar-Terrestrial Physics vol 69 no 15 pp 1825ndash1832 2007
[15] N Ai R Chen and H Li ldquoTo the fractal geomorphologyrdquoGeography and Territorial Research vol 15 no 1 pp 92ndash961999
[16] M Li L Zhu and H Long ldquoAnalysis of several problems in theapplication of fractal in geomorphologyrdquo Earthquake Researchvol 25 no 2 pp 155ndash162 2002
[17] J Shu D Tan and J Wu ldquoFractal structure of Chinese stockmarketrdquo Journal of Southwest Jiao Tong University vol 38 no 2pp 212ndash215 2003
[18] Y Lin ldquoFractal and its application in securitymarketrdquo EconomicBusiness vol 8 pp 44ndash46 2001
[19] Z Peng C Li and H Li ldquoResearch on enterprises fractalmanagement model in knowledge economy era rdquo BusinessResearch vol 255 no 10 pp 33ndash35 2002
[20] W Zeng The Dimension Calculation of Several Special Self-Similar Sets Central China Normal University Wuhan China2007
[21] K J FalconerTheGeometry of Fractal Sets vol 85 ofCambridgeTracts inMathematics CambridgeUniversity Press CambridgeUK 1986
[22] Preiss andDavid ldquoNondifferentiable functionsrdquo Transactions ofthe American Mathematical Society vol 17 no 3 pp 301ndash3251916
[23] B B Mandelbrot Les Objets Fractals Flammarion 1984[24] X Gaoan Fractal Dimension Seismological Press Beijing
China 1989[25] T Erneux and E L Reiss ldquoBrussellator isolasrdquo SIAM Journal on
Applied Mathematics vol 43 no 6 pp 1240ndash1246 1983[26] G Nicolis ldquoPatterns of spatio-temporal organization in chem-
ical and biochemical kineticsrdquo American Mathematical Societyvol 8 pp 33ndash58 1974
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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OptimizationJournal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 5
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80
806040200minus20minus40minus80
(f)
Figure 4 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus022 (b) 119896 = minus038 (c)119896 = minus054 (d) 119896 = minus070 (e) 119896 = minus086 (f) 119896 = minus1
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899 + 2119896) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 119896) (1 minus 1205741199092119899 + 119896)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(9)
When the modulus of the eigenvalues 12058211 12058212 of theJacobi matrix at the fixed point is less than 1 the fixed pointis attractive
100381610038161003816100381612058211121003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198601 + 1198611) plusmn radic(1198601 minus 1198611)2 + 4119862111986312
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (10)
6 Discrete Dynamics in Nature and Society
13
135
14
145
Control parameter k
Box-
coun
ting
dim
ensio
n
minus1 minus09 minus08 minus07 minus06 minus05 minus04 minus03 minus02
Figure 5 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus1 to minus022
that is
10038161003816100381610038161003816100381610038161003816(1198601 + 1198611) plusmn radic(1198601 minus 1198611)2 + 41198621119863110038161003816100381610038161003816100381610038161003816 lt 2 (11)
By Theorem 4 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example take 119886 = 007 119887 = 002 and 120574 = 001 insystem (6) and the initial Julia set is shown in Figure 1 thenwe get 119909lowast = 7 119910lowast = 27 By Theorem 4 the range of thevalue of 119896 is minus19895 lt 119896 lt 04695
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 006 to 045 in Figure 2 and we find that thetrend of the change of the Julia sets is obvious In Figure 3the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from minus1 to minus022 in Figure 4to illustrate the change of the Julia sets in feedback controlmethod In Figure 5 the box-counting dimensions of thecontrolled Julia sets are computed in this control method
In feedback control method the contraction of the leftand the lower parts is faster than the right and the upperparts when the interval of control parameter 119896 is between006 and 045 In addition the complexity of the boundaryof the Julia set significantly decreases and the lower partrapidly contracts When the interval of control parameter 119896is between minus1 and minus022 the complexity of the boundary ofthe Julia set significantly decreases and the Julia set tends tobe centrally symmetric In general with the absolute value of119896 increasing the Julia set contracts to the center gradually
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing the
change generally shows a monotonic decreasing trend Par-ticularly when the control parameter 119896 is in the intervalfrom 016 to 037 and minus09 to minus04 the monotonic changeof box-counting dimensions is obvious which indicates theeffectiveness of this control method on the Julia set ofBrusselator model
32 Optimal Control Method Take 119906119899 = 119896(119891(119909119899 119910119899) minus119909119899) V119899 = 119896(119892(119909119899 119910119899) minus 119910119899) with 119891(119909119899 119910119899) = 119886 + (1 minus 119887 minus120574)119909119899 + 1205741199092119899119910119899 119892(119909119899 119910119899) = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 where 119896 is thecontrol parameter Then the controlled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899+ 119896 (119891 (119909119899 119910119899) minus 119909119899)
119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (119892 (119909119899 119910119899) minus 119910119899) (12)
Theorem 5 Let1198602 = (1minus119887minus120574+2120574119909lowast119910lowast)(119896+1)minus119896 1198612 = (1minus120574119909lowast2)(119896+1)minus1198961198622 = 120574119909lowast2(119896+1) and1198632 = (119887minus2120574119909lowast119910lowast)(119896+1)where (119909lowast 119910lowast) denotes the fixed point of the controlled system(12) If |(1198602+1198612)plusmnradic(1198602 minus 1198612)2 + 411986221198632| lt 2 the fixed pointof the system is attractive
Proof Write 2(119909 119910) = (119886 + (1 minus 119887minus 120574)119909+ 1205741199092119910+119896(119891(119909 119910) minus119909) 119910 + 119887119909 minus 1205741199092119910 + 119896(119892(119909 119910) minus 119910)) The Jacobi matrix of thecontrolled system (12) is
1198632= [(1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896 1205741199092119899 (119896 + 1)
(119887 minus 2120574119909119899119910119899) (119896 + 1) (1 minus 1205741199092119899) (119896 + 1) minus 119896](13)
and its eigenmatrix is
1198632 minus 120582119864 = [(1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896 minus 120582 1205741199092119899 (119896 + 1)(119887 minus 2120574119909119899119910119899) (119896 + 1) (1 minus 1205741199092119899) (119896 + 1) minus 119896 minus 120582] (14)
Discrete Dynamics in Nature and Society 7
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 6 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 064 (b) 119896 = 104 (c) 119896 = 144(d) 119896 = 184 (e) 119896 = 224 (f) 119896 = 26
8 Discrete Dynamics in Nature and Society
13606 08 1 12 14 16 18 2 22 24 26
137
138
139
14
141
142
143
144
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 7 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from 064 to 26
And the characteristic equation is
1205822 minus ((2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899) (119896 + 1) minus 2119896) 120582+ ((1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896)sdot ((1 minus 1205741199092119899) (119896 + 1) minus 119896) minus 1205741199092119899 (119887 minus 2120574119909119899119910119899)sdot (119896 + 1)2 = 0
(15)
When themodulus of the eigenvalue 12058221 12058222 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058221221003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 4119862211986322
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (16)
that is10038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 41198622119863210038161003816100381610038161003816100381610038161003816 lt 2 (17)
By Theorem 5 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus1 lt 119896 lt190659
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 064 to 26 in Figure 6 to illustrate thechange of the Julia sets in optimal controlmethod In Figure 7the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In optimal control method the effective controlled inter-val of 119896 is from minus1 to 1906 But when the interval of 119896 is from064 to 26 this method has the best controlled effectivenessIn this interval with the absolute values of 119896 increasing the
Julia sets gradually contract to the center with nearly the samespeed and no significant change in the overall shape of theJulia sets occurs
From the perspective of the change of box-countingdimensions box-counting dimensions of the Julia sets gen-erally show a monotonic decreasing trend with the absolutevalues of 119896 increasing For the reason that the complexityof the Julia set can be depicted by box-counting dimensionwhen the complexity of the Julia set decreases with theabsolute value of 119896 increasing it indicates that this controlmethod has great control effectiveness
33 Gradient Control Method Take 119906119899 = 119896(1199092119899 minus 119909lowast2) V119899 =119896(1199102119899 minus 119910lowast2) with the control parameter 119896 Therefore thecontrolled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119896 (1199092119899 minus 119909lowast2) 119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (1199102119899 minus 119910lowast2) (18)
Theorem 6 Let 1198603 = 1 minus 119887 minus 120574 + 2120574119909lowast119910lowast + 2119896119909lowast 1198613 =1 minus 120574119909lowast2 + 2119896119910lowast 1198623 = 120574119909lowast2 and 1198633 = 119887 minus 2120574119909lowast119910lowast where(119909lowast 119910lowast) denotes the fixed point of the controlled system (18) If|(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 411986231198633| lt 2 the fixed point of thesystem is attractive
Proof Write 3(119909 119910) = (119886 + (1 minus 119887 minus 120574)119909 + 1205741199092119910 + 119896(1199092 minus119909lowast2) 119910 + 119887119909 minus 1205741199092119910 + 119896(1199102 minus 119910lowast2)) The Jacobi matrix of thecontrolled system (18) is
1198633 = [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 1205741199092119899119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899] (19)
Discrete Dynamics in Nature and Society 9
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 8 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus00005 (b) 119896 = minus0004 (c)119896 = minus0008 (d) 119896 = minus0012 (e) 119896 = minus0016 (f) 119896 = minus0020
10 Discrete Dynamics in Nature and Society
136
137
138
139
14
141
142
143
144
145
minus002
minus0018
minus0016
minus0014
minus0012
minus001
minus0008
minus0006
minus0004
minus0002 0
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 9 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus002 to minus00005
and its eigenmatrix is
1198633 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899 minus 120582] (20)
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 1205741199092119899 + 2119896119910119899) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899) (1 minus 1205741199092119899 + 2119896119910119899)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(21)
When themodulus of the eigenvalue 12058231 12058232 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058231321003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 4119862311986332
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (22)
that is10038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 41198623119863310038161003816100381610038161003816100381610038161003816 lt 2 (23)
By Theorem 6 we know that the Brusselator model canbe controlled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus26420 lt 119896 lt08561
Six simulation diagrams are chosen corresponding to thevalues of 119896 from minus00005 to minus002 in Figure 8 to illustratethe change of the Julia sets in gradient control method In
Figure 9 the box-counting dimensions of the controlled Juliasets are computed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from 00005 to 002 inFigure 10 to illustrate the change of the Julia sets in gradientcontrol method In Figure 11 the box-counting dimensionsof the controlled Julia sets are computed in this controlmethod
In gradient control method we consider two groups ofinterval of the parameter 119896 It is worth noticing that thecontraction of the left and the lower parts of the Julia set arefaster than the right and the upper parts when the interval ofcontrol parameter 119896 is from minus00005 to minus002 while the rightand the upper parts are faster than the left and the lower partswhen the interval of control parameter 119896 is from 00005 to002 In general with the absolute values of 119896 increasing theJulia sets contract to the center gradually and the complexityof the boundary of the Julia sets decreases
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing thebox-counting dimensions of the Julia sets generally show amonotonic decreasing trend Particularly when the controlparameter 119896 is in the range from minus0016 to minus00075 and from00075 to 0016 the change of box-counting dimensions isobviouslymonotonic which indicates the effectiveness of thismethod on the Julia set of the Brusselator model
4 Conclusion
Fractal theory is a hot topic in the research of nonlinearscience Describing the change of chemical elements in thechemical reaction process Brusselator model an importantclass of reaction diffusion equations is significant in thestudy of chaotic and fractal behaviors of nonlinear differentialequations In technological applications it is often requiredthat the behavior and performance of the system can becontrolled effectively
Discrete Dynamics in Nature and Society 11
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 10 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 00005 (b) 119896 = 00045 (c)119896 = 00085 (d) 119896 = 00125 (e) 119896 = 00165 (f) 119896 = 00200
12 Discrete Dynamics in Nature and Society
135
136
137
138
139
14
141
142
143
144
1450
0002
0004
0006
0008
001
0012
0014
0016
0018
002
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 11 The change of box-counting dimensions of the Julia setsof the controlled system when 119896 is from 00005 to 002
In this paper feedback control method optimal controlmethod and gradient control method are taken to controlthe Julia set of Brusselator model respectively and thebox-counting dimensions of the Julia set are calculated Ineach control method when the absolute value of controlparameter 119896 increases discretely the box-counting dimensionof Julia set decreases and the Julia set contracts to the centergradually The decrease of box-counting dimension is nearlymonotonic which indicates that the complexity of the Juliaset of the system is declined graduallyThus when the controlparameter 119896 is selected the Julia set of Brusselator modelcould be controlled Most importantly the three controlmethods have consistency of conclusion and effectiveness
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61403231 and 11501328) and ChinaPostdoctoral Science Foundation (no 2016M592188)
References
[1] X Wang Fractal Mechanism of Generalized M-J Sets DalianUniversity of Technology Press Dalian China 2002
[2] G Chen and X Dong ldquoFrom chaos to order methodologiesperspectives and applicationsrdquo Methodologies Perspectives ampApplications World Scientific vol 31 no 2 pp 113ndash122 1998
[3] B B Mandelbrot Fractal Form Chance and Dimension W HFreeman San Francisco Calif USA 1977
[4] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Co San Francisco Calif USA 1982
[5] L Pitulice D Craciun E Vilaseca et al ldquoFractal dimension ofthe trajectory of a single particle diffusing in crowded mediardquoRomanian Journal of Physics vol 61 no 7-8 pp 1276ndash1286 2016
[6] L Pitulice E Vilaseca I Pastor et al ldquoMonte Carlo simulationsof enzymatic reactions in crowdedmedia Effect of the enzyme-obstacle relative sizerdquo Mathematical Biosciences vol 251 no 1pp 72ndash82 2014
[7] M Lewis and D C Rees ldquoFractal surfaces of proteinsrdquo Sciencevol 230 no 4730 pp 1163ndash1165 1985
[8] A Isvoran L Pitulice C T Craescu and A Chiriac ldquoFractalaspects of calcium binding protein structuresrdquo Chaos Solitonsamp Fractals vol 35 no 5 pp 960ndash966 2008
[9] P J Bentley ldquoFractal proteinsrdquo Genetic Programming amp Evolv-able Machines vol 5 no 1 pp 71ndash101 2004
[10] M Zamir ldquoFractal dimensions and multifractility in vascularbranchingrdquo Journal of Theoretical Biology vol 212 no 2 pp183ndash190 2001
[11] R L Devaney and J P Eckmann ldquoAn introduction to chaoticdynamical systemsrdquo Mathematical Gazette vol 74 no 468 p72 1990
[12] Z Huang ldquoFractal characteristics of turbulencerdquo Advances inMechanics vol 30 no 4 pp 581ndash596 2000
[13] M S Movahed and E Hermanis ldquoFractal analysis of riverflow fluctuationsrdquo Physica A Statistical Mechanics and ItsApplications vol 387 no 4 pp 915ndash932 2008
[14] N Yonaiguchi Y Ida M Hayakawa and S Masuda ldquoFractalanalysis for VHF electromagnetic noises and the identificationof preseismic signature of an earthquakerdquo Journal of Atmo-spheric and Solar-Terrestrial Physics vol 69 no 15 pp 1825ndash1832 2007
[15] N Ai R Chen and H Li ldquoTo the fractal geomorphologyrdquoGeography and Territorial Research vol 15 no 1 pp 92ndash961999
[16] M Li L Zhu and H Long ldquoAnalysis of several problems in theapplication of fractal in geomorphologyrdquo Earthquake Researchvol 25 no 2 pp 155ndash162 2002
[17] J Shu D Tan and J Wu ldquoFractal structure of Chinese stockmarketrdquo Journal of Southwest Jiao Tong University vol 38 no 2pp 212ndash215 2003
[18] Y Lin ldquoFractal and its application in securitymarketrdquo EconomicBusiness vol 8 pp 44ndash46 2001
[19] Z Peng C Li and H Li ldquoResearch on enterprises fractalmanagement model in knowledge economy era rdquo BusinessResearch vol 255 no 10 pp 33ndash35 2002
[20] W Zeng The Dimension Calculation of Several Special Self-Similar Sets Central China Normal University Wuhan China2007
[21] K J FalconerTheGeometry of Fractal Sets vol 85 ofCambridgeTracts inMathematics CambridgeUniversity Press CambridgeUK 1986
[22] Preiss andDavid ldquoNondifferentiable functionsrdquo Transactions ofthe American Mathematical Society vol 17 no 3 pp 301ndash3251916
[23] B B Mandelbrot Les Objets Fractals Flammarion 1984[24] X Gaoan Fractal Dimension Seismological Press Beijing
China 1989[25] T Erneux and E L Reiss ldquoBrussellator isolasrdquo SIAM Journal on
Applied Mathematics vol 43 no 6 pp 1240ndash1246 1983[26] G Nicolis ldquoPatterns of spatio-temporal organization in chem-
ical and biochemical kineticsrdquo American Mathematical Societyvol 8 pp 33ndash58 1974
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Discrete Dynamics in Nature and Society
13
135
14
145
Control parameter k
Box-
coun
ting
dim
ensio
n
minus1 minus09 minus08 minus07 minus06 minus05 minus04 minus03 minus02
Figure 5 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus1 to minus022
that is
10038161003816100381610038161003816100381610038161003816(1198601 + 1198611) plusmn radic(1198601 minus 1198611)2 + 41198621119863110038161003816100381610038161003816100381610038161003816 lt 2 (11)
By Theorem 4 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example take 119886 = 007 119887 = 002 and 120574 = 001 insystem (6) and the initial Julia set is shown in Figure 1 thenwe get 119909lowast = 7 119910lowast = 27 By Theorem 4 the range of thevalue of 119896 is minus19895 lt 119896 lt 04695
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 006 to 045 in Figure 2 and we find that thetrend of the change of the Julia sets is obvious In Figure 3the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from minus1 to minus022 in Figure 4to illustrate the change of the Julia sets in feedback controlmethod In Figure 5 the box-counting dimensions of thecontrolled Julia sets are computed in this control method
In feedback control method the contraction of the leftand the lower parts is faster than the right and the upperparts when the interval of control parameter 119896 is between006 and 045 In addition the complexity of the boundaryof the Julia set significantly decreases and the lower partrapidly contracts When the interval of control parameter 119896is between minus1 and minus022 the complexity of the boundary ofthe Julia set significantly decreases and the Julia set tends tobe centrally symmetric In general with the absolute value of119896 increasing the Julia set contracts to the center gradually
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing the
change generally shows a monotonic decreasing trend Par-ticularly when the control parameter 119896 is in the intervalfrom 016 to 037 and minus09 to minus04 the monotonic changeof box-counting dimensions is obvious which indicates theeffectiveness of this control method on the Julia set ofBrusselator model
32 Optimal Control Method Take 119906119899 = 119896(119891(119909119899 119910119899) minus119909119899) V119899 = 119896(119892(119909119899 119910119899) minus 119910119899) with 119891(119909119899 119910119899) = 119886 + (1 minus 119887 minus120574)119909119899 + 1205741199092119899119910119899 119892(119909119899 119910119899) = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 where 119896 is thecontrol parameter Then the controlled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899+ 119896 (119891 (119909119899 119910119899) minus 119909119899)
119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (119892 (119909119899 119910119899) minus 119910119899) (12)
Theorem 5 Let1198602 = (1minus119887minus120574+2120574119909lowast119910lowast)(119896+1)minus119896 1198612 = (1minus120574119909lowast2)(119896+1)minus1198961198622 = 120574119909lowast2(119896+1) and1198632 = (119887minus2120574119909lowast119910lowast)(119896+1)where (119909lowast 119910lowast) denotes the fixed point of the controlled system(12) If |(1198602+1198612)plusmnradic(1198602 minus 1198612)2 + 411986221198632| lt 2 the fixed pointof the system is attractive
Proof Write 2(119909 119910) = (119886 + (1 minus 119887minus 120574)119909+ 1205741199092119910+119896(119891(119909 119910) minus119909) 119910 + 119887119909 minus 1205741199092119910 + 119896(119892(119909 119910) minus 119910)) The Jacobi matrix of thecontrolled system (12) is
1198632= [(1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896 1205741199092119899 (119896 + 1)
(119887 minus 2120574119909119899119910119899) (119896 + 1) (1 minus 1205741199092119899) (119896 + 1) minus 119896](13)
and its eigenmatrix is
1198632 minus 120582119864 = [(1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896 minus 120582 1205741199092119899 (119896 + 1)(119887 minus 2120574119909119899119910119899) (119896 + 1) (1 minus 1205741199092119899) (119896 + 1) minus 119896 minus 120582] (14)
Discrete Dynamics in Nature and Society 7
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 6 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 064 (b) 119896 = 104 (c) 119896 = 144(d) 119896 = 184 (e) 119896 = 224 (f) 119896 = 26
8 Discrete Dynamics in Nature and Society
13606 08 1 12 14 16 18 2 22 24 26
137
138
139
14
141
142
143
144
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 7 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from 064 to 26
And the characteristic equation is
1205822 minus ((2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899) (119896 + 1) minus 2119896) 120582+ ((1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896)sdot ((1 minus 1205741199092119899) (119896 + 1) minus 119896) minus 1205741199092119899 (119887 minus 2120574119909119899119910119899)sdot (119896 + 1)2 = 0
(15)
When themodulus of the eigenvalue 12058221 12058222 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058221221003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 4119862211986322
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (16)
that is10038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 41198622119863210038161003816100381610038161003816100381610038161003816 lt 2 (17)
By Theorem 5 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus1 lt 119896 lt190659
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 064 to 26 in Figure 6 to illustrate thechange of the Julia sets in optimal controlmethod In Figure 7the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In optimal control method the effective controlled inter-val of 119896 is from minus1 to 1906 But when the interval of 119896 is from064 to 26 this method has the best controlled effectivenessIn this interval with the absolute values of 119896 increasing the
Julia sets gradually contract to the center with nearly the samespeed and no significant change in the overall shape of theJulia sets occurs
From the perspective of the change of box-countingdimensions box-counting dimensions of the Julia sets gen-erally show a monotonic decreasing trend with the absolutevalues of 119896 increasing For the reason that the complexityof the Julia set can be depicted by box-counting dimensionwhen the complexity of the Julia set decreases with theabsolute value of 119896 increasing it indicates that this controlmethod has great control effectiveness
33 Gradient Control Method Take 119906119899 = 119896(1199092119899 minus 119909lowast2) V119899 =119896(1199102119899 minus 119910lowast2) with the control parameter 119896 Therefore thecontrolled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119896 (1199092119899 minus 119909lowast2) 119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (1199102119899 minus 119910lowast2) (18)
Theorem 6 Let 1198603 = 1 minus 119887 minus 120574 + 2120574119909lowast119910lowast + 2119896119909lowast 1198613 =1 minus 120574119909lowast2 + 2119896119910lowast 1198623 = 120574119909lowast2 and 1198633 = 119887 minus 2120574119909lowast119910lowast where(119909lowast 119910lowast) denotes the fixed point of the controlled system (18) If|(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 411986231198633| lt 2 the fixed point of thesystem is attractive
Proof Write 3(119909 119910) = (119886 + (1 minus 119887 minus 120574)119909 + 1205741199092119910 + 119896(1199092 minus119909lowast2) 119910 + 119887119909 minus 1205741199092119910 + 119896(1199102 minus 119910lowast2)) The Jacobi matrix of thecontrolled system (18) is
1198633 = [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 1205741199092119899119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899] (19)
Discrete Dynamics in Nature and Society 9
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 8 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus00005 (b) 119896 = minus0004 (c)119896 = minus0008 (d) 119896 = minus0012 (e) 119896 = minus0016 (f) 119896 = minus0020
10 Discrete Dynamics in Nature and Society
136
137
138
139
14
141
142
143
144
145
minus002
minus0018
minus0016
minus0014
minus0012
minus001
minus0008
minus0006
minus0004
minus0002 0
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 9 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus002 to minus00005
and its eigenmatrix is
1198633 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899 minus 120582] (20)
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 1205741199092119899 + 2119896119910119899) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899) (1 minus 1205741199092119899 + 2119896119910119899)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(21)
When themodulus of the eigenvalue 12058231 12058232 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058231321003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 4119862311986332
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (22)
that is10038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 41198623119863310038161003816100381610038161003816100381610038161003816 lt 2 (23)
By Theorem 6 we know that the Brusselator model canbe controlled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus26420 lt 119896 lt08561
Six simulation diagrams are chosen corresponding to thevalues of 119896 from minus00005 to minus002 in Figure 8 to illustratethe change of the Julia sets in gradient control method In
Figure 9 the box-counting dimensions of the controlled Juliasets are computed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from 00005 to 002 inFigure 10 to illustrate the change of the Julia sets in gradientcontrol method In Figure 11 the box-counting dimensionsof the controlled Julia sets are computed in this controlmethod
In gradient control method we consider two groups ofinterval of the parameter 119896 It is worth noticing that thecontraction of the left and the lower parts of the Julia set arefaster than the right and the upper parts when the interval ofcontrol parameter 119896 is from minus00005 to minus002 while the rightand the upper parts are faster than the left and the lower partswhen the interval of control parameter 119896 is from 00005 to002 In general with the absolute values of 119896 increasing theJulia sets contract to the center gradually and the complexityof the boundary of the Julia sets decreases
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing thebox-counting dimensions of the Julia sets generally show amonotonic decreasing trend Particularly when the controlparameter 119896 is in the range from minus0016 to minus00075 and from00075 to 0016 the change of box-counting dimensions isobviouslymonotonic which indicates the effectiveness of thismethod on the Julia set of the Brusselator model
4 Conclusion
Fractal theory is a hot topic in the research of nonlinearscience Describing the change of chemical elements in thechemical reaction process Brusselator model an importantclass of reaction diffusion equations is significant in thestudy of chaotic and fractal behaviors of nonlinear differentialequations In technological applications it is often requiredthat the behavior and performance of the system can becontrolled effectively
Discrete Dynamics in Nature and Society 11
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 10 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 00005 (b) 119896 = 00045 (c)119896 = 00085 (d) 119896 = 00125 (e) 119896 = 00165 (f) 119896 = 00200
12 Discrete Dynamics in Nature and Society
135
136
137
138
139
14
141
142
143
144
1450
0002
0004
0006
0008
001
0012
0014
0016
0018
002
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 11 The change of box-counting dimensions of the Julia setsof the controlled system when 119896 is from 00005 to 002
In this paper feedback control method optimal controlmethod and gradient control method are taken to controlthe Julia set of Brusselator model respectively and thebox-counting dimensions of the Julia set are calculated Ineach control method when the absolute value of controlparameter 119896 increases discretely the box-counting dimensionof Julia set decreases and the Julia set contracts to the centergradually The decrease of box-counting dimension is nearlymonotonic which indicates that the complexity of the Juliaset of the system is declined graduallyThus when the controlparameter 119896 is selected the Julia set of Brusselator modelcould be controlled Most importantly the three controlmethods have consistency of conclusion and effectiveness
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61403231 and 11501328) and ChinaPostdoctoral Science Foundation (no 2016M592188)
References
[1] X Wang Fractal Mechanism of Generalized M-J Sets DalianUniversity of Technology Press Dalian China 2002
[2] G Chen and X Dong ldquoFrom chaos to order methodologiesperspectives and applicationsrdquo Methodologies Perspectives ampApplications World Scientific vol 31 no 2 pp 113ndash122 1998
[3] B B Mandelbrot Fractal Form Chance and Dimension W HFreeman San Francisco Calif USA 1977
[4] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Co San Francisco Calif USA 1982
[5] L Pitulice D Craciun E Vilaseca et al ldquoFractal dimension ofthe trajectory of a single particle diffusing in crowded mediardquoRomanian Journal of Physics vol 61 no 7-8 pp 1276ndash1286 2016
[6] L Pitulice E Vilaseca I Pastor et al ldquoMonte Carlo simulationsof enzymatic reactions in crowdedmedia Effect of the enzyme-obstacle relative sizerdquo Mathematical Biosciences vol 251 no 1pp 72ndash82 2014
[7] M Lewis and D C Rees ldquoFractal surfaces of proteinsrdquo Sciencevol 230 no 4730 pp 1163ndash1165 1985
[8] A Isvoran L Pitulice C T Craescu and A Chiriac ldquoFractalaspects of calcium binding protein structuresrdquo Chaos Solitonsamp Fractals vol 35 no 5 pp 960ndash966 2008
[9] P J Bentley ldquoFractal proteinsrdquo Genetic Programming amp Evolv-able Machines vol 5 no 1 pp 71ndash101 2004
[10] M Zamir ldquoFractal dimensions and multifractility in vascularbranchingrdquo Journal of Theoretical Biology vol 212 no 2 pp183ndash190 2001
[11] R L Devaney and J P Eckmann ldquoAn introduction to chaoticdynamical systemsrdquo Mathematical Gazette vol 74 no 468 p72 1990
[12] Z Huang ldquoFractal characteristics of turbulencerdquo Advances inMechanics vol 30 no 4 pp 581ndash596 2000
[13] M S Movahed and E Hermanis ldquoFractal analysis of riverflow fluctuationsrdquo Physica A Statistical Mechanics and ItsApplications vol 387 no 4 pp 915ndash932 2008
[14] N Yonaiguchi Y Ida M Hayakawa and S Masuda ldquoFractalanalysis for VHF electromagnetic noises and the identificationof preseismic signature of an earthquakerdquo Journal of Atmo-spheric and Solar-Terrestrial Physics vol 69 no 15 pp 1825ndash1832 2007
[15] N Ai R Chen and H Li ldquoTo the fractal geomorphologyrdquoGeography and Territorial Research vol 15 no 1 pp 92ndash961999
[16] M Li L Zhu and H Long ldquoAnalysis of several problems in theapplication of fractal in geomorphologyrdquo Earthquake Researchvol 25 no 2 pp 155ndash162 2002
[17] J Shu D Tan and J Wu ldquoFractal structure of Chinese stockmarketrdquo Journal of Southwest Jiao Tong University vol 38 no 2pp 212ndash215 2003
[18] Y Lin ldquoFractal and its application in securitymarketrdquo EconomicBusiness vol 8 pp 44ndash46 2001
[19] Z Peng C Li and H Li ldquoResearch on enterprises fractalmanagement model in knowledge economy era rdquo BusinessResearch vol 255 no 10 pp 33ndash35 2002
[20] W Zeng The Dimension Calculation of Several Special Self-Similar Sets Central China Normal University Wuhan China2007
[21] K J FalconerTheGeometry of Fractal Sets vol 85 ofCambridgeTracts inMathematics CambridgeUniversity Press CambridgeUK 1986
[22] Preiss andDavid ldquoNondifferentiable functionsrdquo Transactions ofthe American Mathematical Society vol 17 no 3 pp 301ndash3251916
[23] B B Mandelbrot Les Objets Fractals Flammarion 1984[24] X Gaoan Fractal Dimension Seismological Press Beijing
China 1989[25] T Erneux and E L Reiss ldquoBrussellator isolasrdquo SIAM Journal on
Applied Mathematics vol 43 no 6 pp 1240ndash1246 1983[26] G Nicolis ldquoPatterns of spatio-temporal organization in chem-
ical and biochemical kineticsrdquo American Mathematical Societyvol 8 pp 33ndash58 1974
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 7
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 6 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 064 (b) 119896 = 104 (c) 119896 = 144(d) 119896 = 184 (e) 119896 = 224 (f) 119896 = 26
8 Discrete Dynamics in Nature and Society
13606 08 1 12 14 16 18 2 22 24 26
137
138
139
14
141
142
143
144
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 7 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from 064 to 26
And the characteristic equation is
1205822 minus ((2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899) (119896 + 1) minus 2119896) 120582+ ((1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896)sdot ((1 minus 1205741199092119899) (119896 + 1) minus 119896) minus 1205741199092119899 (119887 minus 2120574119909119899119910119899)sdot (119896 + 1)2 = 0
(15)
When themodulus of the eigenvalue 12058221 12058222 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058221221003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 4119862211986322
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (16)
that is10038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 41198622119863210038161003816100381610038161003816100381610038161003816 lt 2 (17)
By Theorem 5 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus1 lt 119896 lt190659
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 064 to 26 in Figure 6 to illustrate thechange of the Julia sets in optimal controlmethod In Figure 7the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In optimal control method the effective controlled inter-val of 119896 is from minus1 to 1906 But when the interval of 119896 is from064 to 26 this method has the best controlled effectivenessIn this interval with the absolute values of 119896 increasing the
Julia sets gradually contract to the center with nearly the samespeed and no significant change in the overall shape of theJulia sets occurs
From the perspective of the change of box-countingdimensions box-counting dimensions of the Julia sets gen-erally show a monotonic decreasing trend with the absolutevalues of 119896 increasing For the reason that the complexityof the Julia set can be depicted by box-counting dimensionwhen the complexity of the Julia set decreases with theabsolute value of 119896 increasing it indicates that this controlmethod has great control effectiveness
33 Gradient Control Method Take 119906119899 = 119896(1199092119899 minus 119909lowast2) V119899 =119896(1199102119899 minus 119910lowast2) with the control parameter 119896 Therefore thecontrolled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119896 (1199092119899 minus 119909lowast2) 119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (1199102119899 minus 119910lowast2) (18)
Theorem 6 Let 1198603 = 1 minus 119887 minus 120574 + 2120574119909lowast119910lowast + 2119896119909lowast 1198613 =1 minus 120574119909lowast2 + 2119896119910lowast 1198623 = 120574119909lowast2 and 1198633 = 119887 minus 2120574119909lowast119910lowast where(119909lowast 119910lowast) denotes the fixed point of the controlled system (18) If|(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 411986231198633| lt 2 the fixed point of thesystem is attractive
Proof Write 3(119909 119910) = (119886 + (1 minus 119887 minus 120574)119909 + 1205741199092119910 + 119896(1199092 minus119909lowast2) 119910 + 119887119909 minus 1205741199092119910 + 119896(1199102 minus 119910lowast2)) The Jacobi matrix of thecontrolled system (18) is
1198633 = [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 1205741199092119899119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899] (19)
Discrete Dynamics in Nature and Society 9
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 8 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus00005 (b) 119896 = minus0004 (c)119896 = minus0008 (d) 119896 = minus0012 (e) 119896 = minus0016 (f) 119896 = minus0020
10 Discrete Dynamics in Nature and Society
136
137
138
139
14
141
142
143
144
145
minus002
minus0018
minus0016
minus0014
minus0012
minus001
minus0008
minus0006
minus0004
minus0002 0
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 9 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus002 to minus00005
and its eigenmatrix is
1198633 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899 minus 120582] (20)
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 1205741199092119899 + 2119896119910119899) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899) (1 minus 1205741199092119899 + 2119896119910119899)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(21)
When themodulus of the eigenvalue 12058231 12058232 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058231321003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 4119862311986332
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (22)
that is10038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 41198623119863310038161003816100381610038161003816100381610038161003816 lt 2 (23)
By Theorem 6 we know that the Brusselator model canbe controlled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus26420 lt 119896 lt08561
Six simulation diagrams are chosen corresponding to thevalues of 119896 from minus00005 to minus002 in Figure 8 to illustratethe change of the Julia sets in gradient control method In
Figure 9 the box-counting dimensions of the controlled Juliasets are computed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from 00005 to 002 inFigure 10 to illustrate the change of the Julia sets in gradientcontrol method In Figure 11 the box-counting dimensionsof the controlled Julia sets are computed in this controlmethod
In gradient control method we consider two groups ofinterval of the parameter 119896 It is worth noticing that thecontraction of the left and the lower parts of the Julia set arefaster than the right and the upper parts when the interval ofcontrol parameter 119896 is from minus00005 to minus002 while the rightand the upper parts are faster than the left and the lower partswhen the interval of control parameter 119896 is from 00005 to002 In general with the absolute values of 119896 increasing theJulia sets contract to the center gradually and the complexityof the boundary of the Julia sets decreases
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing thebox-counting dimensions of the Julia sets generally show amonotonic decreasing trend Particularly when the controlparameter 119896 is in the range from minus0016 to minus00075 and from00075 to 0016 the change of box-counting dimensions isobviouslymonotonic which indicates the effectiveness of thismethod on the Julia set of the Brusselator model
4 Conclusion
Fractal theory is a hot topic in the research of nonlinearscience Describing the change of chemical elements in thechemical reaction process Brusselator model an importantclass of reaction diffusion equations is significant in thestudy of chaotic and fractal behaviors of nonlinear differentialequations In technological applications it is often requiredthat the behavior and performance of the system can becontrolled effectively
Discrete Dynamics in Nature and Society 11
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 10 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 00005 (b) 119896 = 00045 (c)119896 = 00085 (d) 119896 = 00125 (e) 119896 = 00165 (f) 119896 = 00200
12 Discrete Dynamics in Nature and Society
135
136
137
138
139
14
141
142
143
144
1450
0002
0004
0006
0008
001
0012
0014
0016
0018
002
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 11 The change of box-counting dimensions of the Julia setsof the controlled system when 119896 is from 00005 to 002
In this paper feedback control method optimal controlmethod and gradient control method are taken to controlthe Julia set of Brusselator model respectively and thebox-counting dimensions of the Julia set are calculated Ineach control method when the absolute value of controlparameter 119896 increases discretely the box-counting dimensionof Julia set decreases and the Julia set contracts to the centergradually The decrease of box-counting dimension is nearlymonotonic which indicates that the complexity of the Juliaset of the system is declined graduallyThus when the controlparameter 119896 is selected the Julia set of Brusselator modelcould be controlled Most importantly the three controlmethods have consistency of conclusion and effectiveness
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61403231 and 11501328) and ChinaPostdoctoral Science Foundation (no 2016M592188)
References
[1] X Wang Fractal Mechanism of Generalized M-J Sets DalianUniversity of Technology Press Dalian China 2002
[2] G Chen and X Dong ldquoFrom chaos to order methodologiesperspectives and applicationsrdquo Methodologies Perspectives ampApplications World Scientific vol 31 no 2 pp 113ndash122 1998
[3] B B Mandelbrot Fractal Form Chance and Dimension W HFreeman San Francisco Calif USA 1977
[4] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Co San Francisco Calif USA 1982
[5] L Pitulice D Craciun E Vilaseca et al ldquoFractal dimension ofthe trajectory of a single particle diffusing in crowded mediardquoRomanian Journal of Physics vol 61 no 7-8 pp 1276ndash1286 2016
[6] L Pitulice E Vilaseca I Pastor et al ldquoMonte Carlo simulationsof enzymatic reactions in crowdedmedia Effect of the enzyme-obstacle relative sizerdquo Mathematical Biosciences vol 251 no 1pp 72ndash82 2014
[7] M Lewis and D C Rees ldquoFractal surfaces of proteinsrdquo Sciencevol 230 no 4730 pp 1163ndash1165 1985
[8] A Isvoran L Pitulice C T Craescu and A Chiriac ldquoFractalaspects of calcium binding protein structuresrdquo Chaos Solitonsamp Fractals vol 35 no 5 pp 960ndash966 2008
[9] P J Bentley ldquoFractal proteinsrdquo Genetic Programming amp Evolv-able Machines vol 5 no 1 pp 71ndash101 2004
[10] M Zamir ldquoFractal dimensions and multifractility in vascularbranchingrdquo Journal of Theoretical Biology vol 212 no 2 pp183ndash190 2001
[11] R L Devaney and J P Eckmann ldquoAn introduction to chaoticdynamical systemsrdquo Mathematical Gazette vol 74 no 468 p72 1990
[12] Z Huang ldquoFractal characteristics of turbulencerdquo Advances inMechanics vol 30 no 4 pp 581ndash596 2000
[13] M S Movahed and E Hermanis ldquoFractal analysis of riverflow fluctuationsrdquo Physica A Statistical Mechanics and ItsApplications vol 387 no 4 pp 915ndash932 2008
[14] N Yonaiguchi Y Ida M Hayakawa and S Masuda ldquoFractalanalysis for VHF electromagnetic noises and the identificationof preseismic signature of an earthquakerdquo Journal of Atmo-spheric and Solar-Terrestrial Physics vol 69 no 15 pp 1825ndash1832 2007
[15] N Ai R Chen and H Li ldquoTo the fractal geomorphologyrdquoGeography and Territorial Research vol 15 no 1 pp 92ndash961999
[16] M Li L Zhu and H Long ldquoAnalysis of several problems in theapplication of fractal in geomorphologyrdquo Earthquake Researchvol 25 no 2 pp 155ndash162 2002
[17] J Shu D Tan and J Wu ldquoFractal structure of Chinese stockmarketrdquo Journal of Southwest Jiao Tong University vol 38 no 2pp 212ndash215 2003
[18] Y Lin ldquoFractal and its application in securitymarketrdquo EconomicBusiness vol 8 pp 44ndash46 2001
[19] Z Peng C Li and H Li ldquoResearch on enterprises fractalmanagement model in knowledge economy era rdquo BusinessResearch vol 255 no 10 pp 33ndash35 2002
[20] W Zeng The Dimension Calculation of Several Special Self-Similar Sets Central China Normal University Wuhan China2007
[21] K J FalconerTheGeometry of Fractal Sets vol 85 ofCambridgeTracts inMathematics CambridgeUniversity Press CambridgeUK 1986
[22] Preiss andDavid ldquoNondifferentiable functionsrdquo Transactions ofthe American Mathematical Society vol 17 no 3 pp 301ndash3251916
[23] B B Mandelbrot Les Objets Fractals Flammarion 1984[24] X Gaoan Fractal Dimension Seismological Press Beijing
China 1989[25] T Erneux and E L Reiss ldquoBrussellator isolasrdquo SIAM Journal on
Applied Mathematics vol 43 no 6 pp 1240ndash1246 1983[26] G Nicolis ldquoPatterns of spatio-temporal organization in chem-
ical and biochemical kineticsrdquo American Mathematical Societyvol 8 pp 33ndash58 1974
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Discrete Dynamics in Nature and Society
13606 08 1 12 14 16 18 2 22 24 26
137
138
139
14
141
142
143
144
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 7 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from 064 to 26
And the characteristic equation is
1205822 minus ((2 minus 119887 minus 120574 + 2120574119909119899119910119899 minus 1205741199092119899) (119896 + 1) minus 2119896) 120582+ ((1 minus 119887 minus 120574 + 2120574119909119899119910119899) (119896 + 1) minus 119896)sdot ((1 minus 1205741199092119899) (119896 + 1) minus 119896) minus 1205741199092119899 (119887 minus 2120574119909119899119910119899)sdot (119896 + 1)2 = 0
(15)
When themodulus of the eigenvalue 12058221 12058222 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058221221003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 4119862211986322
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (16)
that is10038161003816100381610038161003816100381610038161003816(1198602 + 1198612) plusmn radic(1198602 minus 1198612)2 + 41198622119863210038161003816100381610038161003816100381610038161003816 lt 2 (17)
By Theorem 5 we know that Brusselator model can becontrolled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus1 lt 119896 lt190659
Six simulation diagrams are chosen corresponding to thevalues of 119896 from 064 to 26 in Figure 6 to illustrate thechange of the Julia sets in optimal controlmethod In Figure 7the box-counting dimensions of the controlled Julia sets arecomputed in this control method
In optimal control method the effective controlled inter-val of 119896 is from minus1 to 1906 But when the interval of 119896 is from064 to 26 this method has the best controlled effectivenessIn this interval with the absolute values of 119896 increasing the
Julia sets gradually contract to the center with nearly the samespeed and no significant change in the overall shape of theJulia sets occurs
From the perspective of the change of box-countingdimensions box-counting dimensions of the Julia sets gen-erally show a monotonic decreasing trend with the absolutevalues of 119896 increasing For the reason that the complexityof the Julia set can be depicted by box-counting dimensionwhen the complexity of the Julia set decreases with theabsolute value of 119896 increasing it indicates that this controlmethod has great control effectiveness
33 Gradient Control Method Take 119906119899 = 119896(1199092119899 minus 119909lowast2) V119899 =119896(1199102119899 minus 119910lowast2) with the control parameter 119896 Therefore thecontrolled system is
119909119899+1 = 119886 + (1 minus 119887 minus 120574) 119909119899 + 1205741199092119899119910119899 + 119896 (1199092119899 minus 119909lowast2) 119910119899+1 = 119910119899 + 119887119909119899 minus 1205741199092119899119910119899 + 119896 (1199102119899 minus 119910lowast2) (18)
Theorem 6 Let 1198603 = 1 minus 119887 minus 120574 + 2120574119909lowast119910lowast + 2119896119909lowast 1198613 =1 minus 120574119909lowast2 + 2119896119910lowast 1198623 = 120574119909lowast2 and 1198633 = 119887 minus 2120574119909lowast119910lowast where(119909lowast 119910lowast) denotes the fixed point of the controlled system (18) If|(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 411986231198633| lt 2 the fixed point of thesystem is attractive
Proof Write 3(119909 119910) = (119886 + (1 minus 119887 minus 120574)119909 + 1205741199092119910 + 119896(1199092 minus119909lowast2) 119910 + 119887119909 minus 1205741199092119910 + 119896(1199102 minus 119910lowast2)) The Jacobi matrix of thecontrolled system (18) is
1198633 = [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 1205741199092119899119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899] (19)
Discrete Dynamics in Nature and Society 9
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 8 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus00005 (b) 119896 = minus0004 (c)119896 = minus0008 (d) 119896 = minus0012 (e) 119896 = minus0016 (f) 119896 = minus0020
10 Discrete Dynamics in Nature and Society
136
137
138
139
14
141
142
143
144
145
minus002
minus0018
minus0016
minus0014
minus0012
minus001
minus0008
minus0006
minus0004
minus0002 0
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 9 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus002 to minus00005
and its eigenmatrix is
1198633 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899 minus 120582] (20)
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 1205741199092119899 + 2119896119910119899) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899) (1 minus 1205741199092119899 + 2119896119910119899)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(21)
When themodulus of the eigenvalue 12058231 12058232 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058231321003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 4119862311986332
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (22)
that is10038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 41198623119863310038161003816100381610038161003816100381610038161003816 lt 2 (23)
By Theorem 6 we know that the Brusselator model canbe controlled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus26420 lt 119896 lt08561
Six simulation diagrams are chosen corresponding to thevalues of 119896 from minus00005 to minus002 in Figure 8 to illustratethe change of the Julia sets in gradient control method In
Figure 9 the box-counting dimensions of the controlled Juliasets are computed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from 00005 to 002 inFigure 10 to illustrate the change of the Julia sets in gradientcontrol method In Figure 11 the box-counting dimensionsof the controlled Julia sets are computed in this controlmethod
In gradient control method we consider two groups ofinterval of the parameter 119896 It is worth noticing that thecontraction of the left and the lower parts of the Julia set arefaster than the right and the upper parts when the interval ofcontrol parameter 119896 is from minus00005 to minus002 while the rightand the upper parts are faster than the left and the lower partswhen the interval of control parameter 119896 is from 00005 to002 In general with the absolute values of 119896 increasing theJulia sets contract to the center gradually and the complexityof the boundary of the Julia sets decreases
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing thebox-counting dimensions of the Julia sets generally show amonotonic decreasing trend Particularly when the controlparameter 119896 is in the range from minus0016 to minus00075 and from00075 to 0016 the change of box-counting dimensions isobviouslymonotonic which indicates the effectiveness of thismethod on the Julia set of the Brusselator model
4 Conclusion
Fractal theory is a hot topic in the research of nonlinearscience Describing the change of chemical elements in thechemical reaction process Brusselator model an importantclass of reaction diffusion equations is significant in thestudy of chaotic and fractal behaviors of nonlinear differentialequations In technological applications it is often requiredthat the behavior and performance of the system can becontrolled effectively
Discrete Dynamics in Nature and Society 11
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 10 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 00005 (b) 119896 = 00045 (c)119896 = 00085 (d) 119896 = 00125 (e) 119896 = 00165 (f) 119896 = 00200
12 Discrete Dynamics in Nature and Society
135
136
137
138
139
14
141
142
143
144
1450
0002
0004
0006
0008
001
0012
0014
0016
0018
002
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 11 The change of box-counting dimensions of the Julia setsof the controlled system when 119896 is from 00005 to 002
In this paper feedback control method optimal controlmethod and gradient control method are taken to controlthe Julia set of Brusselator model respectively and thebox-counting dimensions of the Julia set are calculated Ineach control method when the absolute value of controlparameter 119896 increases discretely the box-counting dimensionof Julia set decreases and the Julia set contracts to the centergradually The decrease of box-counting dimension is nearlymonotonic which indicates that the complexity of the Juliaset of the system is declined graduallyThus when the controlparameter 119896 is selected the Julia set of Brusselator modelcould be controlled Most importantly the three controlmethods have consistency of conclusion and effectiveness
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61403231 and 11501328) and ChinaPostdoctoral Science Foundation (no 2016M592188)
References
[1] X Wang Fractal Mechanism of Generalized M-J Sets DalianUniversity of Technology Press Dalian China 2002
[2] G Chen and X Dong ldquoFrom chaos to order methodologiesperspectives and applicationsrdquo Methodologies Perspectives ampApplications World Scientific vol 31 no 2 pp 113ndash122 1998
[3] B B Mandelbrot Fractal Form Chance and Dimension W HFreeman San Francisco Calif USA 1977
[4] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Co San Francisco Calif USA 1982
[5] L Pitulice D Craciun E Vilaseca et al ldquoFractal dimension ofthe trajectory of a single particle diffusing in crowded mediardquoRomanian Journal of Physics vol 61 no 7-8 pp 1276ndash1286 2016
[6] L Pitulice E Vilaseca I Pastor et al ldquoMonte Carlo simulationsof enzymatic reactions in crowdedmedia Effect of the enzyme-obstacle relative sizerdquo Mathematical Biosciences vol 251 no 1pp 72ndash82 2014
[7] M Lewis and D C Rees ldquoFractal surfaces of proteinsrdquo Sciencevol 230 no 4730 pp 1163ndash1165 1985
[8] A Isvoran L Pitulice C T Craescu and A Chiriac ldquoFractalaspects of calcium binding protein structuresrdquo Chaos Solitonsamp Fractals vol 35 no 5 pp 960ndash966 2008
[9] P J Bentley ldquoFractal proteinsrdquo Genetic Programming amp Evolv-able Machines vol 5 no 1 pp 71ndash101 2004
[10] M Zamir ldquoFractal dimensions and multifractility in vascularbranchingrdquo Journal of Theoretical Biology vol 212 no 2 pp183ndash190 2001
[11] R L Devaney and J P Eckmann ldquoAn introduction to chaoticdynamical systemsrdquo Mathematical Gazette vol 74 no 468 p72 1990
[12] Z Huang ldquoFractal characteristics of turbulencerdquo Advances inMechanics vol 30 no 4 pp 581ndash596 2000
[13] M S Movahed and E Hermanis ldquoFractal analysis of riverflow fluctuationsrdquo Physica A Statistical Mechanics and ItsApplications vol 387 no 4 pp 915ndash932 2008
[14] N Yonaiguchi Y Ida M Hayakawa and S Masuda ldquoFractalanalysis for VHF electromagnetic noises and the identificationof preseismic signature of an earthquakerdquo Journal of Atmo-spheric and Solar-Terrestrial Physics vol 69 no 15 pp 1825ndash1832 2007
[15] N Ai R Chen and H Li ldquoTo the fractal geomorphologyrdquoGeography and Territorial Research vol 15 no 1 pp 92ndash961999
[16] M Li L Zhu and H Long ldquoAnalysis of several problems in theapplication of fractal in geomorphologyrdquo Earthquake Researchvol 25 no 2 pp 155ndash162 2002
[17] J Shu D Tan and J Wu ldquoFractal structure of Chinese stockmarketrdquo Journal of Southwest Jiao Tong University vol 38 no 2pp 212ndash215 2003
[18] Y Lin ldquoFractal and its application in securitymarketrdquo EconomicBusiness vol 8 pp 44ndash46 2001
[19] Z Peng C Li and H Li ldquoResearch on enterprises fractalmanagement model in knowledge economy era rdquo BusinessResearch vol 255 no 10 pp 33ndash35 2002
[20] W Zeng The Dimension Calculation of Several Special Self-Similar Sets Central China Normal University Wuhan China2007
[21] K J FalconerTheGeometry of Fractal Sets vol 85 ofCambridgeTracts inMathematics CambridgeUniversity Press CambridgeUK 1986
[22] Preiss andDavid ldquoNondifferentiable functionsrdquo Transactions ofthe American Mathematical Society vol 17 no 3 pp 301ndash3251916
[23] B B Mandelbrot Les Objets Fractals Flammarion 1984[24] X Gaoan Fractal Dimension Seismological Press Beijing
China 1989[25] T Erneux and E L Reiss ldquoBrussellator isolasrdquo SIAM Journal on
Applied Mathematics vol 43 no 6 pp 1240ndash1246 1983[26] G Nicolis ldquoPatterns of spatio-temporal organization in chem-
ical and biochemical kineticsrdquo American Mathematical Societyvol 8 pp 33ndash58 1974
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 9
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 8 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = minus00005 (b) 119896 = minus0004 (c)119896 = minus0008 (d) 119896 = minus0012 (e) 119896 = minus0016 (f) 119896 = minus0020
10 Discrete Dynamics in Nature and Society
136
137
138
139
14
141
142
143
144
145
minus002
minus0018
minus0016
minus0014
minus0012
minus001
minus0008
minus0006
minus0004
minus0002 0
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 9 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus002 to minus00005
and its eigenmatrix is
1198633 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899 minus 120582] (20)
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 1205741199092119899 + 2119896119910119899) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899) (1 minus 1205741199092119899 + 2119896119910119899)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(21)
When themodulus of the eigenvalue 12058231 12058232 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058231321003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 4119862311986332
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (22)
that is10038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 41198623119863310038161003816100381610038161003816100381610038161003816 lt 2 (23)
By Theorem 6 we know that the Brusselator model canbe controlled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus26420 lt 119896 lt08561
Six simulation diagrams are chosen corresponding to thevalues of 119896 from minus00005 to minus002 in Figure 8 to illustratethe change of the Julia sets in gradient control method In
Figure 9 the box-counting dimensions of the controlled Juliasets are computed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from 00005 to 002 inFigure 10 to illustrate the change of the Julia sets in gradientcontrol method In Figure 11 the box-counting dimensionsof the controlled Julia sets are computed in this controlmethod
In gradient control method we consider two groups ofinterval of the parameter 119896 It is worth noticing that thecontraction of the left and the lower parts of the Julia set arefaster than the right and the upper parts when the interval ofcontrol parameter 119896 is from minus00005 to minus002 while the rightand the upper parts are faster than the left and the lower partswhen the interval of control parameter 119896 is from 00005 to002 In general with the absolute values of 119896 increasing theJulia sets contract to the center gradually and the complexityof the boundary of the Julia sets decreases
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing thebox-counting dimensions of the Julia sets generally show amonotonic decreasing trend Particularly when the controlparameter 119896 is in the range from minus0016 to minus00075 and from00075 to 0016 the change of box-counting dimensions isobviouslymonotonic which indicates the effectiveness of thismethod on the Julia set of the Brusselator model
4 Conclusion
Fractal theory is a hot topic in the research of nonlinearscience Describing the change of chemical elements in thechemical reaction process Brusselator model an importantclass of reaction diffusion equations is significant in thestudy of chaotic and fractal behaviors of nonlinear differentialequations In technological applications it is often requiredthat the behavior and performance of the system can becontrolled effectively
Discrete Dynamics in Nature and Society 11
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 10 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 00005 (b) 119896 = 00045 (c)119896 = 00085 (d) 119896 = 00125 (e) 119896 = 00165 (f) 119896 = 00200
12 Discrete Dynamics in Nature and Society
135
136
137
138
139
14
141
142
143
144
1450
0002
0004
0006
0008
001
0012
0014
0016
0018
002
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 11 The change of box-counting dimensions of the Julia setsof the controlled system when 119896 is from 00005 to 002
In this paper feedback control method optimal controlmethod and gradient control method are taken to controlthe Julia set of Brusselator model respectively and thebox-counting dimensions of the Julia set are calculated Ineach control method when the absolute value of controlparameter 119896 increases discretely the box-counting dimensionof Julia set decreases and the Julia set contracts to the centergradually The decrease of box-counting dimension is nearlymonotonic which indicates that the complexity of the Juliaset of the system is declined graduallyThus when the controlparameter 119896 is selected the Julia set of Brusselator modelcould be controlled Most importantly the three controlmethods have consistency of conclusion and effectiveness
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61403231 and 11501328) and ChinaPostdoctoral Science Foundation (no 2016M592188)
References
[1] X Wang Fractal Mechanism of Generalized M-J Sets DalianUniversity of Technology Press Dalian China 2002
[2] G Chen and X Dong ldquoFrom chaos to order methodologiesperspectives and applicationsrdquo Methodologies Perspectives ampApplications World Scientific vol 31 no 2 pp 113ndash122 1998
[3] B B Mandelbrot Fractal Form Chance and Dimension W HFreeman San Francisco Calif USA 1977
[4] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Co San Francisco Calif USA 1982
[5] L Pitulice D Craciun E Vilaseca et al ldquoFractal dimension ofthe trajectory of a single particle diffusing in crowded mediardquoRomanian Journal of Physics vol 61 no 7-8 pp 1276ndash1286 2016
[6] L Pitulice E Vilaseca I Pastor et al ldquoMonte Carlo simulationsof enzymatic reactions in crowdedmedia Effect of the enzyme-obstacle relative sizerdquo Mathematical Biosciences vol 251 no 1pp 72ndash82 2014
[7] M Lewis and D C Rees ldquoFractal surfaces of proteinsrdquo Sciencevol 230 no 4730 pp 1163ndash1165 1985
[8] A Isvoran L Pitulice C T Craescu and A Chiriac ldquoFractalaspects of calcium binding protein structuresrdquo Chaos Solitonsamp Fractals vol 35 no 5 pp 960ndash966 2008
[9] P J Bentley ldquoFractal proteinsrdquo Genetic Programming amp Evolv-able Machines vol 5 no 1 pp 71ndash101 2004
[10] M Zamir ldquoFractal dimensions and multifractility in vascularbranchingrdquo Journal of Theoretical Biology vol 212 no 2 pp183ndash190 2001
[11] R L Devaney and J P Eckmann ldquoAn introduction to chaoticdynamical systemsrdquo Mathematical Gazette vol 74 no 468 p72 1990
[12] Z Huang ldquoFractal characteristics of turbulencerdquo Advances inMechanics vol 30 no 4 pp 581ndash596 2000
[13] M S Movahed and E Hermanis ldquoFractal analysis of riverflow fluctuationsrdquo Physica A Statistical Mechanics and ItsApplications vol 387 no 4 pp 915ndash932 2008
[14] N Yonaiguchi Y Ida M Hayakawa and S Masuda ldquoFractalanalysis for VHF electromagnetic noises and the identificationof preseismic signature of an earthquakerdquo Journal of Atmo-spheric and Solar-Terrestrial Physics vol 69 no 15 pp 1825ndash1832 2007
[15] N Ai R Chen and H Li ldquoTo the fractal geomorphologyrdquoGeography and Territorial Research vol 15 no 1 pp 92ndash961999
[16] M Li L Zhu and H Long ldquoAnalysis of several problems in theapplication of fractal in geomorphologyrdquo Earthquake Researchvol 25 no 2 pp 155ndash162 2002
[17] J Shu D Tan and J Wu ldquoFractal structure of Chinese stockmarketrdquo Journal of Southwest Jiao Tong University vol 38 no 2pp 212ndash215 2003
[18] Y Lin ldquoFractal and its application in securitymarketrdquo EconomicBusiness vol 8 pp 44ndash46 2001
[19] Z Peng C Li and H Li ldquoResearch on enterprises fractalmanagement model in knowledge economy era rdquo BusinessResearch vol 255 no 10 pp 33ndash35 2002
[20] W Zeng The Dimension Calculation of Several Special Self-Similar Sets Central China Normal University Wuhan China2007
[21] K J FalconerTheGeometry of Fractal Sets vol 85 ofCambridgeTracts inMathematics CambridgeUniversity Press CambridgeUK 1986
[22] Preiss andDavid ldquoNondifferentiable functionsrdquo Transactions ofthe American Mathematical Society vol 17 no 3 pp 301ndash3251916
[23] B B Mandelbrot Les Objets Fractals Flammarion 1984[24] X Gaoan Fractal Dimension Seismological Press Beijing
China 1989[25] T Erneux and E L Reiss ldquoBrussellator isolasrdquo SIAM Journal on
Applied Mathematics vol 43 no 6 pp 1240ndash1246 1983[26] G Nicolis ldquoPatterns of spatio-temporal organization in chem-
ical and biochemical kineticsrdquo American Mathematical Societyvol 8 pp 33ndash58 1974
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Discrete Dynamics in Nature and Society
136
137
138
139
14
141
142
143
144
145
minus002
minus0018
minus0016
minus0014
minus0012
minus001
minus0008
minus0006
minus0004
minus0002 0
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 9 The change of box-counting dimensions of the Julia sets of the controlled system when 119896 is from minus002 to minus00005
and its eigenmatrix is
1198633 minus 120582119864= [1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 120582 1205741199092119899
119887 minus 2120574119909119899119910119899 1 minus 1205741199092119899 + 2119896119910119899 minus 120582] (20)
And the characteristic equation is
1205822 minus (2 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899 minus 1205741199092119899 + 2119896119910119899) 120582+ (1 minus 119887 minus 120574 + 2120574119909119899119910119899 + 2119896119909119899) (1 minus 1205741199092119899 + 2119896119910119899)minus 1205741199092119899 (119887 minus 2120574119909119899119910119899) = 0
(21)
When themodulus of the eigenvalue 12058231 12058232 of the Jacobimatrix at the fixed point is less than 1 the fixed point isattractive
100381610038161003816100381612058231321003816100381610038161003816 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 4119862311986332
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816lt 1 (22)
that is10038161003816100381610038161003816100381610038161003816(1198603 + 1198613) plusmn radic(1198603 minus 1198613)2 + 41198623119863310038161003816100381610038161003816100381610038161003816 lt 2 (23)
By Theorem 6 we know that the Brusselator model canbe controlled by selecting the value of 119896 which satisfies thecondition and then the control of the Julia set can be realized
For example we take the values of system parameters119886 = 007 119887 = 002 and 120574 = 001 which are the same asSection 31 Then the range of the value of 119896 is minus26420 lt 119896 lt08561
Six simulation diagrams are chosen corresponding to thevalues of 119896 from minus00005 to minus002 in Figure 8 to illustratethe change of the Julia sets in gradient control method In
Figure 9 the box-counting dimensions of the controlled Juliasets are computed in this control method
In the same control six simulation diagrams are chosencorresponding to the values of 119896 from 00005 to 002 inFigure 10 to illustrate the change of the Julia sets in gradientcontrol method In Figure 11 the box-counting dimensionsof the controlled Julia sets are computed in this controlmethod
In gradient control method we consider two groups ofinterval of the parameter 119896 It is worth noticing that thecontraction of the left and the lower parts of the Julia set arefaster than the right and the upper parts when the interval ofcontrol parameter 119896 is from minus00005 to minus002 while the rightand the upper parts are faster than the left and the lower partswhen the interval of control parameter 119896 is from 00005 to002 In general with the absolute values of 119896 increasing theJulia sets contract to the center gradually and the complexityof the boundary of the Julia sets decreases
From the perspective of the change of box-countingdimensions with the absolute values of 119896 increasing thebox-counting dimensions of the Julia sets generally show amonotonic decreasing trend Particularly when the controlparameter 119896 is in the range from minus0016 to minus00075 and from00075 to 0016 the change of box-counting dimensions isobviouslymonotonic which indicates the effectiveness of thismethod on the Julia set of the Brusselator model
4 Conclusion
Fractal theory is a hot topic in the research of nonlinearscience Describing the change of chemical elements in thechemical reaction process Brusselator model an importantclass of reaction diffusion equations is significant in thestudy of chaotic and fractal behaviors of nonlinear differentialequations In technological applications it is often requiredthat the behavior and performance of the system can becontrolled effectively
Discrete Dynamics in Nature and Society 11
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 10 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 00005 (b) 119896 = 00045 (c)119896 = 00085 (d) 119896 = 00125 (e) 119896 = 00165 (f) 119896 = 00200
12 Discrete Dynamics in Nature and Society
135
136
137
138
139
14
141
142
143
144
1450
0002
0004
0006
0008
001
0012
0014
0016
0018
002
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 11 The change of box-counting dimensions of the Julia setsof the controlled system when 119896 is from 00005 to 002
In this paper feedback control method optimal controlmethod and gradient control method are taken to controlthe Julia set of Brusselator model respectively and thebox-counting dimensions of the Julia set are calculated Ineach control method when the absolute value of controlparameter 119896 increases discretely the box-counting dimensionof Julia set decreases and the Julia set contracts to the centergradually The decrease of box-counting dimension is nearlymonotonic which indicates that the complexity of the Juliaset of the system is declined graduallyThus when the controlparameter 119896 is selected the Julia set of Brusselator modelcould be controlled Most importantly the three controlmethods have consistency of conclusion and effectiveness
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61403231 and 11501328) and ChinaPostdoctoral Science Foundation (no 2016M592188)
References
[1] X Wang Fractal Mechanism of Generalized M-J Sets DalianUniversity of Technology Press Dalian China 2002
[2] G Chen and X Dong ldquoFrom chaos to order methodologiesperspectives and applicationsrdquo Methodologies Perspectives ampApplications World Scientific vol 31 no 2 pp 113ndash122 1998
[3] B B Mandelbrot Fractal Form Chance and Dimension W HFreeman San Francisco Calif USA 1977
[4] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Co San Francisco Calif USA 1982
[5] L Pitulice D Craciun E Vilaseca et al ldquoFractal dimension ofthe trajectory of a single particle diffusing in crowded mediardquoRomanian Journal of Physics vol 61 no 7-8 pp 1276ndash1286 2016
[6] L Pitulice E Vilaseca I Pastor et al ldquoMonte Carlo simulationsof enzymatic reactions in crowdedmedia Effect of the enzyme-obstacle relative sizerdquo Mathematical Biosciences vol 251 no 1pp 72ndash82 2014
[7] M Lewis and D C Rees ldquoFractal surfaces of proteinsrdquo Sciencevol 230 no 4730 pp 1163ndash1165 1985
[8] A Isvoran L Pitulice C T Craescu and A Chiriac ldquoFractalaspects of calcium binding protein structuresrdquo Chaos Solitonsamp Fractals vol 35 no 5 pp 960ndash966 2008
[9] P J Bentley ldquoFractal proteinsrdquo Genetic Programming amp Evolv-able Machines vol 5 no 1 pp 71ndash101 2004
[10] M Zamir ldquoFractal dimensions and multifractility in vascularbranchingrdquo Journal of Theoretical Biology vol 212 no 2 pp183ndash190 2001
[11] R L Devaney and J P Eckmann ldquoAn introduction to chaoticdynamical systemsrdquo Mathematical Gazette vol 74 no 468 p72 1990
[12] Z Huang ldquoFractal characteristics of turbulencerdquo Advances inMechanics vol 30 no 4 pp 581ndash596 2000
[13] M S Movahed and E Hermanis ldquoFractal analysis of riverflow fluctuationsrdquo Physica A Statistical Mechanics and ItsApplications vol 387 no 4 pp 915ndash932 2008
[14] N Yonaiguchi Y Ida M Hayakawa and S Masuda ldquoFractalanalysis for VHF electromagnetic noises and the identificationof preseismic signature of an earthquakerdquo Journal of Atmo-spheric and Solar-Terrestrial Physics vol 69 no 15 pp 1825ndash1832 2007
[15] N Ai R Chen and H Li ldquoTo the fractal geomorphologyrdquoGeography and Territorial Research vol 15 no 1 pp 92ndash961999
[16] M Li L Zhu and H Long ldquoAnalysis of several problems in theapplication of fractal in geomorphologyrdquo Earthquake Researchvol 25 no 2 pp 155ndash162 2002
[17] J Shu D Tan and J Wu ldquoFractal structure of Chinese stockmarketrdquo Journal of Southwest Jiao Tong University vol 38 no 2pp 212ndash215 2003
[18] Y Lin ldquoFractal and its application in securitymarketrdquo EconomicBusiness vol 8 pp 44ndash46 2001
[19] Z Peng C Li and H Li ldquoResearch on enterprises fractalmanagement model in knowledge economy era rdquo BusinessResearch vol 255 no 10 pp 33ndash35 2002
[20] W Zeng The Dimension Calculation of Several Special Self-Similar Sets Central China Normal University Wuhan China2007
[21] K J FalconerTheGeometry of Fractal Sets vol 85 ofCambridgeTracts inMathematics CambridgeUniversity Press CambridgeUK 1986
[22] Preiss andDavid ldquoNondifferentiable functionsrdquo Transactions ofthe American Mathematical Society vol 17 no 3 pp 301ndash3251916
[23] B B Mandelbrot Les Objets Fractals Flammarion 1984[24] X Gaoan Fractal Dimension Seismological Press Beijing
China 1989[25] T Erneux and E L Reiss ldquoBrussellator isolasrdquo SIAM Journal on
Applied Mathematics vol 43 no 6 pp 1240ndash1246 1983[26] G Nicolis ldquoPatterns of spatio-temporal organization in chem-
ical and biochemical kineticsrdquo American Mathematical Societyvol 8 pp 33ndash58 1974
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 11
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(a)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(b)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(c)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(d)
minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(e)minus60
80
60
40
20
0
minus20
minus40
minus60
minus80806040200minus20minus40minus80
(f)
Figure 10 The change of the Julia sets of the controlled system when 119886 = 007 119887 = 002 and 120574 = 001 (a) 119896 = 00005 (b) 119896 = 00045 (c)119896 = 00085 (d) 119896 = 00125 (e) 119896 = 00165 (f) 119896 = 00200
12 Discrete Dynamics in Nature and Society
135
136
137
138
139
14
141
142
143
144
1450
0002
0004
0006
0008
001
0012
0014
0016
0018
002
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 11 The change of box-counting dimensions of the Julia setsof the controlled system when 119896 is from 00005 to 002
In this paper feedback control method optimal controlmethod and gradient control method are taken to controlthe Julia set of Brusselator model respectively and thebox-counting dimensions of the Julia set are calculated Ineach control method when the absolute value of controlparameter 119896 increases discretely the box-counting dimensionof Julia set decreases and the Julia set contracts to the centergradually The decrease of box-counting dimension is nearlymonotonic which indicates that the complexity of the Juliaset of the system is declined graduallyThus when the controlparameter 119896 is selected the Julia set of Brusselator modelcould be controlled Most importantly the three controlmethods have consistency of conclusion and effectiveness
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61403231 and 11501328) and ChinaPostdoctoral Science Foundation (no 2016M592188)
References
[1] X Wang Fractal Mechanism of Generalized M-J Sets DalianUniversity of Technology Press Dalian China 2002
[2] G Chen and X Dong ldquoFrom chaos to order methodologiesperspectives and applicationsrdquo Methodologies Perspectives ampApplications World Scientific vol 31 no 2 pp 113ndash122 1998
[3] B B Mandelbrot Fractal Form Chance and Dimension W HFreeman San Francisco Calif USA 1977
[4] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Co San Francisco Calif USA 1982
[5] L Pitulice D Craciun E Vilaseca et al ldquoFractal dimension ofthe trajectory of a single particle diffusing in crowded mediardquoRomanian Journal of Physics vol 61 no 7-8 pp 1276ndash1286 2016
[6] L Pitulice E Vilaseca I Pastor et al ldquoMonte Carlo simulationsof enzymatic reactions in crowdedmedia Effect of the enzyme-obstacle relative sizerdquo Mathematical Biosciences vol 251 no 1pp 72ndash82 2014
[7] M Lewis and D C Rees ldquoFractal surfaces of proteinsrdquo Sciencevol 230 no 4730 pp 1163ndash1165 1985
[8] A Isvoran L Pitulice C T Craescu and A Chiriac ldquoFractalaspects of calcium binding protein structuresrdquo Chaos Solitonsamp Fractals vol 35 no 5 pp 960ndash966 2008
[9] P J Bentley ldquoFractal proteinsrdquo Genetic Programming amp Evolv-able Machines vol 5 no 1 pp 71ndash101 2004
[10] M Zamir ldquoFractal dimensions and multifractility in vascularbranchingrdquo Journal of Theoretical Biology vol 212 no 2 pp183ndash190 2001
[11] R L Devaney and J P Eckmann ldquoAn introduction to chaoticdynamical systemsrdquo Mathematical Gazette vol 74 no 468 p72 1990
[12] Z Huang ldquoFractal characteristics of turbulencerdquo Advances inMechanics vol 30 no 4 pp 581ndash596 2000
[13] M S Movahed and E Hermanis ldquoFractal analysis of riverflow fluctuationsrdquo Physica A Statistical Mechanics and ItsApplications vol 387 no 4 pp 915ndash932 2008
[14] N Yonaiguchi Y Ida M Hayakawa and S Masuda ldquoFractalanalysis for VHF electromagnetic noises and the identificationof preseismic signature of an earthquakerdquo Journal of Atmo-spheric and Solar-Terrestrial Physics vol 69 no 15 pp 1825ndash1832 2007
[15] N Ai R Chen and H Li ldquoTo the fractal geomorphologyrdquoGeography and Territorial Research vol 15 no 1 pp 92ndash961999
[16] M Li L Zhu and H Long ldquoAnalysis of several problems in theapplication of fractal in geomorphologyrdquo Earthquake Researchvol 25 no 2 pp 155ndash162 2002
[17] J Shu D Tan and J Wu ldquoFractal structure of Chinese stockmarketrdquo Journal of Southwest Jiao Tong University vol 38 no 2pp 212ndash215 2003
[18] Y Lin ldquoFractal and its application in securitymarketrdquo EconomicBusiness vol 8 pp 44ndash46 2001
[19] Z Peng C Li and H Li ldquoResearch on enterprises fractalmanagement model in knowledge economy era rdquo BusinessResearch vol 255 no 10 pp 33ndash35 2002
[20] W Zeng The Dimension Calculation of Several Special Self-Similar Sets Central China Normal University Wuhan China2007
[21] K J FalconerTheGeometry of Fractal Sets vol 85 ofCambridgeTracts inMathematics CambridgeUniversity Press CambridgeUK 1986
[22] Preiss andDavid ldquoNondifferentiable functionsrdquo Transactions ofthe American Mathematical Society vol 17 no 3 pp 301ndash3251916
[23] B B Mandelbrot Les Objets Fractals Flammarion 1984[24] X Gaoan Fractal Dimension Seismological Press Beijing
China 1989[25] T Erneux and E L Reiss ldquoBrussellator isolasrdquo SIAM Journal on
Applied Mathematics vol 43 no 6 pp 1240ndash1246 1983[26] G Nicolis ldquoPatterns of spatio-temporal organization in chem-
ical and biochemical kineticsrdquo American Mathematical Societyvol 8 pp 33ndash58 1974
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Discrete Dynamics in Nature and Society
135
136
137
138
139
14
141
142
143
144
1450
0002
0004
0006
0008
001
0012
0014
0016
0018
002
Control parameter k
Box-
coun
ting
dim
ensio
n
Figure 11 The change of box-counting dimensions of the Julia setsof the controlled system when 119896 is from 00005 to 002
In this paper feedback control method optimal controlmethod and gradient control method are taken to controlthe Julia set of Brusselator model respectively and thebox-counting dimensions of the Julia set are calculated Ineach control method when the absolute value of controlparameter 119896 increases discretely the box-counting dimensionof Julia set decreases and the Julia set contracts to the centergradually The decrease of box-counting dimension is nearlymonotonic which indicates that the complexity of the Juliaset of the system is declined graduallyThus when the controlparameter 119896 is selected the Julia set of Brusselator modelcould be controlled Most importantly the three controlmethods have consistency of conclusion and effectiveness
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (nos 61403231 and 11501328) and ChinaPostdoctoral Science Foundation (no 2016M592188)
References
[1] X Wang Fractal Mechanism of Generalized M-J Sets DalianUniversity of Technology Press Dalian China 2002
[2] G Chen and X Dong ldquoFrom chaos to order methodologiesperspectives and applicationsrdquo Methodologies Perspectives ampApplications World Scientific vol 31 no 2 pp 113ndash122 1998
[3] B B Mandelbrot Fractal Form Chance and Dimension W HFreeman San Francisco Calif USA 1977
[4] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Co San Francisco Calif USA 1982
[5] L Pitulice D Craciun E Vilaseca et al ldquoFractal dimension ofthe trajectory of a single particle diffusing in crowded mediardquoRomanian Journal of Physics vol 61 no 7-8 pp 1276ndash1286 2016
[6] L Pitulice E Vilaseca I Pastor et al ldquoMonte Carlo simulationsof enzymatic reactions in crowdedmedia Effect of the enzyme-obstacle relative sizerdquo Mathematical Biosciences vol 251 no 1pp 72ndash82 2014
[7] M Lewis and D C Rees ldquoFractal surfaces of proteinsrdquo Sciencevol 230 no 4730 pp 1163ndash1165 1985
[8] A Isvoran L Pitulice C T Craescu and A Chiriac ldquoFractalaspects of calcium binding protein structuresrdquo Chaos Solitonsamp Fractals vol 35 no 5 pp 960ndash966 2008
[9] P J Bentley ldquoFractal proteinsrdquo Genetic Programming amp Evolv-able Machines vol 5 no 1 pp 71ndash101 2004
[10] M Zamir ldquoFractal dimensions and multifractility in vascularbranchingrdquo Journal of Theoretical Biology vol 212 no 2 pp183ndash190 2001
[11] R L Devaney and J P Eckmann ldquoAn introduction to chaoticdynamical systemsrdquo Mathematical Gazette vol 74 no 468 p72 1990
[12] Z Huang ldquoFractal characteristics of turbulencerdquo Advances inMechanics vol 30 no 4 pp 581ndash596 2000
[13] M S Movahed and E Hermanis ldquoFractal analysis of riverflow fluctuationsrdquo Physica A Statistical Mechanics and ItsApplications vol 387 no 4 pp 915ndash932 2008
[14] N Yonaiguchi Y Ida M Hayakawa and S Masuda ldquoFractalanalysis for VHF electromagnetic noises and the identificationof preseismic signature of an earthquakerdquo Journal of Atmo-spheric and Solar-Terrestrial Physics vol 69 no 15 pp 1825ndash1832 2007
[15] N Ai R Chen and H Li ldquoTo the fractal geomorphologyrdquoGeography and Territorial Research vol 15 no 1 pp 92ndash961999
[16] M Li L Zhu and H Long ldquoAnalysis of several problems in theapplication of fractal in geomorphologyrdquo Earthquake Researchvol 25 no 2 pp 155ndash162 2002
[17] J Shu D Tan and J Wu ldquoFractal structure of Chinese stockmarketrdquo Journal of Southwest Jiao Tong University vol 38 no 2pp 212ndash215 2003
[18] Y Lin ldquoFractal and its application in securitymarketrdquo EconomicBusiness vol 8 pp 44ndash46 2001
[19] Z Peng C Li and H Li ldquoResearch on enterprises fractalmanagement model in knowledge economy era rdquo BusinessResearch vol 255 no 10 pp 33ndash35 2002
[20] W Zeng The Dimension Calculation of Several Special Self-Similar Sets Central China Normal University Wuhan China2007
[21] K J FalconerTheGeometry of Fractal Sets vol 85 ofCambridgeTracts inMathematics CambridgeUniversity Press CambridgeUK 1986
[22] Preiss andDavid ldquoNondifferentiable functionsrdquo Transactions ofthe American Mathematical Society vol 17 no 3 pp 301ndash3251916
[23] B B Mandelbrot Les Objets Fractals Flammarion 1984[24] X Gaoan Fractal Dimension Seismological Press Beijing
China 1989[25] T Erneux and E L Reiss ldquoBrussellator isolasrdquo SIAM Journal on
Applied Mathematics vol 43 no 6 pp 1240ndash1246 1983[26] G Nicolis ldquoPatterns of spatio-temporal organization in chem-
ical and biochemical kineticsrdquo American Mathematical Societyvol 8 pp 33ndash58 1974
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Discrete Dynamics in Nature and Society 13
[27] I Prigogine and R Lefever ldquoSymmetry breaking instabilities indissipative systems IIrdquoThe Journal of Chemical Physics vol 48no 4 pp 1695ndash1700 1968
[28] FW Schneider ldquoPeriodic perturbations of chemical oscillatorsexperimentsrdquo Annual Review of Physical Chemistry vol 36 no36 pp 347ndash378 2003
[29] R Lefever M Herschkowitz-Kaufman and J M Turner ldquoThesteady-state solutions of the Brusselator modelrdquo Physics LettersA vol 60 no 3 pp 389ndash396 1977
[30] P K Vani G A Ramanujam and P Kaliappan ldquoPainlevkanalysis and particular solutions of a coupled nonlinear reactiondiffusion systemrdquo Journal of Physics A Mathematical andGeneral vol 26 no 3 pp L97ndashL99 1993
[31] A Pickering ldquoA new truncation in Painleve analysisrdquo Journal ofPhysics A Mathematical and General vol 26 no 17 pp 4395ndash4405 1998
[32] I Ndayirinde and W Malfliet ldquoNew special solutions of thelsquoBrusselatorrsquo reaction modelrdquo Journal of Physics A Mathemat-ical and General vol 30 no 14 pp 5151ndash5157 1997
[33] A L Larsen ldquoWeiss approach to a pair of coupled nonlinearreaction-diffusion equationsrdquo Physics Letters A vol 179 no 4-5 pp 284ndash290 1993
[34] Y Zhang Applications and Control of Fractals ShandongUniversity 2008
[35] K J Falconer ldquoFractal geometrymdashmathematical foundationsand applicationsrdquo Biometrics vol 46 no 4 p 499 1990
[36] A Y Pogromsky ldquoIntroduction to control of oscillations andchaosrdquoModern Language Journal vol 23 no 92 pp 3ndash5 2015
[37] Y Qin and X Zeng ldquoQualitative study of the Brussels oscillatorequation in Biochemistryrdquo Chinese Science Bulletin vol 25 no8 pp 337ndash339 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of