Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 167065 10 pageshttpdxdoiorg1011552013167065

Research ArticleDelayed Feedback Control and BifurcationAnalysis of an Autonomy System

Zhen Wang1 Huitao Zhao23 and Xiangyu Kong1

1 Institute of Information and System Computation Science Beifang University of Nationalities Yinchuan Ningxia 750021 China2Department of Mathematics and Information Science Zhoukou Normal University Zhoukou Henan 466001 China3Department of Applied Mathematics Kunming University of Science and Technology Kunming Yunnan 650093 China

Correspondence should be addressed to Zhen Wang jen715163com

Received 28 February 2013 Accepted 26 March 2013

Academic Editor Yisheng Song

Copyright copy 2013 Zhen Wang 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

An autonomy system with time-delayed feedback is studied by using the theory of functional differential equation and Hassardrsquosmethod the conditions onwhich zero equilibrium exists andHopf bifurcation occurs are given the qualities of theHopf bifurcationare also studied Finally several numerical simulations are given which indicate that when the delay passes through certain criticalvalues chaotic oscillation is converted into a stable state or a stable periodic orbit

1 Introduction

Since the first chaotic attractor was found by Lorsenz in 1963the theory of chaos has been developing rapidly The topicsof chaos and chaotic control are growing rapidly in manydifferent fields such as biological systems and ecological andchemical systems [1ndash5] The desirability of chaos depends onthe particular application Therefore it is important that thechaotic response of a system can be controlled

Many researchers have proposed chaos control and syn-chronization schemes in recent years [6ndash11] among whichdelayed feedback controller (DFC) is an effective method forchaos control it has been receiving considerable attentionrecently [11ndash14]The basic idea of DFC is to realize a continu-ous control for a dynamical system by applying a feedbacksignal which is proportional to the difference between thedynamical variable119883(119905) and its delayed value [14] Choosingthe appropriate feedback strength and time delay can makethe system stable The delayed feedback control methoddoes not require any computer analysis and can be simplyimplemented in various experiments For example Song andWei in [15] investigated the chaos phenomena of Chenrsquossystem using the method of delayed feedback control Theirresults show that when 119870 taking some value taking thedelay 120591 as the bifurcation parameter when 120591 passes through

a certain critical value the stability of the equilibrium willbe changed from unstable to stable chaos vanishes and aperiodic solution emerges

Let us consider a system of nonlinear autonomous differ-ential equations as follows

= 119886119909 minus 119910

119910 = 119909 minus 119887119910 minus 119911

= 119886119909 + 4119911 (119910 minus 2)

(1)

When 119886 = 02 119887 = 002 the zero equilibrium of system (1) isunstable and system (1) is chaotic (see Figure 1) For controlof chaos we add a time-delayed force119870(119910(119905)minus119910(119905minus120591)) to thesecond equation of system (1) that is the following delayedfeedback control system

= 119886119909 minus 119910

119910 = 119909 minus 119887119910 minus 119911 + 119870 (119910 (119905) minus 119910 (119905 minus 120591))

= 119886119909 + 4119911 (119910 minus 2)

(2)

When 119870 = 0 or 120591 = 0 we can see that system (2) is thesame as system (1) In this paper by taking 120591 as bifurcationparameter we will show that when 119870 takes some value with

2 Abstract and Applied Analysis

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

119905

119905

119905

05

119909

minus5

05

minus5

05

minus5

119910119911

119911

02

4

119910minus2

minus4

02

4

minus2minus4

minus1

01

2

34

119909

Figure 1 The trajectories and phase graphs of system (1) with 119886 = 02 and 119887 = 002

the increasing of 120591 the stability of zero equilibrium of system(2) will change and a family of periodic orbits bifurcates fromzero equilibrium

This paper is organized as follows In Section 2 we firstfocus on the stability and Hopf bifurcation of the zero equi-librium of system (2) In Section 3 we derive the directionand stability of Hopf bifurcation by using normal form andcentral manifold theory Finally in Section 4 an example isgiven for showing the effects of chaotic control

2 Local Stability and Delay-InducedHopf Bifurcations

In this section by analyzing the characteristic equation ofthe linearized system of system (2) at the equilibriums weinvestigate the stability of the equilibriums and the existenceof local Hopf bifurcations occurring at the equilibriums

System (2) has two equilibriums 1198640(0 0 0) and 119864

lowast(119909lowast

119910lowast 119911lowast) where

119909lowast=

8 (1 minus 119886119887) minus 119886

4119886 (1 minus 119886119887) 119910

lowast= 119886119909lowast 119911

lowast= (1 minus 119886119887) 119909

lowast

(3)

The linearization of system (2) at 1198640is

= 119886119909 minus 119910

119910 = 119909 minus 119887119910 minus 119911 + 119870 (119910 (119905) minus 119910 (119905 minus 120591))

= 119886119909 minus 8119911

(4)

whose characteristic equation is

1205823+ 11988621205822+ 1198861120582 + 1198860+ (11988721205822+ 1198871120582 + 1198870) 119890minus120582120591

= 0 (5)

where 1198862= 8+119887minus119886minus119870 119886

1= 8119887minus8119870minus119886119887+119886119870minus8119886+1 119886

0=

8119886119870 minus 8119886119887 minus 119886 + 8 1198872= 119870 119887

1= (8 minus 119886)119870 119887

0= minus8119886119870

andsum2119894=0

1198872

119894= 0Thus we need to study the distribution of the

roots of the third-degree exponential polynomial equationFor this end we first introduce the following simple resultwhich was proved by Ruan and Wei [16] using Rouchersquostheorem

Lemma 1 Consider the exponential polynomial

119875 (120582 119890minus1205821205911 119890

minus120582120591119898)

= 120582119899+ 119901(0)

1120582119899minus1

+ sdot sdot sdot + 119901(0)

119899minus1120582 + 119901(0)

119899

+ [119901(1)

1120582119899minus1

+ sdot sdot sdot + 119901(1)

119899minus1120582 + 119901(1)

119899] 119890minus1205821205911

+ sdot sdot sdot + [119901(119898)

1120582119899minus1

+ sdot sdot sdot + 119901(119898)

119899minus1120582 + 119901(119898)

119899] 119890minus120582120591119898

(6)

where 120591119894ge 0 (119894 = 1 2 119898) and 119901

(119894)

119895(119894 = 0 1 119898 119895 =

1 2 119899) are constants As (1205911 1205912 120591

119898) vary the sum of

the order of the zeros of 119875(120582 119890minus1205821205911 119890minus120582120591119898) on the open righthalf-plane can change only if a zero appears on or crosses theimaginary axis

Obviously 119894120596 (120596 gt 0) is a root of (5) if and only if 120596satisfies

minus 1205963119894 minus 11988621205962+ 1198861120596119894 + 119886

0

+ (minus11988721205962+ 1198871120596119894 + 119887

0) (cos120596120591 minus sin120596120591) = 0

(7)

Separating the real and imaginary parts we have

11988621205962minus 1198860= (1198870minus 11988721205962) cos120596120591 + 119887

1120596 sin120596120591

minus1205963+ 1198861120596 = (119887

0minus 11988721205962) sin120596120591 minus 119887

1120596 cos120596120591

(8)

which is equivalent to

1205966+ (1198862

2minus 1198872

2minus 21198861) 1205964

+ (1198862

1minus 211988601198862minus 1198872

1+ 211988701198872) 1205962+ 1198862

0minus 1198872

0= 0

(9)

Let 119911 = 1205962 and then (9) becomes

1199113+ 1199011199112+ 119902119911 + 119903 = 0 (10)

where 119901 = 1198862

2minus 1198872

2minus 21198861 119902 = 119886

2

1minus 211988601198862minus 1198872

1+ 211988701198872 and

119903 = 1198862

0minus 1198872

0

Denote

ℎ (119911) = 1199113+ 1199011199112+ 119902119911 + 119903 (11)

Then we have the following lemma

Abstract and Applied Analysis 3

Lemma 2 For the polynomial equation (10) one has thefollowing results

(i) If 119903 lt 0 then (10) has at least one positive root(ii) If 119903 ge 0 and 998779 = 119901

2minus 3119902 le 0 then (10) has no positive

roots(iii) If 119903 ge 0 and998779 = 119901

2minus3119902 gt 0 then (10) has positive roots

if and only if 119911lowast1= (13)(minus119901 +radic998779) gt 0 and ℎ(119911lowast

1) le 0

Suppose that (10) has positive roots without loss ofgenerality we assume that it has three positive roots definedby 1199111 1199112 and 119911

3 respectively Then (9) has three positive

roots

1205961= radic1199111 120596

2= radic1199112 120596

3= radic1199113 (12)

Thus we have

cos120596120591 =

11988711205962(1205962minus 1198861) minus (119886

21205962minus 1198860) (11988721205962minus 1198870)

(11988721205962 minus 119887

0)2+ 11988711205962

(13)

If we denote

120591(119895)

119896=

1

120596119896

arccos(11988711205962(1205962minus 1198861) minus (119886

21205962minus 1198860) (11988721205962minus 1198870)

(11988721205962 minus 119887

0)2

+ 11988711205962

)

+2119895120587 119896 = 1 2 3 119895 = 0 1

(14)

then plusmn119894120596119896is a pair of purely imaginary roots of (9) with 120591

(119895)

119896

Define

1205910= 120591(0)

1198960

= min119896isin123

120591(0)

119896 120596

0= 1205961198960

(15)

Note that when 120591 = 0 (5) becomes

1205823+ (1198862+ 1198872) 1205822+ (1198861+ 1198871) 120582 + 119886

0+ 1198870= 0 (16)

Therefore applying Lemmas 1 and 2 to (5) we obtain thefollowing lemma

Lemma 3 For the third-degree transcendental equation (5)one has the following results

(i) If 119903 ge 0 and 998779 = 1199012minus 3119902 ⩽ 0 then all roots with

positive real parts of (5) have the same sum to those ofthe polynomial equation (16) for all 120591 ge 0

(ii) If either 119903 lt 0 or 119903 ge 0 and 998779 = 1199012minus 3119902 gt 0 119911

lowast

1gt 0

and ℎ(119911lowast

1) lt 0 then all roots with positive real parts

of (5) have the same sum to those of the polynomialequation (16) for 120591 isin [0 120591

0)

Let 120582(120591) = 120572(120591) + 119894120596(120591) be the root of (9) near 120591 = 120591(119895)

119896

satisfying

120572 (120591(119895)

119896) = 0 120596 (120591

(119895)

119896) = 120596119896 (17)

Then it is easy to verify the following transversality condition

Lemma 4 Suppose that 119911119896= 1205962

119896and ℎ1015840(119911

119896) = 0 where ℎ(119911) is

defined by (15) Then 119889(Re120582(120591(119895)119896))119889120591 = 0 and 119889(Re120582(120591(119895)

119896))

119889120591 and ℎ1015840(119911119896) have the same sign

Now we study the characteristic equation (5) of system(4) Applying Lemmas 3 and 4 to (5) we have the followingtheorem

Theorem 5 Let 120591(119895)119896

and 1205960 1205910be defined by (14) and (15)

respectively Then consider the following

(i) If 119903 ge 0 and 998779 = 1199012minus 3119902 le 0 then all roots with

positive real parts of (5) have the same sum to those ofthe polynomial equation (16) for all 120591 ge 0

(ii) If either 119903 lt 0 or 119903 ge 0 and 998779 = 1199012minus 3119902 gt 0 119911lowast

1gt 0

and ℎ(119911lowast

1) lt 0 then ℎ(119911) has at least one positive root

119911119896 and all roots with positive real parts of (5) have the

same sum to those of the polynomial equation (16) for120591 isin [0 120591

0)

(iii) If the conditions of (ii) are satisfied and ℎ1015840(119911119896) = 0 then

system (2) exhibits Hopf bifurcation at the equilibrium1198640for 120591 = 120591

(119895)

119896

3 Stability and Direction ofBifurcating Periodic Orbits

In the previous section we obtain the conditions underwhichfamily periodic solutions bifurcate from the equilibrium119864

0at

the critical value of 120591 As pointed by in Hassard et al [17] it isinteresting to determine the direction stability and period ofthese periodic solutions Following the ideal of Hassard et alwe derive the explicit formulae for determining the propertiesof the Hopf bifurcation at the critical value of 120591 using thenormal form and the centermanifold theoryThroughout thissection we always assume that system (2) undergoes Hopfbifurcations at the equilibrium 119864

0for 120591 = 120591

119896 and then plusmn119894120596

119896is

corresponding to purely imaginary roots of the characteristicequation at the equilibrium 119864

0

Letting 120591 = 120591119905 120591 = 120591119896+ 120583 system (2) is transformed into

an FDE in 119862 = 119862([minus1 0] 1198773) as

= 119871120583(119906119905) + 119891 (120583 119906

119905) (18)

where 119906(119905) = (119909(119905) 119910(119905) 119911(119905))119879isin 1198773 and 119871

120583 119862 rarr 119877 119891

119877 times 119862 rarr 119877 are given respectively by

119871120583(120601) = (120591

119896+ 120583)(

119886 minus1 0

1 119870 minus 119887 minus1

119886 0 minus8

)(

1206011(0)

1206012(0)

1206013(0)

)

+ (120591119896+ 120583)(

0 0 0

0 minus119870 0

0 0 0

)(

1206011(minus1)

1206012(minus1)

1206013 (minus1)

)

(19)

119891 (120591 120601) = (120591119896+ 120583)(

0

0

41206012(0) 1206013(0)

) (20)

4 Abstract and Applied Analysis

By the Riesz representation theorem there exists a function120578(120579 120583) of bounded variation for 120579 isin [minus1 0] such that

119871120583120601 = int

0

minus1

119889120578 (120579 0) 120601 (120579) for120601 isin 119862 (21)

In fact we can choose

120578 (120579 120583) = (120591119896+ 120583)(

119886 minus1 0

1 119870 minus 119887 minus1

119886 0 minus8

)120575 (120579)

minus (120591119896+ 120583)(

0 0 0

0 minus119870 0

0 0 0

)120575 (120579 + 1)

(22)

where 120575 is Dirac-delta functionFor 120601 isin 119862

1([minus1 0] 119877

3) define

119860 (120583) 120601 =

119889120601 (120579)

119889120579 120579 isin [minus1 0)

int

0

minus1

119889120578 (120583 119904) 120601 (119904) 120579 = 0

119877 (120583) 120601 =

0 120579 isin [minus1 0)

119891 (120583 120601) 120579 = 0

(23)

Then when 120579 = 0 system (18) is equivalent to

119905= 119860 (120583) 119906

119905+ 119877 (120583) 119906

119905 (24)

where 119906119905(120579) = 119906(119905 + 120579) for 120579 isin [minus1 0] For 120595 isin 119862

1([0 1]

(1198773)lowast) define

119860lowast120595 (119904) =

minus119889120595 (119904)

119889119904 119904 isin (0 1]

int

0

minus1

119889120578119879(119905 0) 120595 (minus119905) 119904 = 0

(25)

and a bilinear inner product

⟨120595 (119904) 120601 (120579)⟩ = 120595 (0) 120601 (0)

minus int

0

minus1

int

120579

120585=0

120595 (120585 minus 120579) 119889120578 (120579) 120601 (120585) 119889120585

(26)

where 120578(120579) = 120578(120579 0) Then 119860 = 119860(0) and 119860lowast are adjoint

operators By the discussion in Section 2 we know thatplusmn119894120596119896120591119896are eigenvalues of 119860 Thus they are also eigenvalues

of 119860lowast We first need to compute the eigenvector of 119860 and 119860lowast

corresponding to 119894120596119896120591119896andminus119894120596

119896120591119896 respectively Suppose that

119902(120579) = (1 120572 120573)119879119890119894120579120596119896120591119896 is the eigenvector of 119860 corresponding

to 119894120596119896120591119896 Then 119860119902(120579) = 119894120596

119896120591119896119902(120579) It follows from the

definition of 119860 and (19) (21) and (22) that

120591119896(

119894120596119896minus 119886 1 0

minus1 119894120596119896minus 119870 + 119887 + 119870119890

minus119894120596119896120591119896 1

minus119886 0 119894120596119896+ 8

)119902 (0)

= (

0

0

0

)

(27)

Thus we can easily get

119902 (0) = (1 120572 120573)119879= (1 119886 minus 119894120596

119896

119886

119894120596119896+ 8

)

119879

(28)

Similarly let 119902lowast(119904) = 119863(1 120572lowast 120573lowast)119890119894119904120596119896120591119896 be the eigenvector of

119860lowast corresponding to minus119894120596

119896120591119896 By the definition of119860lowast and (19)

(21) and (22) we can compute

119902lowast(119904) = 119863 (1 120572

lowast 120573lowast) 119890119894119904120596119896120591119896

= 119863(1minus (119894120596119896+ 119886) (119894120596

119896minus 8)

119894120596119896minus 8 + 119886

minus119894120596119896minus 119886

119894120596119896minus 8 + 119886

) 119890119894119904120596119896120591119896

(29)

In order to assure ⟨119902lowast(119904) 119902(120579)⟩ = 1 we need to determine thevalue of119863 From (26) we have

⟨119902lowast(119904) 119902 (120579)⟩ = 119863(1 120572

lowast 120573lowast

) (1 120572 120573)119879

minus int

0

minus1

int

120579

120585=0

119863(1 120572lowast 120573lowast

) 119890minus119894(120585minus120579)120596

119896120591119896119889120578

times (120579) (1 120572 120573)119879119890119894120585120596119896120591119896119889120585

= 1198631 + 120572120572lowast+ 120573120573lowast

minus int

0

minus1

(1 120572lowast 120573lowast

) 120579119890119894120579120596119896120591119896119889120578

times (120579) (1 120572 120573)119879

= 1198631 + 120572120572lowast+ 120573120573lowast

minus 119870120591119896120572120572lowast119890minus119894120596119896120591119896

(30)

Thus we can choose119863 as

119863 =1

1 + 120572120572lowast+ 120573120573lowast

minus 119870120591119896120572120572lowast119890119894120596119896120591119896

(31)

In the following we first compute the coordinates to describethe center manifold 119862

0at 120583 = 0 Let 119906

119905be the solution of (18)

when 120583 = 0 Define

119911 (119905) = ⟨119902lowast 119906119905⟩ 119882 (119905 120579) = 119906

119905(120579) minus 2Re 119911 (119905) 119902 (120579)

(32)

On the center manifold 1198620 we have

119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)

= 11988220(120579)

1199112

2+11988211(120579) 119911119911

+11988202 (120579)

1199112

2+11988230 (120579)

1199113

6+ sdot sdot sdot

(33)

where 119911 and 119911 are local coordinates for center manifold 1198620

in the direction of 119902 and 119902lowast Note that 119882 is real if 119906

119905is real

Abstract and Applied Analysis 5

We consider only real solutions For the solution 119906119905isin 1198620of

(18) since 120583 = 0 we have

= 119894120596119896120591119896119911

+ ⟨119902lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579) + 2Re 119911 (119905) 119902 (120579))⟩

= 119894120596119896120591119896119911

+ 119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) + 2Re 119911 (119905) 119902 (0))

≜ 119894120596119896120591119896119911 + 119902lowast(0) 1198910(119911 119911) = 119894120596

119896120591119896119911 + 119892 (119911 119911)

(34)

where

119892 (119911 119911) = 119902lowast(0) 1198910 (119911 119911)

= 11989220(120579)

1199112

2+ 11989211(120579) 119911119911 + 119892

02(120579)

1199112

2+ sdot sdot sdot

(35)

By (32) we have 119906119905(120579) = (119906

1119905(120579) 1199062119905(120579) 1199063119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199061119905 (0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

1199062119905 (0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2+119882(2)

11(0) 119911119911

+119882(2)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

1199063119905 (0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

(36)

It follows together with (20) that

119892 (119911 119911) = 119902lowast(0) 1198910 (119911 119911)

= 119863120591119896(1 120572lowast 120573lowast

)(

0

0

41199062119905(0) 1199063119905(0)

)

= 4119863120591119896120573lowast

[119911120572 + 119911 120572 +119882(2)

20(0)

1199112

2+119882(2)

11(0) 119911119911

+119882(2)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)]

times [119911120573 + 119911120573 +119882(3)

20(0)

1199112

2+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)]

(37)

Comparing the coefficients with (35) we have

11989220

= 8119863120591119896120573lowast

120572120573

11989211

= 8119863120591119896120573lowast

(120572120573 + 120572120573)

11989202

= 8119863120591119896120573lowast

120572120573

11989221

= 4119863120591119896120573lowast

[2120572119882(3)

11(0) + 120572119882

(3)

20(0)

+120573119882(2)

20(0) + 2120573119882

(2)

11(0)]

(38)

In order to determine 11989221 in the sequel we need to compute

11988220(120579) and119882

11(120579) From (24) and (32) we have

= 119905minus 119902 minus 119902

=

119860119882 minus 2Re 119902lowast (0) 1198910119902 (120579) 120579 isin [0 1)

119860119882 minus 2Re 119902lowast (0) 1198910119902 (120579) + 1198910 120579 = 0

≜ 119860119882 +119867 (119911 119911 120579)

(39)

where

119867(119911 119911 120579) = 11986720 (120579)

1199112

2+ 11986711 (120579) 119911119911 + 119867

02 (120579)1199112

2+ sdot sdot sdot

(40)

Notice that near the origin on the centermanifold1198620 we have

= 119882119911 + 119882

119911 (41)

Thus we have

(119860 minus 2119894120591119896120596119896119868)11988220 (120579) = minus119867

20 (120579)

11986011988211(120579) = minus119867

11(120579)

(42)

Since (39) for 120579 isin [minus1 0) we have

119867(119911 119911 120579) = minus 119902lowast(0) 1198910119902 (120579) minus 119902

lowast(0) 1198910119902 (120579)

= minus 119892119902 (120579) minus 119892 119902 (120579)

(43)

Comparing the coefficients with (40) gives that

11986720(120579) = minus119892

20119902 (120579) minus 119892

02119902 (120579)

11986711 (120579) = minus119892

11119902 (120579) minus 119892

11119902 (120579)

(44)

From (43) (46) and the definition of 119860 we can get

20(120579) = 2119894120591

11989612059611989611988220(120579) + 119892

20119902 (120579) + 119892

02119902 (120579) (45)

Noticing that 119902(120579) = 119902(0)119890119894120591119896120596119896120579 we have

11988220(120579) =

11989411989220

120591119896120596119896

119902 (0) 119890119894120591119896120596119896120579+

11989411989202

3120591119896120596119896

119902 (0) 119890minus119894120591119896120596119896120579

+ 11986411198902119894120591119896120596119896120579

(46)

6 Abstract and Applied Analysis

051

152

253

Max(119910)

0 5 10 15 20120591

minus1

0minus05

Figure 2 The bifurcation graph with 120591 when 119870 = minus1

where 1198641= (119864(1)

1 119864(2)

1 119864(3)

1) isin 1198773 is a constant vector In the

same way we can also obtain

11988211 (120579) = minus

11989411989211

120591119896120596119896

119902 (0) 119890119894120591119896120596119896120579+

11989411989211

120591119896120596119896

119902 (0) 119890minus119894120591119896120596119896120579+ 1198642 (47)

where 1198642= (119864(1)

2 119864(2)

2 119864(3)

2) isin 1198773 is also a constant vector

In what follows we will seek appropriate 1198641and 119864

2 From

the definition of 119860 and (42) we obtain

int

0

minus1

119889120578 (120579)11988220(120579) = 2119894120591

11989612059611989611988220(0) minus 119867

20(0) (48)

int

0

minus1

119889120578 (120579)11988211 (120579) = minus11986711 (0) (49)

where 120578(120579) = 120578(0 120579) From (39) and (40) we have

11986720(0) = minus119892

20119902 (0) minus 119892

02119902 (0) + 2120591

119896(

0

0

4120572120573

) (50)

11986711 (0) = minus119892

11119902 (0) minus 119892

11119902 (0) + 2120591

119896(

0

0

4Re 120572120573) (51)

Substituting (46) and (50) into (48) and noticing that

(119894120596119896120591119896119868 minus int

0

minus1

119890119894120596119896120591119896120579119889120578 (120579)) 119902 (0) = 0

(minus119894120596119896120591119896119868 minus int

0

minus1

119890minus119894120596119896120591119896120579119889120578 (120579)) 119902 (0) = 0

(52)

we obtain

(2119894120596119896120591119896119868 minus int

0

minus1

1198902119894120596119896120591119896120579119889120578 (120579))119864

1= 2120591119896(

0

0

4120572120573

) (53)

which leads to

(

2119894120596119896minus 119886 1 0

minus1 2119894120596119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896 1

minus119886 0 2119894120596119896+ 8

)1198641

= 2(

0

0

4120572120573

)

(54)

It follows that

119864(1)

1=

8

119860120572120573

119864(2)

1=

8

119860120572120573 (2119894120596

119896minus 119886)

119864(3)

1=

8

119860120572120573 [(2119894120596

119896minus 119886) (2119894120596

119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896) + 1]

(55)

where119860 = (2119894120596

119896+ 8)

times [(2119894120596119896minus 119886) (2119894120596

119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896) + 1] minus 119886

(56)

Similarly substituting (47) and (51) into (49) we can get

(

minus119886 1 0

minus1 119887 1

minus119886 0 8

)1198642= 2(

0

0

4Re 120572120573) (57)

Thus we have

119864(1)

2=

8

119861Re 120572120573

119864(2)

2=

8

119861Re 120572120573

119864(3)

2=

8

119861Re 120572120573 (1 minus 119886119887)

(58)

where

119861 = 8 minus 119886 minus 8119886119887 (59)

Thus we can determine11988220(120579) and119882

11(120579) Furthermore we

can determine each 119892119894119895 Therefore each 119892

119894119895is determined by

the parameters and delay in (2) Thus we can compute thefollowing values

1198881(0) =

119894

2120596119896120591119896

(1198922011989211minus 2

1003816100381610038161003816119892111003816100381610038161003816

2minus1

3

1003816100381610038161003816119892021003816100381610038161003816

2) +

11989221

2

1205832= minus

Re 1198881(0)

Re 1205821015840 (120591119896)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(120591119896)

120596119896120591119896

1205732= 2Re 119888

1(0)

(60)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

119896 that is 120583

2

determines the directions of theHopf bifurcation if1205832gt 0 (lt

0) then theHopf bifurcation is supercritical (subcritical) andthe bifurcation exists for 120591 gt 120591

0(lt 1205910) 1205732determines the

stability of the bifurcation periodic solutions the bifurcatingperiodic solutions are stable (unstable) if 120573

2lt 0 (gt 0) and119879

2

determines the period of the bifurcating periodic solutionsthe period increases (decreases) if 119879

2gt 0 (lt 0)

Abstract and Applied Analysis 7

0 24

4

0

0

051

152

2

25

0 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

119909

119909

minus5

05

minus5

5

minus5

119910

119910

119911

119911

minus05

minus2 minus2minus4 minus4

119905

119905

119905

Figure 3 The trajectories and phase graphs of system (61) with 120591 = 07362 when 119870 = minus1

02

40

24

002

0406

080 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

0

119909

minus5

5

minus5

1

minus1

119910

119910

119911

119911

minus02

minus2 minus2minus4 minus4

119905

119905

119905

119909

Figure 4 The trajectories and phase graphs of system (61) with 120591 = 13497 when 119870 = minus1

4 Numerical Simulations andApplication to Control of Chaos

In this section we apply the results we get in the previoussections to system (61) for the purpose of control of chaosFrom Section 2 we know that under certain conditions afamily of periodic solutions bifurcates from the steady statesof system (61) at some critical values of 120591 and the stability ofthe steady state may be changed along with the increase of 120591If the bifurcating periodic solution is orbitally asymptoticallystable or some steady state becomes local stable then chaosmay vanish Following this idea we consider the followingdelayed feedback control system for an autonomy system

= 02119909 minus 119910

119910 = 119909 minus 002119910 minus 119911 + 119870 (119910 (119905) minus 119910 (119905 minus 120591))

= 02119909 + 4119911 (119910 minus 2)

(61)

which has two equilibriums 1198640(0 0 0) and 119864lowast(9749 19498

9710) When 120591 = 0 or 119870 = 0 1198640is unstable and system

(61) is chaotic (see Figure 1)The corresponding characteristicequation of system (61) at 119864

0is

1205823+ (782 minus 119870) 120582

2minus (0444 + 78119870) 120582 + 7768

+ 16119870 + (1205822+ 78120582 minus 16)119870119890

minus120582120591= 0

(62)

When 120591 = 0 (62) has one negative root and a pair of complexroots with positive parts By the discussion of Section 2we can get 119901 = 620404 minus 004119870 119902 = minus1212944 minus

25616119870 and 119903 = 603418 + 248576119870 and then we cancompute 998779 = 42129 + 27216119870 + 00016119870

2 and it is easyto know that 998779 gt 0 and when 119870 ge minus24275 119911lowast

1gt 0 always

holdsThen byTheorem 5 we have the following conclusion

Conclusion 1 If 119870 lt minus24275 or 119870 ge minus24275 and ℎ(119911lowast

1) lt 0

then there exist some 1205910gt 0 such that when 120591 isin [0 120591

0) 1198640is

always unstable

For example we fix119870 = minus1 Through computing we canget 119911lowast1= 10693 gt 0 and ℎ(119911lowast

1) = minus192713 lt 0 Furthermore

we can compute

8 Abstract and Applied Analysis

0 50 100 150 200

05

02

001 0 1 2

01

2

0

00501

0 50 100 150 200

0 50 100 150 200

119909

119909

minus5119910

119910

minus2

119911

119911

minus01

minus01

minus1 minus1

minus2minus2

minus3

minus005

119905

119905

119905

Figure 5 The trajectories and phase graphs of system (61) with 120591 = 26815 when 119870 = minus1

005

115

20 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

119911

minus05

4

02

119910 minus2

minus4

02

4

119909

minus2minus4

119905

119905

119905

Figure 6 The trajectories and phase graphs of system (61) with 120591 = 63804 when 119870 = minus1

1199111≐ 03714185849 120596

1≐ 06094

120591(119895)

1= 16243 +

2119895120587

1205961

ℎ1015840(1199111) = minus713282

1199112≐ 1494030027 120596

2≐ 12223

120591(119895)

2= 03768 +

2119895120587

1205962

ℎ1015840(1199112) = 725266

(63)

Thus from Lemma 4 we have

Re 120582 (120591(119895)1)

119889120591lt 0

Re 120582 (120591(119895)2)

119889120591gt 0

(64)

In addition notice that

120591(0)

2= 03768 lt 120591

(0)

1= 16243 lt 120591

(1)

2

= 55173 lt 120591(2)

2= 106577 lt 120591

(1)

1

= 119340 lt 120591(3)

2= 157982 lt sdot sdot sdot

(65)

Thus from (64) and (65) we have the following conclusionabout the stability of the zero equilibrium of system (61) andHopf bifurcations

Conclusion 2 Suppose that 120591(119895)119896 119896 = 1 2 119895 = 0 1 is

defined by (63)

(i) When 120591 isin [0 120591(0)

1) cup (120591(1)

2 +infin) the equilibrium 119864

0is

unstable (see Figures 3 4 6 7 and 8)(ii) When 120591 isin (120591

(0)

1 120591(1)

2) the equilibrium 119864

0is asymptot-

ically stable (see Figure 5)(iii) System (61) undergoes a Hopf bifurcation at the

equilibrium 1198640when 120591 = 120591

(119895)

119896(see Figures 3 4

and 6)

Thus if we take 120591 isin (120591(0)

1 120591(1)

2) the stability of zero

equilibrium of the systemwill change from unstable to stableFor this example there are more complicated dynamicalphenomena occurs Bifurcation diagram is plotted with theparameter 120591 when 119870 = minus1 (Figure 2) In Figure 2 we can seethat chaos vanishes via a period-doubling bifurcation when120591 varying from 0 to 03768 Figure 3 shows that when 120591 =

07362 a period-2 orbits bifurcating from zero equilibriumFigure 4 shows when 120591 = 13497 a period-1 periodicalorbits bifurcating from zero equilibrium In Figure 2 a stablewindow is obtained from 120591 = 16243 to 120591 = 55173 which isdemonstrated by Figure 5 with 120591 = 26815 When 120591 gt 55173Figure 2 shows that periodic orbits and strange attractors also

Abstract and Applied Analysis 9

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

005

115

2

119911

minus05

4

02

119910 minus2

minus4

02

4

119909minus2

minus4

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

119905

119905

119905

Figure 7 The trajectories and phase graphs of system (61) with 120591 = 73006 when 119870 = minus1

05

119909

minus5

05

minus5

05

minus5

119910119911

119911

4

02

119910 minus2

minus4

02

4

119909minus2minus4

minus1

012345

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

119905

119905

119905

Figure 8 The trajectories and phase graphs of system (61) with 120591 = 137423 when 119870 = minus1

occur and system (61) through Hopf bifurcation route stillgoes to chaos and it is demonstrated by Figures 6ndash8

5 Discussion

In this paper an autonomy system with time-delayed feed-back is studied by using the theory of functional differentialequation and Hassardrsquos method Illustrating with numericalsimulations we show that delayed feedback controller (DFC)is an effective method for chaos control

In fact system (2) has more complicated dynamicalbehaves From Figure 2 we can see that chaos degenerationprocess the orbits of system (2) continuously degenerate toa periodic solution region through reverse period-doublingbifurcation With the increasing of 120591 chaos occurs againthrough Hopf bifurcation route However the numericalsimulations indicate that there were points of similaritybetween this route to chaos and quasiperiodicity route tochaos [18] in our future research we will consider this topicAnd in this paper we only give detailed analysis on thetrivial equilibrium 119864

0 the detailed analysis of the interior

equilibrium 119864lowast is the second interested topic in our future

research

Acknowledgments

The authors would like to thank the anonymous refereefor the very helpful suggestions and comments which ledto the improvement of the original paper And this workis supported by 2013 Scientific Research Project of BeifangUniversity of Nationalities (2013XYZ021) Institute of Infor-mation and System Computation Science of Beifang Uni-versity (13xyb01) Science and Technology Department ofHenan Province (122300410417) and Education Departmentof Henan Province (13A110108)

References

[1] G Corso and F B Rizzato ldquoHamiltonian bifurcations leading tochaos in low-energy relativistic wave- particle systemrdquo PhysicaD vol 80 pp 296ndash306 1995

[2] M Saleem and T Agrawal ldquoChaos in a tumor growth modelwith delayed responses of the immune systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 891095 16 pages2012

[3] U Kumar ldquoA retrospection of chaotic phenomene in electricalsystemsrdquo Active and Passive Electronic Components vol 21 pp1ndash15 1998

10 Abstract and Applied Analysis

[4] Z Chen X Zhang and Q Bi ldquoBifurcations and chaos ofcoupled electrical circuitsrdquo Nonlinear Analysis vol 9 no 3 pp1158ndash1168 2008

[5] S Hallegatte M Ghil P Dumas and J Hourcade ldquoBusinesscycles bifurcations and chaos in a neo-classical model withinvestment dynamicsrdquo Journal of Economic Behavior and Orga-nization vol 67 pp 57ndash77 2008

[6] DGhoshA R Chowdhury andP Saha ldquoBifurcation continua-tion chaos and chaos control in nonlinear Bloch systemrdquo Com-munications inNonlinear Science andNumerical Simulation vol13 no 8 pp 1461ndash1471 2008

[7] A El-Gohary andA S Al-Ruzaiza ldquoChaos and adaptive controlin two prey one predator system with nonlinear feedbackrdquoChaos Solitons and Fractals vol 34 no 2 pp 443ndash453 2007

[8] H Salarieh and A Alasty ldquoControl of stochastic chaos usingsliding mode methodrdquo Journal of Computational and AppliedMathematics vol 225 no 1 pp 135ndash145 2009

[9] J Guan ldquoBifurcation analysis and chaos control in genesiosystem with delayed feedbackrdquo ISRNMathematical Physics vol2012 Article ID 843962 12 pages 2012

[10] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquoNonlinear Analysis vol 9 no 4 pp 1800ndash18102008

[11] N Vasegh and A Sedigh ldquoDelayed feedback control of time-delayed chaotic systems analytical approach at Hopf bifurca-tionrdquo Physics Letters A vol 372 pp 5110ndash5114 2008

[12] J Lu Z Ma and L Li ldquoDouble delayed feedback control for thestabilization of unstable steady states in chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 7 pp 3037ndash3045 2009

[13] M Bleich and J Socolar ldquoStability of periodic orbits controlledby time-delay feedbackrdquo Physics Letters A vol 210 pp 87ndash941996

[14] Y Ding W Jiang and H Wang ldquoDelayed feedback controland bifurcation analysis of Rossler chaotic systemrdquo NonlinearDynamics vol 61 no 4 pp 707ndash715 2010

[15] Y Song and J Wei ldquoBifurcation analysis for Chenrsquos system withdelayed feedback and its application to control of chaosrdquoChaosSolitons and Fractals vol 22 no 1 pp 75ndash91 2004

[16] S Ruan and J Wei ldquoOn the zeros of transcendental functionswith applications to stability of delay differential equations withtwo delaysrdquo Dynamics of Continuous Discrete and ImpulsiveSystems A vol 10 no 6 pp 863ndash874 2003

[17] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge UK 1981

[18] I Fuente L Martinez J Veguillas and J AguirregabirialdquoQuasiperiodicity route to chaos in a biochemical systemrdquoBiophysical Journal vol 71 pp 2375ndash2379 1998

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

2 Abstract and Applied Analysis

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

119905

119905

119905

05

119909

minus5

05

minus5

05

minus5

119910119911

119911

02

4

119910minus2

minus4

02

4

minus2minus4

minus1

01

2

34

119909

Figure 1 The trajectories and phase graphs of system (1) with 119886 = 02 and 119887 = 002

the increasing of 120591 the stability of zero equilibrium of system(2) will change and a family of periodic orbits bifurcates fromzero equilibrium

This paper is organized as follows In Section 2 we firstfocus on the stability and Hopf bifurcation of the zero equi-librium of system (2) In Section 3 we derive the directionand stability of Hopf bifurcation by using normal form andcentral manifold theory Finally in Section 4 an example isgiven for showing the effects of chaotic control

2 Local Stability and Delay-InducedHopf Bifurcations

In this section by analyzing the characteristic equation ofthe linearized system of system (2) at the equilibriums weinvestigate the stability of the equilibriums and the existenceof local Hopf bifurcations occurring at the equilibriums

System (2) has two equilibriums 1198640(0 0 0) and 119864

lowast(119909lowast

119910lowast 119911lowast) where

119909lowast=

8 (1 minus 119886119887) minus 119886

4119886 (1 minus 119886119887) 119910

lowast= 119886119909lowast 119911

lowast= (1 minus 119886119887) 119909

lowast

(3)

The linearization of system (2) at 1198640is

= 119886119909 minus 119910

119910 = 119909 minus 119887119910 minus 119911 + 119870 (119910 (119905) minus 119910 (119905 minus 120591))

= 119886119909 minus 8119911

(4)

whose characteristic equation is

1205823+ 11988621205822+ 1198861120582 + 1198860+ (11988721205822+ 1198871120582 + 1198870) 119890minus120582120591

= 0 (5)

where 1198862= 8+119887minus119886minus119870 119886

1= 8119887minus8119870minus119886119887+119886119870minus8119886+1 119886

0=

8119886119870 minus 8119886119887 minus 119886 + 8 1198872= 119870 119887

1= (8 minus 119886)119870 119887

0= minus8119886119870

andsum2119894=0

1198872

119894= 0Thus we need to study the distribution of the

roots of the third-degree exponential polynomial equationFor this end we first introduce the following simple resultwhich was proved by Ruan and Wei [16] using Rouchersquostheorem

Lemma 1 Consider the exponential polynomial

119875 (120582 119890minus1205821205911 119890

minus120582120591119898)

= 120582119899+ 119901(0)

1120582119899minus1

+ sdot sdot sdot + 119901(0)

119899minus1120582 + 119901(0)

119899

+ [119901(1)

1120582119899minus1

+ sdot sdot sdot + 119901(1)

119899minus1120582 + 119901(1)

119899] 119890minus1205821205911

+ sdot sdot sdot + [119901(119898)

1120582119899minus1

+ sdot sdot sdot + 119901(119898)

119899minus1120582 + 119901(119898)

119899] 119890minus120582120591119898

(6)

where 120591119894ge 0 (119894 = 1 2 119898) and 119901

(119894)

119895(119894 = 0 1 119898 119895 =

1 2 119899) are constants As (1205911 1205912 120591

119898) vary the sum of

the order of the zeros of 119875(120582 119890minus1205821205911 119890minus120582120591119898) on the open righthalf-plane can change only if a zero appears on or crosses theimaginary axis

Obviously 119894120596 (120596 gt 0) is a root of (5) if and only if 120596satisfies

minus 1205963119894 minus 11988621205962+ 1198861120596119894 + 119886

0

+ (minus11988721205962+ 1198871120596119894 + 119887

0) (cos120596120591 minus sin120596120591) = 0

(7)

Separating the real and imaginary parts we have

11988621205962minus 1198860= (1198870minus 11988721205962) cos120596120591 + 119887

1120596 sin120596120591

minus1205963+ 1198861120596 = (119887

0minus 11988721205962) sin120596120591 minus 119887

1120596 cos120596120591

(8)

which is equivalent to

1205966+ (1198862

2minus 1198872

2minus 21198861) 1205964

+ (1198862

1minus 211988601198862minus 1198872

1+ 211988701198872) 1205962+ 1198862

0minus 1198872

0= 0

(9)

Let 119911 = 1205962 and then (9) becomes

1199113+ 1199011199112+ 119902119911 + 119903 = 0 (10)

where 119901 = 1198862

2minus 1198872

2minus 21198861 119902 = 119886

2

1minus 211988601198862minus 1198872

1+ 211988701198872 and

119903 = 1198862

0minus 1198872

0

Denote

ℎ (119911) = 1199113+ 1199011199112+ 119902119911 + 119903 (11)

Then we have the following lemma

Abstract and Applied Analysis 3

Lemma 2 For the polynomial equation (10) one has thefollowing results

(i) If 119903 lt 0 then (10) has at least one positive root(ii) If 119903 ge 0 and 998779 = 119901

2minus 3119902 le 0 then (10) has no positive

roots(iii) If 119903 ge 0 and998779 = 119901

2minus3119902 gt 0 then (10) has positive roots

if and only if 119911lowast1= (13)(minus119901 +radic998779) gt 0 and ℎ(119911lowast

1) le 0

Suppose that (10) has positive roots without loss ofgenerality we assume that it has three positive roots definedby 1199111 1199112 and 119911

3 respectively Then (9) has three positive

roots

1205961= radic1199111 120596

2= radic1199112 120596

3= radic1199113 (12)

Thus we have

cos120596120591 =

11988711205962(1205962minus 1198861) minus (119886

21205962minus 1198860) (11988721205962minus 1198870)

(11988721205962 minus 119887

0)2+ 11988711205962

(13)

If we denote

120591(119895)

119896=

1

120596119896

arccos(11988711205962(1205962minus 1198861) minus (119886

21205962minus 1198860) (11988721205962minus 1198870)

(11988721205962 minus 119887

0)2

+ 11988711205962

)

+2119895120587 119896 = 1 2 3 119895 = 0 1

(14)

then plusmn119894120596119896is a pair of purely imaginary roots of (9) with 120591

(119895)

119896

Define

1205910= 120591(0)

1198960

= min119896isin123

120591(0)

119896 120596

0= 1205961198960

(15)

Note that when 120591 = 0 (5) becomes

1205823+ (1198862+ 1198872) 1205822+ (1198861+ 1198871) 120582 + 119886

0+ 1198870= 0 (16)

Therefore applying Lemmas 1 and 2 to (5) we obtain thefollowing lemma

Lemma 3 For the third-degree transcendental equation (5)one has the following results

(i) If 119903 ge 0 and 998779 = 1199012minus 3119902 ⩽ 0 then all roots with

positive real parts of (5) have the same sum to those ofthe polynomial equation (16) for all 120591 ge 0

(ii) If either 119903 lt 0 or 119903 ge 0 and 998779 = 1199012minus 3119902 gt 0 119911

lowast

1gt 0

and ℎ(119911lowast

1) lt 0 then all roots with positive real parts

of (5) have the same sum to those of the polynomialequation (16) for 120591 isin [0 120591

0)

Let 120582(120591) = 120572(120591) + 119894120596(120591) be the root of (9) near 120591 = 120591(119895)

119896

satisfying

120572 (120591(119895)

119896) = 0 120596 (120591

(119895)

119896) = 120596119896 (17)

Then it is easy to verify the following transversality condition

Lemma 4 Suppose that 119911119896= 1205962

119896and ℎ1015840(119911

119896) = 0 where ℎ(119911) is

defined by (15) Then 119889(Re120582(120591(119895)119896))119889120591 = 0 and 119889(Re120582(120591(119895)

119896))

119889120591 and ℎ1015840(119911119896) have the same sign

Now we study the characteristic equation (5) of system(4) Applying Lemmas 3 and 4 to (5) we have the followingtheorem

Theorem 5 Let 120591(119895)119896

and 1205960 1205910be defined by (14) and (15)

respectively Then consider the following

(i) If 119903 ge 0 and 998779 = 1199012minus 3119902 le 0 then all roots with

positive real parts of (5) have the same sum to those ofthe polynomial equation (16) for all 120591 ge 0

(ii) If either 119903 lt 0 or 119903 ge 0 and 998779 = 1199012minus 3119902 gt 0 119911lowast

1gt 0

and ℎ(119911lowast

1) lt 0 then ℎ(119911) has at least one positive root

119911119896 and all roots with positive real parts of (5) have the

same sum to those of the polynomial equation (16) for120591 isin [0 120591

0)

(iii) If the conditions of (ii) are satisfied and ℎ1015840(119911119896) = 0 then

system (2) exhibits Hopf bifurcation at the equilibrium1198640for 120591 = 120591

(119895)

119896

3 Stability and Direction ofBifurcating Periodic Orbits

In the previous section we obtain the conditions underwhichfamily periodic solutions bifurcate from the equilibrium119864

0at

the critical value of 120591 As pointed by in Hassard et al [17] it isinteresting to determine the direction stability and period ofthese periodic solutions Following the ideal of Hassard et alwe derive the explicit formulae for determining the propertiesof the Hopf bifurcation at the critical value of 120591 using thenormal form and the centermanifold theoryThroughout thissection we always assume that system (2) undergoes Hopfbifurcations at the equilibrium 119864

0for 120591 = 120591

119896 and then plusmn119894120596

119896is

corresponding to purely imaginary roots of the characteristicequation at the equilibrium 119864

0

Letting 120591 = 120591119905 120591 = 120591119896+ 120583 system (2) is transformed into

an FDE in 119862 = 119862([minus1 0] 1198773) as

= 119871120583(119906119905) + 119891 (120583 119906

119905) (18)

where 119906(119905) = (119909(119905) 119910(119905) 119911(119905))119879isin 1198773 and 119871

120583 119862 rarr 119877 119891

119877 times 119862 rarr 119877 are given respectively by

119871120583(120601) = (120591

119896+ 120583)(

119886 minus1 0

1 119870 minus 119887 minus1

119886 0 minus8

)(

1206011(0)

1206012(0)

1206013(0)

)

+ (120591119896+ 120583)(

0 0 0

0 minus119870 0

0 0 0

)(

1206011(minus1)

1206012(minus1)

1206013 (minus1)

)

(19)

119891 (120591 120601) = (120591119896+ 120583)(

0

0

41206012(0) 1206013(0)

) (20)

4 Abstract and Applied Analysis

By the Riesz representation theorem there exists a function120578(120579 120583) of bounded variation for 120579 isin [minus1 0] such that

119871120583120601 = int

0

minus1

119889120578 (120579 0) 120601 (120579) for120601 isin 119862 (21)

In fact we can choose

120578 (120579 120583) = (120591119896+ 120583)(

119886 minus1 0

1 119870 minus 119887 minus1

119886 0 minus8

)120575 (120579)

minus (120591119896+ 120583)(

0 0 0

0 minus119870 0

0 0 0

)120575 (120579 + 1)

(22)

where 120575 is Dirac-delta functionFor 120601 isin 119862

1([minus1 0] 119877

3) define

119860 (120583) 120601 =

119889120601 (120579)

119889120579 120579 isin [minus1 0)

int

0

minus1

119889120578 (120583 119904) 120601 (119904) 120579 = 0

119877 (120583) 120601 =

0 120579 isin [minus1 0)

119891 (120583 120601) 120579 = 0

(23)

Then when 120579 = 0 system (18) is equivalent to

119905= 119860 (120583) 119906

119905+ 119877 (120583) 119906

119905 (24)

where 119906119905(120579) = 119906(119905 + 120579) for 120579 isin [minus1 0] For 120595 isin 119862

1([0 1]

(1198773)lowast) define

119860lowast120595 (119904) =

minus119889120595 (119904)

119889119904 119904 isin (0 1]

int

0

minus1

119889120578119879(119905 0) 120595 (minus119905) 119904 = 0

(25)

and a bilinear inner product

⟨120595 (119904) 120601 (120579)⟩ = 120595 (0) 120601 (0)

minus int

0

minus1

int

120579

120585=0

120595 (120585 minus 120579) 119889120578 (120579) 120601 (120585) 119889120585

(26)

where 120578(120579) = 120578(120579 0) Then 119860 = 119860(0) and 119860lowast are adjoint

operators By the discussion in Section 2 we know thatplusmn119894120596119896120591119896are eigenvalues of 119860 Thus they are also eigenvalues

of 119860lowast We first need to compute the eigenvector of 119860 and 119860lowast

corresponding to 119894120596119896120591119896andminus119894120596

119896120591119896 respectively Suppose that

119902(120579) = (1 120572 120573)119879119890119894120579120596119896120591119896 is the eigenvector of 119860 corresponding

to 119894120596119896120591119896 Then 119860119902(120579) = 119894120596

119896120591119896119902(120579) It follows from the

definition of 119860 and (19) (21) and (22) that

120591119896(

119894120596119896minus 119886 1 0

minus1 119894120596119896minus 119870 + 119887 + 119870119890

minus119894120596119896120591119896 1

minus119886 0 119894120596119896+ 8

)119902 (0)

= (

0

0

0

)

(27)

Thus we can easily get

119902 (0) = (1 120572 120573)119879= (1 119886 minus 119894120596

119896

119886

119894120596119896+ 8

)

119879

(28)

Similarly let 119902lowast(119904) = 119863(1 120572lowast 120573lowast)119890119894119904120596119896120591119896 be the eigenvector of

119860lowast corresponding to minus119894120596

119896120591119896 By the definition of119860lowast and (19)

(21) and (22) we can compute

119902lowast(119904) = 119863 (1 120572

lowast 120573lowast) 119890119894119904120596119896120591119896

= 119863(1minus (119894120596119896+ 119886) (119894120596

119896minus 8)

119894120596119896minus 8 + 119886

minus119894120596119896minus 119886

119894120596119896minus 8 + 119886

) 119890119894119904120596119896120591119896

(29)

In order to assure ⟨119902lowast(119904) 119902(120579)⟩ = 1 we need to determine thevalue of119863 From (26) we have

⟨119902lowast(119904) 119902 (120579)⟩ = 119863(1 120572

lowast 120573lowast

) (1 120572 120573)119879

minus int

0

minus1

int

120579

120585=0

119863(1 120572lowast 120573lowast

) 119890minus119894(120585minus120579)120596

119896120591119896119889120578

times (120579) (1 120572 120573)119879119890119894120585120596119896120591119896119889120585

= 1198631 + 120572120572lowast+ 120573120573lowast

minus int

0

minus1

(1 120572lowast 120573lowast

) 120579119890119894120579120596119896120591119896119889120578

times (120579) (1 120572 120573)119879

= 1198631 + 120572120572lowast+ 120573120573lowast

minus 119870120591119896120572120572lowast119890minus119894120596119896120591119896

(30)

Thus we can choose119863 as

119863 =1

1 + 120572120572lowast+ 120573120573lowast

minus 119870120591119896120572120572lowast119890119894120596119896120591119896

(31)

In the following we first compute the coordinates to describethe center manifold 119862

0at 120583 = 0 Let 119906

119905be the solution of (18)

when 120583 = 0 Define

119911 (119905) = ⟨119902lowast 119906119905⟩ 119882 (119905 120579) = 119906

119905(120579) minus 2Re 119911 (119905) 119902 (120579)

(32)

On the center manifold 1198620 we have

119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)

= 11988220(120579)

1199112

2+11988211(120579) 119911119911

+11988202 (120579)

1199112

2+11988230 (120579)

1199113

6+ sdot sdot sdot

(33)

where 119911 and 119911 are local coordinates for center manifold 1198620

in the direction of 119902 and 119902lowast Note that 119882 is real if 119906

119905is real

Abstract and Applied Analysis 5

We consider only real solutions For the solution 119906119905isin 1198620of

(18) since 120583 = 0 we have

= 119894120596119896120591119896119911

+ ⟨119902lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579) + 2Re 119911 (119905) 119902 (120579))⟩

= 119894120596119896120591119896119911

+ 119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) + 2Re 119911 (119905) 119902 (0))

≜ 119894120596119896120591119896119911 + 119902lowast(0) 1198910(119911 119911) = 119894120596

119896120591119896119911 + 119892 (119911 119911)

(34)

where

119892 (119911 119911) = 119902lowast(0) 1198910 (119911 119911)

= 11989220(120579)

1199112

2+ 11989211(120579) 119911119911 + 119892

02(120579)

1199112

2+ sdot sdot sdot

(35)

By (32) we have 119906119905(120579) = (119906

1119905(120579) 1199062119905(120579) 1199063119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199061119905 (0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

1199062119905 (0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2+119882(2)

11(0) 119911119911

+119882(2)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

1199063119905 (0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

(36)

It follows together with (20) that

119892 (119911 119911) = 119902lowast(0) 1198910 (119911 119911)

= 119863120591119896(1 120572lowast 120573lowast

)(

0

0

41199062119905(0) 1199063119905(0)

)

= 4119863120591119896120573lowast

[119911120572 + 119911 120572 +119882(2)

20(0)

1199112

2+119882(2)

11(0) 119911119911

+119882(2)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)]

times [119911120573 + 119911120573 +119882(3)

20(0)

1199112

2+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)]

(37)

Comparing the coefficients with (35) we have

11989220

= 8119863120591119896120573lowast

120572120573

11989211

= 8119863120591119896120573lowast

(120572120573 + 120572120573)

11989202

= 8119863120591119896120573lowast

120572120573

11989221

= 4119863120591119896120573lowast

[2120572119882(3)

11(0) + 120572119882

(3)

20(0)

+120573119882(2)

20(0) + 2120573119882

(2)

11(0)]

(38)

In order to determine 11989221 in the sequel we need to compute

11988220(120579) and119882

11(120579) From (24) and (32) we have

= 119905minus 119902 minus 119902

=

119860119882 minus 2Re 119902lowast (0) 1198910119902 (120579) 120579 isin [0 1)

119860119882 minus 2Re 119902lowast (0) 1198910119902 (120579) + 1198910 120579 = 0

≜ 119860119882 +119867 (119911 119911 120579)

(39)

where

119867(119911 119911 120579) = 11986720 (120579)

1199112

2+ 11986711 (120579) 119911119911 + 119867

02 (120579)1199112

2+ sdot sdot sdot

(40)

Notice that near the origin on the centermanifold1198620 we have

= 119882119911 + 119882

119911 (41)

Thus we have

(119860 minus 2119894120591119896120596119896119868)11988220 (120579) = minus119867

20 (120579)

11986011988211(120579) = minus119867

11(120579)

(42)

Since (39) for 120579 isin [minus1 0) we have

119867(119911 119911 120579) = minus 119902lowast(0) 1198910119902 (120579) minus 119902

lowast(0) 1198910119902 (120579)

= minus 119892119902 (120579) minus 119892 119902 (120579)

(43)

Comparing the coefficients with (40) gives that

11986720(120579) = minus119892

20119902 (120579) minus 119892

02119902 (120579)

11986711 (120579) = minus119892

11119902 (120579) minus 119892

11119902 (120579)

(44)

From (43) (46) and the definition of 119860 we can get

20(120579) = 2119894120591

11989612059611989611988220(120579) + 119892

20119902 (120579) + 119892

02119902 (120579) (45)

Noticing that 119902(120579) = 119902(0)119890119894120591119896120596119896120579 we have

11988220(120579) =

11989411989220

120591119896120596119896

119902 (0) 119890119894120591119896120596119896120579+

11989411989202

3120591119896120596119896

119902 (0) 119890minus119894120591119896120596119896120579

+ 11986411198902119894120591119896120596119896120579

(46)

6 Abstract and Applied Analysis

051

152

253

Max(119910)

0 5 10 15 20120591

minus1

0minus05

Figure 2 The bifurcation graph with 120591 when 119870 = minus1

where 1198641= (119864(1)

1 119864(2)

1 119864(3)

1) isin 1198773 is a constant vector In the

same way we can also obtain

11988211 (120579) = minus

11989411989211

120591119896120596119896

119902 (0) 119890119894120591119896120596119896120579+

11989411989211

120591119896120596119896

119902 (0) 119890minus119894120591119896120596119896120579+ 1198642 (47)

where 1198642= (119864(1)

2 119864(2)

2 119864(3)

2) isin 1198773 is also a constant vector

In what follows we will seek appropriate 1198641and 119864

2 From

the definition of 119860 and (42) we obtain

int

0

minus1

119889120578 (120579)11988220(120579) = 2119894120591

11989612059611989611988220(0) minus 119867

20(0) (48)

int

0

minus1

119889120578 (120579)11988211 (120579) = minus11986711 (0) (49)

where 120578(120579) = 120578(0 120579) From (39) and (40) we have

11986720(0) = minus119892

20119902 (0) minus 119892

02119902 (0) + 2120591

119896(

0

0

4120572120573

) (50)

11986711 (0) = minus119892

11119902 (0) minus 119892

11119902 (0) + 2120591

119896(

0

0

4Re 120572120573) (51)

Substituting (46) and (50) into (48) and noticing that

(119894120596119896120591119896119868 minus int

0

minus1

119890119894120596119896120591119896120579119889120578 (120579)) 119902 (0) = 0

(minus119894120596119896120591119896119868 minus int

0

minus1

119890minus119894120596119896120591119896120579119889120578 (120579)) 119902 (0) = 0

(52)

we obtain

(2119894120596119896120591119896119868 minus int

0

minus1

1198902119894120596119896120591119896120579119889120578 (120579))119864

1= 2120591119896(

0

0

4120572120573

) (53)

which leads to

(

2119894120596119896minus 119886 1 0

minus1 2119894120596119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896 1

minus119886 0 2119894120596119896+ 8

)1198641

= 2(

0

0

4120572120573

)

(54)

It follows that

119864(1)

1=

8

119860120572120573

119864(2)

1=

8

119860120572120573 (2119894120596

119896minus 119886)

119864(3)

1=

8

119860120572120573 [(2119894120596

119896minus 119886) (2119894120596

119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896) + 1]

(55)

where119860 = (2119894120596

119896+ 8)

times [(2119894120596119896minus 119886) (2119894120596

119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896) + 1] minus 119886

(56)

Similarly substituting (47) and (51) into (49) we can get

(

minus119886 1 0

minus1 119887 1

minus119886 0 8

)1198642= 2(

0

0

4Re 120572120573) (57)

Thus we have

119864(1)

2=

8

119861Re 120572120573

119864(2)

2=

8

119861Re 120572120573

119864(3)

2=

8

119861Re 120572120573 (1 minus 119886119887)

(58)

where

119861 = 8 minus 119886 minus 8119886119887 (59)

Thus we can determine11988220(120579) and119882

11(120579) Furthermore we

can determine each 119892119894119895 Therefore each 119892

119894119895is determined by

the parameters and delay in (2) Thus we can compute thefollowing values

1198881(0) =

119894

2120596119896120591119896

(1198922011989211minus 2

1003816100381610038161003816119892111003816100381610038161003816

2minus1

3

1003816100381610038161003816119892021003816100381610038161003816

2) +

11989221

2

1205832= minus

Re 1198881(0)

Re 1205821015840 (120591119896)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(120591119896)

120596119896120591119896

1205732= 2Re 119888

1(0)

(60)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

119896 that is 120583

2

determines the directions of theHopf bifurcation if1205832gt 0 (lt

0) then theHopf bifurcation is supercritical (subcritical) andthe bifurcation exists for 120591 gt 120591

0(lt 1205910) 1205732determines the

stability of the bifurcation periodic solutions the bifurcatingperiodic solutions are stable (unstable) if 120573

2lt 0 (gt 0) and119879

2

determines the period of the bifurcating periodic solutionsthe period increases (decreases) if 119879

2gt 0 (lt 0)

Abstract and Applied Analysis 7

0 24

4

0

0

051

152

2

25

0 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

119909

119909

minus5

05

minus5

5

minus5

119910

119910

119911

119911

minus05

minus2 minus2minus4 minus4

119905

119905

119905

Figure 3 The trajectories and phase graphs of system (61) with 120591 = 07362 when 119870 = minus1

02

40

24

002

0406

080 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

0

119909

minus5

5

minus5

1

minus1

119910

119910

119911

119911

minus02

minus2 minus2minus4 minus4

119905

119905

119905

119909

Figure 4 The trajectories and phase graphs of system (61) with 120591 = 13497 when 119870 = minus1

4 Numerical Simulations andApplication to Control of Chaos

In this section we apply the results we get in the previoussections to system (61) for the purpose of control of chaosFrom Section 2 we know that under certain conditions afamily of periodic solutions bifurcates from the steady statesof system (61) at some critical values of 120591 and the stability ofthe steady state may be changed along with the increase of 120591If the bifurcating periodic solution is orbitally asymptoticallystable or some steady state becomes local stable then chaosmay vanish Following this idea we consider the followingdelayed feedback control system for an autonomy system

= 02119909 minus 119910

119910 = 119909 minus 002119910 minus 119911 + 119870 (119910 (119905) minus 119910 (119905 minus 120591))

= 02119909 + 4119911 (119910 minus 2)

(61)

which has two equilibriums 1198640(0 0 0) and 119864lowast(9749 19498

9710) When 120591 = 0 or 119870 = 0 1198640is unstable and system

(61) is chaotic (see Figure 1)The corresponding characteristicequation of system (61) at 119864

0is

1205823+ (782 minus 119870) 120582

2minus (0444 + 78119870) 120582 + 7768

+ 16119870 + (1205822+ 78120582 minus 16)119870119890

minus120582120591= 0

(62)

When 120591 = 0 (62) has one negative root and a pair of complexroots with positive parts By the discussion of Section 2we can get 119901 = 620404 minus 004119870 119902 = minus1212944 minus

25616119870 and 119903 = 603418 + 248576119870 and then we cancompute 998779 = 42129 + 27216119870 + 00016119870

2 and it is easyto know that 998779 gt 0 and when 119870 ge minus24275 119911lowast

1gt 0 always

holdsThen byTheorem 5 we have the following conclusion

Conclusion 1 If 119870 lt minus24275 or 119870 ge minus24275 and ℎ(119911lowast

1) lt 0

then there exist some 1205910gt 0 such that when 120591 isin [0 120591

0) 1198640is

always unstable

For example we fix119870 = minus1 Through computing we canget 119911lowast1= 10693 gt 0 and ℎ(119911lowast

1) = minus192713 lt 0 Furthermore

we can compute

8 Abstract and Applied Analysis

0 50 100 150 200

05

02

001 0 1 2

01

2

0

00501

0 50 100 150 200

0 50 100 150 200

119909

119909

minus5119910

119910

minus2

119911

119911

minus01

minus01

minus1 minus1

minus2minus2

minus3

minus005

119905

119905

119905

Figure 5 The trajectories and phase graphs of system (61) with 120591 = 26815 when 119870 = minus1

005

115

20 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

119911

minus05

4

02

119910 minus2

minus4

02

4

119909

minus2minus4

119905

119905

119905

Figure 6 The trajectories and phase graphs of system (61) with 120591 = 63804 when 119870 = minus1

1199111≐ 03714185849 120596

1≐ 06094

120591(119895)

1= 16243 +

2119895120587

1205961

ℎ1015840(1199111) = minus713282

1199112≐ 1494030027 120596

2≐ 12223

120591(119895)

2= 03768 +

2119895120587

1205962

ℎ1015840(1199112) = 725266

(63)

Thus from Lemma 4 we have

Re 120582 (120591(119895)1)

119889120591lt 0

Re 120582 (120591(119895)2)

119889120591gt 0

(64)

In addition notice that

120591(0)

2= 03768 lt 120591

(0)

1= 16243 lt 120591

(1)

2

= 55173 lt 120591(2)

2= 106577 lt 120591

(1)

1

= 119340 lt 120591(3)

2= 157982 lt sdot sdot sdot

(65)

Thus from (64) and (65) we have the following conclusionabout the stability of the zero equilibrium of system (61) andHopf bifurcations

Conclusion 2 Suppose that 120591(119895)119896 119896 = 1 2 119895 = 0 1 is

defined by (63)

(i) When 120591 isin [0 120591(0)

1) cup (120591(1)

2 +infin) the equilibrium 119864

0is

unstable (see Figures 3 4 6 7 and 8)(ii) When 120591 isin (120591

(0)

1 120591(1)

2) the equilibrium 119864

0is asymptot-

ically stable (see Figure 5)(iii) System (61) undergoes a Hopf bifurcation at the

equilibrium 1198640when 120591 = 120591

(119895)

119896(see Figures 3 4

and 6)

Thus if we take 120591 isin (120591(0)

1 120591(1)

2) the stability of zero

equilibrium of the systemwill change from unstable to stableFor this example there are more complicated dynamicalphenomena occurs Bifurcation diagram is plotted with theparameter 120591 when 119870 = minus1 (Figure 2) In Figure 2 we can seethat chaos vanishes via a period-doubling bifurcation when120591 varying from 0 to 03768 Figure 3 shows that when 120591 =

07362 a period-2 orbits bifurcating from zero equilibriumFigure 4 shows when 120591 = 13497 a period-1 periodicalorbits bifurcating from zero equilibrium In Figure 2 a stablewindow is obtained from 120591 = 16243 to 120591 = 55173 which isdemonstrated by Figure 5 with 120591 = 26815 When 120591 gt 55173Figure 2 shows that periodic orbits and strange attractors also

Abstract and Applied Analysis 9

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

005

115

2

119911

minus05

4

02

119910 minus2

minus4

02

4

119909minus2

minus4

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

119905

119905

119905

Figure 7 The trajectories and phase graphs of system (61) with 120591 = 73006 when 119870 = minus1

05

119909

minus5

05

minus5

05

minus5

119910119911

119911

4

02

119910 minus2

minus4

02

4

119909minus2minus4

minus1

012345

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

119905

119905

119905

Figure 8 The trajectories and phase graphs of system (61) with 120591 = 137423 when 119870 = minus1

occur and system (61) through Hopf bifurcation route stillgoes to chaos and it is demonstrated by Figures 6ndash8

5 Discussion

In this paper an autonomy system with time-delayed feed-back is studied by using the theory of functional differentialequation and Hassardrsquos method Illustrating with numericalsimulations we show that delayed feedback controller (DFC)is an effective method for chaos control

In fact system (2) has more complicated dynamicalbehaves From Figure 2 we can see that chaos degenerationprocess the orbits of system (2) continuously degenerate toa periodic solution region through reverse period-doublingbifurcation With the increasing of 120591 chaos occurs againthrough Hopf bifurcation route However the numericalsimulations indicate that there were points of similaritybetween this route to chaos and quasiperiodicity route tochaos [18] in our future research we will consider this topicAnd in this paper we only give detailed analysis on thetrivial equilibrium 119864

0 the detailed analysis of the interior

equilibrium 119864lowast is the second interested topic in our future

research

Acknowledgments

The authors would like to thank the anonymous refereefor the very helpful suggestions and comments which ledto the improvement of the original paper And this workis supported by 2013 Scientific Research Project of BeifangUniversity of Nationalities (2013XYZ021) Institute of Infor-mation and System Computation Science of Beifang Uni-versity (13xyb01) Science and Technology Department ofHenan Province (122300410417) and Education Departmentof Henan Province (13A110108)

References

[1] G Corso and F B Rizzato ldquoHamiltonian bifurcations leading tochaos in low-energy relativistic wave- particle systemrdquo PhysicaD vol 80 pp 296ndash306 1995

[2] M Saleem and T Agrawal ldquoChaos in a tumor growth modelwith delayed responses of the immune systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 891095 16 pages2012

[3] U Kumar ldquoA retrospection of chaotic phenomene in electricalsystemsrdquo Active and Passive Electronic Components vol 21 pp1ndash15 1998

10 Abstract and Applied Analysis

[4] Z Chen X Zhang and Q Bi ldquoBifurcations and chaos ofcoupled electrical circuitsrdquo Nonlinear Analysis vol 9 no 3 pp1158ndash1168 2008

[5] S Hallegatte M Ghil P Dumas and J Hourcade ldquoBusinesscycles bifurcations and chaos in a neo-classical model withinvestment dynamicsrdquo Journal of Economic Behavior and Orga-nization vol 67 pp 57ndash77 2008

[6] DGhoshA R Chowdhury andP Saha ldquoBifurcation continua-tion chaos and chaos control in nonlinear Bloch systemrdquo Com-munications inNonlinear Science andNumerical Simulation vol13 no 8 pp 1461ndash1471 2008

[7] A El-Gohary andA S Al-Ruzaiza ldquoChaos and adaptive controlin two prey one predator system with nonlinear feedbackrdquoChaos Solitons and Fractals vol 34 no 2 pp 443ndash453 2007

[8] H Salarieh and A Alasty ldquoControl of stochastic chaos usingsliding mode methodrdquo Journal of Computational and AppliedMathematics vol 225 no 1 pp 135ndash145 2009

[9] J Guan ldquoBifurcation analysis and chaos control in genesiosystem with delayed feedbackrdquo ISRNMathematical Physics vol2012 Article ID 843962 12 pages 2012

[10] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquoNonlinear Analysis vol 9 no 4 pp 1800ndash18102008

[11] N Vasegh and A Sedigh ldquoDelayed feedback control of time-delayed chaotic systems analytical approach at Hopf bifurca-tionrdquo Physics Letters A vol 372 pp 5110ndash5114 2008

[12] J Lu Z Ma and L Li ldquoDouble delayed feedback control for thestabilization of unstable steady states in chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 7 pp 3037ndash3045 2009

[13] M Bleich and J Socolar ldquoStability of periodic orbits controlledby time-delay feedbackrdquo Physics Letters A vol 210 pp 87ndash941996

[14] Y Ding W Jiang and H Wang ldquoDelayed feedback controland bifurcation analysis of Rossler chaotic systemrdquo NonlinearDynamics vol 61 no 4 pp 707ndash715 2010

[15] Y Song and J Wei ldquoBifurcation analysis for Chenrsquos system withdelayed feedback and its application to control of chaosrdquoChaosSolitons and Fractals vol 22 no 1 pp 75ndash91 2004

[16] S Ruan and J Wei ldquoOn the zeros of transcendental functionswith applications to stability of delay differential equations withtwo delaysrdquo Dynamics of Continuous Discrete and ImpulsiveSystems A vol 10 no 6 pp 863ndash874 2003

[17] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge UK 1981

[18] I Fuente L Martinez J Veguillas and J AguirregabirialdquoQuasiperiodicity route to chaos in a biochemical systemrdquoBiophysical Journal vol 71 pp 2375ndash2379 1998

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

Abstract and Applied Analysis 3

Lemma 2 For the polynomial equation (10) one has thefollowing results

(i) If 119903 lt 0 then (10) has at least one positive root(ii) If 119903 ge 0 and 998779 = 119901

2minus 3119902 le 0 then (10) has no positive

roots(iii) If 119903 ge 0 and998779 = 119901

2minus3119902 gt 0 then (10) has positive roots

if and only if 119911lowast1= (13)(minus119901 +radic998779) gt 0 and ℎ(119911lowast

1) le 0

Suppose that (10) has positive roots without loss ofgenerality we assume that it has three positive roots definedby 1199111 1199112 and 119911

3 respectively Then (9) has three positive

roots

1205961= radic1199111 120596

2= radic1199112 120596

3= radic1199113 (12)

Thus we have

cos120596120591 =

11988711205962(1205962minus 1198861) minus (119886

21205962minus 1198860) (11988721205962minus 1198870)

(11988721205962 minus 119887

0)2+ 11988711205962

(13)

If we denote

120591(119895)

119896=

1

120596119896

arccos(11988711205962(1205962minus 1198861) minus (119886

21205962minus 1198860) (11988721205962minus 1198870)

(11988721205962 minus 119887

0)2

+ 11988711205962

)

+2119895120587 119896 = 1 2 3 119895 = 0 1

(14)

then plusmn119894120596119896is a pair of purely imaginary roots of (9) with 120591

(119895)

119896

Define

1205910= 120591(0)

1198960

= min119896isin123

120591(0)

119896 120596

0= 1205961198960

(15)

Note that when 120591 = 0 (5) becomes

1205823+ (1198862+ 1198872) 1205822+ (1198861+ 1198871) 120582 + 119886

0+ 1198870= 0 (16)

Therefore applying Lemmas 1 and 2 to (5) we obtain thefollowing lemma

Lemma 3 For the third-degree transcendental equation (5)one has the following results

(i) If 119903 ge 0 and 998779 = 1199012minus 3119902 ⩽ 0 then all roots with

positive real parts of (5) have the same sum to those ofthe polynomial equation (16) for all 120591 ge 0

(ii) If either 119903 lt 0 or 119903 ge 0 and 998779 = 1199012minus 3119902 gt 0 119911

lowast

1gt 0

and ℎ(119911lowast

1) lt 0 then all roots with positive real parts

of (5) have the same sum to those of the polynomialequation (16) for 120591 isin [0 120591

0)

Let 120582(120591) = 120572(120591) + 119894120596(120591) be the root of (9) near 120591 = 120591(119895)

119896

satisfying

120572 (120591(119895)

119896) = 0 120596 (120591

(119895)

119896) = 120596119896 (17)

Then it is easy to verify the following transversality condition

Lemma 4 Suppose that 119911119896= 1205962

119896and ℎ1015840(119911

119896) = 0 where ℎ(119911) is

defined by (15) Then 119889(Re120582(120591(119895)119896))119889120591 = 0 and 119889(Re120582(120591(119895)

119896))

119889120591 and ℎ1015840(119911119896) have the same sign

Now we study the characteristic equation (5) of system(4) Applying Lemmas 3 and 4 to (5) we have the followingtheorem

Theorem 5 Let 120591(119895)119896

and 1205960 1205910be defined by (14) and (15)

respectively Then consider the following

(i) If 119903 ge 0 and 998779 = 1199012minus 3119902 le 0 then all roots with

positive real parts of (5) have the same sum to those ofthe polynomial equation (16) for all 120591 ge 0

(ii) If either 119903 lt 0 or 119903 ge 0 and 998779 = 1199012minus 3119902 gt 0 119911lowast

1gt 0

and ℎ(119911lowast

1) lt 0 then ℎ(119911) has at least one positive root

119911119896 and all roots with positive real parts of (5) have the

same sum to those of the polynomial equation (16) for120591 isin [0 120591

0)

(iii) If the conditions of (ii) are satisfied and ℎ1015840(119911119896) = 0 then

system (2) exhibits Hopf bifurcation at the equilibrium1198640for 120591 = 120591

(119895)

119896

3 Stability and Direction ofBifurcating Periodic Orbits

In the previous section we obtain the conditions underwhichfamily periodic solutions bifurcate from the equilibrium119864

0at

the critical value of 120591 As pointed by in Hassard et al [17] it isinteresting to determine the direction stability and period ofthese periodic solutions Following the ideal of Hassard et alwe derive the explicit formulae for determining the propertiesof the Hopf bifurcation at the critical value of 120591 using thenormal form and the centermanifold theoryThroughout thissection we always assume that system (2) undergoes Hopfbifurcations at the equilibrium 119864

0for 120591 = 120591

119896 and then plusmn119894120596

119896is

corresponding to purely imaginary roots of the characteristicequation at the equilibrium 119864

0

Letting 120591 = 120591119905 120591 = 120591119896+ 120583 system (2) is transformed into

an FDE in 119862 = 119862([minus1 0] 1198773) as

= 119871120583(119906119905) + 119891 (120583 119906

119905) (18)

where 119906(119905) = (119909(119905) 119910(119905) 119911(119905))119879isin 1198773 and 119871

120583 119862 rarr 119877 119891

119877 times 119862 rarr 119877 are given respectively by

119871120583(120601) = (120591

119896+ 120583)(

119886 minus1 0

1 119870 minus 119887 minus1

119886 0 minus8

)(

1206011(0)

1206012(0)

1206013(0)

)

+ (120591119896+ 120583)(

0 0 0

0 minus119870 0

0 0 0

)(

1206011(minus1)

1206012(minus1)

1206013 (minus1)

)

(19)

119891 (120591 120601) = (120591119896+ 120583)(

0

0

41206012(0) 1206013(0)

) (20)

4 Abstract and Applied Analysis

By the Riesz representation theorem there exists a function120578(120579 120583) of bounded variation for 120579 isin [minus1 0] such that

119871120583120601 = int

0

minus1

119889120578 (120579 0) 120601 (120579) for120601 isin 119862 (21)

In fact we can choose

120578 (120579 120583) = (120591119896+ 120583)(

119886 minus1 0

1 119870 minus 119887 minus1

119886 0 minus8

)120575 (120579)

minus (120591119896+ 120583)(

0 0 0

0 minus119870 0

0 0 0

)120575 (120579 + 1)

(22)

where 120575 is Dirac-delta functionFor 120601 isin 119862

1([minus1 0] 119877

3) define

119860 (120583) 120601 =

119889120601 (120579)

119889120579 120579 isin [minus1 0)

int

0

minus1

119889120578 (120583 119904) 120601 (119904) 120579 = 0

119877 (120583) 120601 =

0 120579 isin [minus1 0)

119891 (120583 120601) 120579 = 0

(23)

Then when 120579 = 0 system (18) is equivalent to

119905= 119860 (120583) 119906

119905+ 119877 (120583) 119906

119905 (24)

where 119906119905(120579) = 119906(119905 + 120579) for 120579 isin [minus1 0] For 120595 isin 119862

1([0 1]

(1198773)lowast) define

119860lowast120595 (119904) =

minus119889120595 (119904)

119889119904 119904 isin (0 1]

int

0

minus1

119889120578119879(119905 0) 120595 (minus119905) 119904 = 0

(25)

and a bilinear inner product

⟨120595 (119904) 120601 (120579)⟩ = 120595 (0) 120601 (0)

minus int

0

minus1

int

120579

120585=0

120595 (120585 minus 120579) 119889120578 (120579) 120601 (120585) 119889120585

(26)

where 120578(120579) = 120578(120579 0) Then 119860 = 119860(0) and 119860lowast are adjoint

operators By the discussion in Section 2 we know thatplusmn119894120596119896120591119896are eigenvalues of 119860 Thus they are also eigenvalues

of 119860lowast We first need to compute the eigenvector of 119860 and 119860lowast

corresponding to 119894120596119896120591119896andminus119894120596

119896120591119896 respectively Suppose that

119902(120579) = (1 120572 120573)119879119890119894120579120596119896120591119896 is the eigenvector of 119860 corresponding

to 119894120596119896120591119896 Then 119860119902(120579) = 119894120596

119896120591119896119902(120579) It follows from the

definition of 119860 and (19) (21) and (22) that

120591119896(

119894120596119896minus 119886 1 0

minus1 119894120596119896minus 119870 + 119887 + 119870119890

minus119894120596119896120591119896 1

minus119886 0 119894120596119896+ 8

)119902 (0)

= (

0

0

0

)

(27)

Thus we can easily get

119902 (0) = (1 120572 120573)119879= (1 119886 minus 119894120596

119896

119886

119894120596119896+ 8

)

119879

(28)

Similarly let 119902lowast(119904) = 119863(1 120572lowast 120573lowast)119890119894119904120596119896120591119896 be the eigenvector of

119860lowast corresponding to minus119894120596

119896120591119896 By the definition of119860lowast and (19)

(21) and (22) we can compute

119902lowast(119904) = 119863 (1 120572

lowast 120573lowast) 119890119894119904120596119896120591119896

= 119863(1minus (119894120596119896+ 119886) (119894120596

119896minus 8)

119894120596119896minus 8 + 119886

minus119894120596119896minus 119886

119894120596119896minus 8 + 119886

) 119890119894119904120596119896120591119896

(29)

In order to assure ⟨119902lowast(119904) 119902(120579)⟩ = 1 we need to determine thevalue of119863 From (26) we have

⟨119902lowast(119904) 119902 (120579)⟩ = 119863(1 120572

lowast 120573lowast

) (1 120572 120573)119879

minus int

0

minus1

int

120579

120585=0

119863(1 120572lowast 120573lowast

) 119890minus119894(120585minus120579)120596

119896120591119896119889120578

times (120579) (1 120572 120573)119879119890119894120585120596119896120591119896119889120585

= 1198631 + 120572120572lowast+ 120573120573lowast

minus int

0

minus1

(1 120572lowast 120573lowast

) 120579119890119894120579120596119896120591119896119889120578

times (120579) (1 120572 120573)119879

= 1198631 + 120572120572lowast+ 120573120573lowast

minus 119870120591119896120572120572lowast119890minus119894120596119896120591119896

(30)

Thus we can choose119863 as

119863 =1

1 + 120572120572lowast+ 120573120573lowast

minus 119870120591119896120572120572lowast119890119894120596119896120591119896

(31)

In the following we first compute the coordinates to describethe center manifold 119862

0at 120583 = 0 Let 119906

119905be the solution of (18)

when 120583 = 0 Define

119911 (119905) = ⟨119902lowast 119906119905⟩ 119882 (119905 120579) = 119906

119905(120579) minus 2Re 119911 (119905) 119902 (120579)

(32)

On the center manifold 1198620 we have

119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)

= 11988220(120579)

1199112

2+11988211(120579) 119911119911

+11988202 (120579)

1199112

2+11988230 (120579)

1199113

6+ sdot sdot sdot

(33)

where 119911 and 119911 are local coordinates for center manifold 1198620

in the direction of 119902 and 119902lowast Note that 119882 is real if 119906

119905is real

Abstract and Applied Analysis 5

We consider only real solutions For the solution 119906119905isin 1198620of

(18) since 120583 = 0 we have

= 119894120596119896120591119896119911

+ ⟨119902lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579) + 2Re 119911 (119905) 119902 (120579))⟩

= 119894120596119896120591119896119911

+ 119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) + 2Re 119911 (119905) 119902 (0))

≜ 119894120596119896120591119896119911 + 119902lowast(0) 1198910(119911 119911) = 119894120596

119896120591119896119911 + 119892 (119911 119911)

(34)

where

119892 (119911 119911) = 119902lowast(0) 1198910 (119911 119911)

= 11989220(120579)

1199112

2+ 11989211(120579) 119911119911 + 119892

02(120579)

1199112

2+ sdot sdot sdot

(35)

By (32) we have 119906119905(120579) = (119906

1119905(120579) 1199062119905(120579) 1199063119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199061119905 (0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

1199062119905 (0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2+119882(2)

11(0) 119911119911

+119882(2)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

1199063119905 (0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

(36)

It follows together with (20) that

119892 (119911 119911) = 119902lowast(0) 1198910 (119911 119911)

= 119863120591119896(1 120572lowast 120573lowast

)(

0

0

41199062119905(0) 1199063119905(0)

)

= 4119863120591119896120573lowast

[119911120572 + 119911 120572 +119882(2)

20(0)

1199112

2+119882(2)

11(0) 119911119911

+119882(2)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)]

times [119911120573 + 119911120573 +119882(3)

20(0)

1199112

2+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)]

(37)

Comparing the coefficients with (35) we have

11989220

= 8119863120591119896120573lowast

120572120573

11989211

= 8119863120591119896120573lowast

(120572120573 + 120572120573)

11989202

= 8119863120591119896120573lowast

120572120573

11989221

= 4119863120591119896120573lowast

[2120572119882(3)

11(0) + 120572119882

(3)

20(0)

+120573119882(2)

20(0) + 2120573119882

(2)

11(0)]

(38)

In order to determine 11989221 in the sequel we need to compute

11988220(120579) and119882

11(120579) From (24) and (32) we have

= 119905minus 119902 minus 119902

=

119860119882 minus 2Re 119902lowast (0) 1198910119902 (120579) 120579 isin [0 1)

119860119882 minus 2Re 119902lowast (0) 1198910119902 (120579) + 1198910 120579 = 0

≜ 119860119882 +119867 (119911 119911 120579)

(39)

where

119867(119911 119911 120579) = 11986720 (120579)

1199112

2+ 11986711 (120579) 119911119911 + 119867

02 (120579)1199112

2+ sdot sdot sdot

(40)

Notice that near the origin on the centermanifold1198620 we have

= 119882119911 + 119882

119911 (41)

Thus we have

(119860 minus 2119894120591119896120596119896119868)11988220 (120579) = minus119867

20 (120579)

11986011988211(120579) = minus119867

11(120579)

(42)

Since (39) for 120579 isin [minus1 0) we have

119867(119911 119911 120579) = minus 119902lowast(0) 1198910119902 (120579) minus 119902

lowast(0) 1198910119902 (120579)

= minus 119892119902 (120579) minus 119892 119902 (120579)

(43)

Comparing the coefficients with (40) gives that

11986720(120579) = minus119892

20119902 (120579) minus 119892

02119902 (120579)

11986711 (120579) = minus119892

11119902 (120579) minus 119892

11119902 (120579)

(44)

From (43) (46) and the definition of 119860 we can get

20(120579) = 2119894120591

11989612059611989611988220(120579) + 119892

20119902 (120579) + 119892

02119902 (120579) (45)

Noticing that 119902(120579) = 119902(0)119890119894120591119896120596119896120579 we have

11988220(120579) =

11989411989220

120591119896120596119896

119902 (0) 119890119894120591119896120596119896120579+

11989411989202

3120591119896120596119896

119902 (0) 119890minus119894120591119896120596119896120579

+ 11986411198902119894120591119896120596119896120579

(46)

6 Abstract and Applied Analysis

051

152

253

Max(119910)

0 5 10 15 20120591

minus1

0minus05

Figure 2 The bifurcation graph with 120591 when 119870 = minus1

where 1198641= (119864(1)

1 119864(2)

1 119864(3)

1) isin 1198773 is a constant vector In the

same way we can also obtain

11988211 (120579) = minus

11989411989211

120591119896120596119896

119902 (0) 119890119894120591119896120596119896120579+

11989411989211

120591119896120596119896

119902 (0) 119890minus119894120591119896120596119896120579+ 1198642 (47)

where 1198642= (119864(1)

2 119864(2)

2 119864(3)

2) isin 1198773 is also a constant vector

In what follows we will seek appropriate 1198641and 119864

2 From

the definition of 119860 and (42) we obtain

int

0

minus1

119889120578 (120579)11988220(120579) = 2119894120591

11989612059611989611988220(0) minus 119867

20(0) (48)

int

0

minus1

119889120578 (120579)11988211 (120579) = minus11986711 (0) (49)

where 120578(120579) = 120578(0 120579) From (39) and (40) we have

11986720(0) = minus119892

20119902 (0) minus 119892

02119902 (0) + 2120591

119896(

0

0

4120572120573

) (50)

11986711 (0) = minus119892

11119902 (0) minus 119892

11119902 (0) + 2120591

119896(

0

0

4Re 120572120573) (51)

Substituting (46) and (50) into (48) and noticing that

(119894120596119896120591119896119868 minus int

0

minus1

119890119894120596119896120591119896120579119889120578 (120579)) 119902 (0) = 0

(minus119894120596119896120591119896119868 minus int

0

minus1

119890minus119894120596119896120591119896120579119889120578 (120579)) 119902 (0) = 0

(52)

we obtain

(2119894120596119896120591119896119868 minus int

0

minus1

1198902119894120596119896120591119896120579119889120578 (120579))119864

1= 2120591119896(

0

0

4120572120573

) (53)

which leads to

(

2119894120596119896minus 119886 1 0

minus1 2119894120596119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896 1

minus119886 0 2119894120596119896+ 8

)1198641

= 2(

0

0

4120572120573

)

(54)

It follows that

119864(1)

1=

8

119860120572120573

119864(2)

1=

8

119860120572120573 (2119894120596

119896minus 119886)

119864(3)

1=

8

119860120572120573 [(2119894120596

119896minus 119886) (2119894120596

119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896) + 1]

(55)

where119860 = (2119894120596

119896+ 8)

times [(2119894120596119896minus 119886) (2119894120596

119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896) + 1] minus 119886

(56)

Similarly substituting (47) and (51) into (49) we can get

(

minus119886 1 0

minus1 119887 1

minus119886 0 8

)1198642= 2(

0

0

4Re 120572120573) (57)

Thus we have

119864(1)

2=

8

119861Re 120572120573

119864(2)

2=

8

119861Re 120572120573

119864(3)

2=

8

119861Re 120572120573 (1 minus 119886119887)

(58)

where

119861 = 8 minus 119886 minus 8119886119887 (59)

Thus we can determine11988220(120579) and119882

11(120579) Furthermore we

can determine each 119892119894119895 Therefore each 119892

119894119895is determined by

the parameters and delay in (2) Thus we can compute thefollowing values

1198881(0) =

119894

2120596119896120591119896

(1198922011989211minus 2

1003816100381610038161003816119892111003816100381610038161003816

2minus1

3

1003816100381610038161003816119892021003816100381610038161003816

2) +

11989221

2

1205832= minus

Re 1198881(0)

Re 1205821015840 (120591119896)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(120591119896)

120596119896120591119896

1205732= 2Re 119888

1(0)

(60)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

119896 that is 120583

2

determines the directions of theHopf bifurcation if1205832gt 0 (lt

0) then theHopf bifurcation is supercritical (subcritical) andthe bifurcation exists for 120591 gt 120591

0(lt 1205910) 1205732determines the

stability of the bifurcation periodic solutions the bifurcatingperiodic solutions are stable (unstable) if 120573

2lt 0 (gt 0) and119879

2

determines the period of the bifurcating periodic solutionsthe period increases (decreases) if 119879

2gt 0 (lt 0)

Abstract and Applied Analysis 7

0 24

4

0

0

051

152

2

25

0 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

119909

119909

minus5

05

minus5

5

minus5

119910

119910

119911

119911

minus05

minus2 minus2minus4 minus4

119905

119905

119905

Figure 3 The trajectories and phase graphs of system (61) with 120591 = 07362 when 119870 = minus1

02

40

24

002

0406

080 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

0

119909

minus5

5

minus5

1

minus1

119910

119910

119911

119911

minus02

minus2 minus2minus4 minus4

119905

119905

119905

119909

Figure 4 The trajectories and phase graphs of system (61) with 120591 = 13497 when 119870 = minus1

4 Numerical Simulations andApplication to Control of Chaos

In this section we apply the results we get in the previoussections to system (61) for the purpose of control of chaosFrom Section 2 we know that under certain conditions afamily of periodic solutions bifurcates from the steady statesof system (61) at some critical values of 120591 and the stability ofthe steady state may be changed along with the increase of 120591If the bifurcating periodic solution is orbitally asymptoticallystable or some steady state becomes local stable then chaosmay vanish Following this idea we consider the followingdelayed feedback control system for an autonomy system

= 02119909 minus 119910

119910 = 119909 minus 002119910 minus 119911 + 119870 (119910 (119905) minus 119910 (119905 minus 120591))

= 02119909 + 4119911 (119910 minus 2)

(61)

which has two equilibriums 1198640(0 0 0) and 119864lowast(9749 19498

9710) When 120591 = 0 or 119870 = 0 1198640is unstable and system

(61) is chaotic (see Figure 1)The corresponding characteristicequation of system (61) at 119864

0is

1205823+ (782 minus 119870) 120582

2minus (0444 + 78119870) 120582 + 7768

+ 16119870 + (1205822+ 78120582 minus 16)119870119890

minus120582120591= 0

(62)

When 120591 = 0 (62) has one negative root and a pair of complexroots with positive parts By the discussion of Section 2we can get 119901 = 620404 minus 004119870 119902 = minus1212944 minus

25616119870 and 119903 = 603418 + 248576119870 and then we cancompute 998779 = 42129 + 27216119870 + 00016119870

2 and it is easyto know that 998779 gt 0 and when 119870 ge minus24275 119911lowast

1gt 0 always

holdsThen byTheorem 5 we have the following conclusion

Conclusion 1 If 119870 lt minus24275 or 119870 ge minus24275 and ℎ(119911lowast

1) lt 0

then there exist some 1205910gt 0 such that when 120591 isin [0 120591

0) 1198640is

always unstable

For example we fix119870 = minus1 Through computing we canget 119911lowast1= 10693 gt 0 and ℎ(119911lowast

1) = minus192713 lt 0 Furthermore

we can compute

8 Abstract and Applied Analysis

0 50 100 150 200

05

02

001 0 1 2

01

2

0

00501

0 50 100 150 200

0 50 100 150 200

119909

119909

minus5119910

119910

minus2

119911

119911

minus01

minus01

minus1 minus1

minus2minus2

minus3

minus005

119905

119905

119905

Figure 5 The trajectories and phase graphs of system (61) with 120591 = 26815 when 119870 = minus1

005

115

20 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

119911

minus05

4

02

119910 minus2

minus4

02

4

119909

minus2minus4

119905

119905

119905

Figure 6 The trajectories and phase graphs of system (61) with 120591 = 63804 when 119870 = minus1

1199111≐ 03714185849 120596

1≐ 06094

120591(119895)

1= 16243 +

2119895120587

1205961

ℎ1015840(1199111) = minus713282

1199112≐ 1494030027 120596

2≐ 12223

120591(119895)

2= 03768 +

2119895120587

1205962

ℎ1015840(1199112) = 725266

(63)

Thus from Lemma 4 we have

Re 120582 (120591(119895)1)

119889120591lt 0

Re 120582 (120591(119895)2)

119889120591gt 0

(64)

In addition notice that

120591(0)

2= 03768 lt 120591

(0)

1= 16243 lt 120591

(1)

2

= 55173 lt 120591(2)

2= 106577 lt 120591

(1)

1

= 119340 lt 120591(3)

2= 157982 lt sdot sdot sdot

(65)

Thus from (64) and (65) we have the following conclusionabout the stability of the zero equilibrium of system (61) andHopf bifurcations

Conclusion 2 Suppose that 120591(119895)119896 119896 = 1 2 119895 = 0 1 is

defined by (63)

(i) When 120591 isin [0 120591(0)

1) cup (120591(1)

2 +infin) the equilibrium 119864

0is

unstable (see Figures 3 4 6 7 and 8)(ii) When 120591 isin (120591

(0)

1 120591(1)

2) the equilibrium 119864

0is asymptot-

ically stable (see Figure 5)(iii) System (61) undergoes a Hopf bifurcation at the

equilibrium 1198640when 120591 = 120591

(119895)

119896(see Figures 3 4

and 6)

Thus if we take 120591 isin (120591(0)

1 120591(1)

2) the stability of zero

equilibrium of the systemwill change from unstable to stableFor this example there are more complicated dynamicalphenomena occurs Bifurcation diagram is plotted with theparameter 120591 when 119870 = minus1 (Figure 2) In Figure 2 we can seethat chaos vanishes via a period-doubling bifurcation when120591 varying from 0 to 03768 Figure 3 shows that when 120591 =

07362 a period-2 orbits bifurcating from zero equilibriumFigure 4 shows when 120591 = 13497 a period-1 periodicalorbits bifurcating from zero equilibrium In Figure 2 a stablewindow is obtained from 120591 = 16243 to 120591 = 55173 which isdemonstrated by Figure 5 with 120591 = 26815 When 120591 gt 55173Figure 2 shows that periodic orbits and strange attractors also

Abstract and Applied Analysis 9

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

005

115

2

119911

minus05

4

02

119910 minus2

minus4

02

4

119909minus2

minus4

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

119905

119905

119905

Figure 7 The trajectories and phase graphs of system (61) with 120591 = 73006 when 119870 = minus1

05

119909

minus5

05

minus5

05

minus5

119910119911

119911

4

02

119910 minus2

minus4

02

4

119909minus2minus4

minus1

012345

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

119905

119905

119905

Figure 8 The trajectories and phase graphs of system (61) with 120591 = 137423 when 119870 = minus1

occur and system (61) through Hopf bifurcation route stillgoes to chaos and it is demonstrated by Figures 6ndash8

5 Discussion

In this paper an autonomy system with time-delayed feed-back is studied by using the theory of functional differentialequation and Hassardrsquos method Illustrating with numericalsimulations we show that delayed feedback controller (DFC)is an effective method for chaos control

In fact system (2) has more complicated dynamicalbehaves From Figure 2 we can see that chaos degenerationprocess the orbits of system (2) continuously degenerate toa periodic solution region through reverse period-doublingbifurcation With the increasing of 120591 chaos occurs againthrough Hopf bifurcation route However the numericalsimulations indicate that there were points of similaritybetween this route to chaos and quasiperiodicity route tochaos [18] in our future research we will consider this topicAnd in this paper we only give detailed analysis on thetrivial equilibrium 119864

0 the detailed analysis of the interior

equilibrium 119864lowast is the second interested topic in our future

research

Acknowledgments

The authors would like to thank the anonymous refereefor the very helpful suggestions and comments which ledto the improvement of the original paper And this workis supported by 2013 Scientific Research Project of BeifangUniversity of Nationalities (2013XYZ021) Institute of Infor-mation and System Computation Science of Beifang Uni-versity (13xyb01) Science and Technology Department ofHenan Province (122300410417) and Education Departmentof Henan Province (13A110108)

References

[1] G Corso and F B Rizzato ldquoHamiltonian bifurcations leading tochaos in low-energy relativistic wave- particle systemrdquo PhysicaD vol 80 pp 296ndash306 1995

[2] M Saleem and T Agrawal ldquoChaos in a tumor growth modelwith delayed responses of the immune systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 891095 16 pages2012

[3] U Kumar ldquoA retrospection of chaotic phenomene in electricalsystemsrdquo Active and Passive Electronic Components vol 21 pp1ndash15 1998

10 Abstract and Applied Analysis

[4] Z Chen X Zhang and Q Bi ldquoBifurcations and chaos ofcoupled electrical circuitsrdquo Nonlinear Analysis vol 9 no 3 pp1158ndash1168 2008

[5] S Hallegatte M Ghil P Dumas and J Hourcade ldquoBusinesscycles bifurcations and chaos in a neo-classical model withinvestment dynamicsrdquo Journal of Economic Behavior and Orga-nization vol 67 pp 57ndash77 2008

[6] DGhoshA R Chowdhury andP Saha ldquoBifurcation continua-tion chaos and chaos control in nonlinear Bloch systemrdquo Com-munications inNonlinear Science andNumerical Simulation vol13 no 8 pp 1461ndash1471 2008

[7] A El-Gohary andA S Al-Ruzaiza ldquoChaos and adaptive controlin two prey one predator system with nonlinear feedbackrdquoChaos Solitons and Fractals vol 34 no 2 pp 443ndash453 2007

[8] H Salarieh and A Alasty ldquoControl of stochastic chaos usingsliding mode methodrdquo Journal of Computational and AppliedMathematics vol 225 no 1 pp 135ndash145 2009

[9] J Guan ldquoBifurcation analysis and chaos control in genesiosystem with delayed feedbackrdquo ISRNMathematical Physics vol2012 Article ID 843962 12 pages 2012

[10] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquoNonlinear Analysis vol 9 no 4 pp 1800ndash18102008

[11] N Vasegh and A Sedigh ldquoDelayed feedback control of time-delayed chaotic systems analytical approach at Hopf bifurca-tionrdquo Physics Letters A vol 372 pp 5110ndash5114 2008

[12] J Lu Z Ma and L Li ldquoDouble delayed feedback control for thestabilization of unstable steady states in chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 7 pp 3037ndash3045 2009

[13] M Bleich and J Socolar ldquoStability of periodic orbits controlledby time-delay feedbackrdquo Physics Letters A vol 210 pp 87ndash941996

[14] Y Ding W Jiang and H Wang ldquoDelayed feedback controland bifurcation analysis of Rossler chaotic systemrdquo NonlinearDynamics vol 61 no 4 pp 707ndash715 2010

[15] Y Song and J Wei ldquoBifurcation analysis for Chenrsquos system withdelayed feedback and its application to control of chaosrdquoChaosSolitons and Fractals vol 22 no 1 pp 75ndash91 2004

[16] S Ruan and J Wei ldquoOn the zeros of transcendental functionswith applications to stability of delay differential equations withtwo delaysrdquo Dynamics of Continuous Discrete and ImpulsiveSystems A vol 10 no 6 pp 863ndash874 2003

[17] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge UK 1981

[18] I Fuente L Martinez J Veguillas and J AguirregabirialdquoQuasiperiodicity route to chaos in a biochemical systemrdquoBiophysical Journal vol 71 pp 2375ndash2379 1998

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

4 Abstract and Applied Analysis

By the Riesz representation theorem there exists a function120578(120579 120583) of bounded variation for 120579 isin [minus1 0] such that

119871120583120601 = int

0

minus1

119889120578 (120579 0) 120601 (120579) for120601 isin 119862 (21)

In fact we can choose

120578 (120579 120583) = (120591119896+ 120583)(

119886 minus1 0

1 119870 minus 119887 minus1

119886 0 minus8

)120575 (120579)

minus (120591119896+ 120583)(

0 0 0

0 minus119870 0

0 0 0

)120575 (120579 + 1)

(22)

where 120575 is Dirac-delta functionFor 120601 isin 119862

1([minus1 0] 119877

3) define

119860 (120583) 120601 =

119889120601 (120579)

119889120579 120579 isin [minus1 0)

int

0

minus1

119889120578 (120583 119904) 120601 (119904) 120579 = 0

119877 (120583) 120601 =

0 120579 isin [minus1 0)

119891 (120583 120601) 120579 = 0

(23)

Then when 120579 = 0 system (18) is equivalent to

119905= 119860 (120583) 119906

119905+ 119877 (120583) 119906

119905 (24)

where 119906119905(120579) = 119906(119905 + 120579) for 120579 isin [minus1 0] For 120595 isin 119862

1([0 1]

(1198773)lowast) define

119860lowast120595 (119904) =

minus119889120595 (119904)

119889119904 119904 isin (0 1]

int

0

minus1

119889120578119879(119905 0) 120595 (minus119905) 119904 = 0

(25)

and a bilinear inner product

⟨120595 (119904) 120601 (120579)⟩ = 120595 (0) 120601 (0)

minus int

0

minus1

int

120579

120585=0

120595 (120585 minus 120579) 119889120578 (120579) 120601 (120585) 119889120585

(26)

where 120578(120579) = 120578(120579 0) Then 119860 = 119860(0) and 119860lowast are adjoint

operators By the discussion in Section 2 we know thatplusmn119894120596119896120591119896are eigenvalues of 119860 Thus they are also eigenvalues

of 119860lowast We first need to compute the eigenvector of 119860 and 119860lowast

corresponding to 119894120596119896120591119896andminus119894120596

119896120591119896 respectively Suppose that

119902(120579) = (1 120572 120573)119879119890119894120579120596119896120591119896 is the eigenvector of 119860 corresponding

to 119894120596119896120591119896 Then 119860119902(120579) = 119894120596

119896120591119896119902(120579) It follows from the

definition of 119860 and (19) (21) and (22) that

120591119896(

119894120596119896minus 119886 1 0

minus1 119894120596119896minus 119870 + 119887 + 119870119890

minus119894120596119896120591119896 1

minus119886 0 119894120596119896+ 8

)119902 (0)

= (

0

0

0

)

(27)

Thus we can easily get

119902 (0) = (1 120572 120573)119879= (1 119886 minus 119894120596

119896

119886

119894120596119896+ 8

)

119879

(28)

Similarly let 119902lowast(119904) = 119863(1 120572lowast 120573lowast)119890119894119904120596119896120591119896 be the eigenvector of

119860lowast corresponding to minus119894120596

119896120591119896 By the definition of119860lowast and (19)

(21) and (22) we can compute

119902lowast(119904) = 119863 (1 120572

lowast 120573lowast) 119890119894119904120596119896120591119896

= 119863(1minus (119894120596119896+ 119886) (119894120596

119896minus 8)

119894120596119896minus 8 + 119886

minus119894120596119896minus 119886

119894120596119896minus 8 + 119886

) 119890119894119904120596119896120591119896

(29)

In order to assure ⟨119902lowast(119904) 119902(120579)⟩ = 1 we need to determine thevalue of119863 From (26) we have

⟨119902lowast(119904) 119902 (120579)⟩ = 119863(1 120572

lowast 120573lowast

) (1 120572 120573)119879

minus int

0

minus1

int

120579

120585=0

119863(1 120572lowast 120573lowast

) 119890minus119894(120585minus120579)120596

119896120591119896119889120578

times (120579) (1 120572 120573)119879119890119894120585120596119896120591119896119889120585

= 1198631 + 120572120572lowast+ 120573120573lowast

minus int

0

minus1

(1 120572lowast 120573lowast

) 120579119890119894120579120596119896120591119896119889120578

times (120579) (1 120572 120573)119879

= 1198631 + 120572120572lowast+ 120573120573lowast

minus 119870120591119896120572120572lowast119890minus119894120596119896120591119896

(30)

Thus we can choose119863 as

119863 =1

1 + 120572120572lowast+ 120573120573lowast

minus 119870120591119896120572120572lowast119890119894120596119896120591119896

(31)

In the following we first compute the coordinates to describethe center manifold 119862

0at 120583 = 0 Let 119906

119905be the solution of (18)

when 120583 = 0 Define

119911 (119905) = ⟨119902lowast 119906119905⟩ 119882 (119905 120579) = 119906

119905(120579) minus 2Re 119911 (119905) 119902 (120579)

(32)

On the center manifold 1198620 we have

119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)

= 11988220(120579)

1199112

2+11988211(120579) 119911119911

+11988202 (120579)

1199112

2+11988230 (120579)

1199113

6+ sdot sdot sdot

(33)

where 119911 and 119911 are local coordinates for center manifold 1198620

in the direction of 119902 and 119902lowast Note that 119882 is real if 119906

119905is real

Abstract and Applied Analysis 5

We consider only real solutions For the solution 119906119905isin 1198620of

(18) since 120583 = 0 we have

= 119894120596119896120591119896119911

+ ⟨119902lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579) + 2Re 119911 (119905) 119902 (120579))⟩

= 119894120596119896120591119896119911

+ 119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) + 2Re 119911 (119905) 119902 (0))

≜ 119894120596119896120591119896119911 + 119902lowast(0) 1198910(119911 119911) = 119894120596

119896120591119896119911 + 119892 (119911 119911)

(34)

where

119892 (119911 119911) = 119902lowast(0) 1198910 (119911 119911)

= 11989220(120579)

1199112

2+ 11989211(120579) 119911119911 + 119892

02(120579)

1199112

2+ sdot sdot sdot

(35)

By (32) we have 119906119905(120579) = (119906

1119905(120579) 1199062119905(120579) 1199063119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199061119905 (0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

1199062119905 (0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2+119882(2)

11(0) 119911119911

+119882(2)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

1199063119905 (0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

(36)

It follows together with (20) that

119892 (119911 119911) = 119902lowast(0) 1198910 (119911 119911)

= 119863120591119896(1 120572lowast 120573lowast

)(

0

0

41199062119905(0) 1199063119905(0)

)

= 4119863120591119896120573lowast

[119911120572 + 119911 120572 +119882(2)

20(0)

1199112

2+119882(2)

11(0) 119911119911

+119882(2)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)]

times [119911120573 + 119911120573 +119882(3)

20(0)

1199112

2+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)]

(37)

Comparing the coefficients with (35) we have

11989220

= 8119863120591119896120573lowast

120572120573

11989211

= 8119863120591119896120573lowast

(120572120573 + 120572120573)

11989202

= 8119863120591119896120573lowast

120572120573

11989221

= 4119863120591119896120573lowast

[2120572119882(3)

11(0) + 120572119882

(3)

20(0)

+120573119882(2)

20(0) + 2120573119882

(2)

11(0)]

(38)

In order to determine 11989221 in the sequel we need to compute

11988220(120579) and119882

11(120579) From (24) and (32) we have

= 119905minus 119902 minus 119902

=

119860119882 minus 2Re 119902lowast (0) 1198910119902 (120579) 120579 isin [0 1)

119860119882 minus 2Re 119902lowast (0) 1198910119902 (120579) + 1198910 120579 = 0

≜ 119860119882 +119867 (119911 119911 120579)

(39)

where

119867(119911 119911 120579) = 11986720 (120579)

1199112

2+ 11986711 (120579) 119911119911 + 119867

02 (120579)1199112

2+ sdot sdot sdot

(40)

Notice that near the origin on the centermanifold1198620 we have

= 119882119911 + 119882

119911 (41)

Thus we have

(119860 minus 2119894120591119896120596119896119868)11988220 (120579) = minus119867

20 (120579)

11986011988211(120579) = minus119867

11(120579)

(42)

Since (39) for 120579 isin [minus1 0) we have

119867(119911 119911 120579) = minus 119902lowast(0) 1198910119902 (120579) minus 119902

lowast(0) 1198910119902 (120579)

= minus 119892119902 (120579) minus 119892 119902 (120579)

(43)

Comparing the coefficients with (40) gives that

11986720(120579) = minus119892

20119902 (120579) minus 119892

02119902 (120579)

11986711 (120579) = minus119892

11119902 (120579) minus 119892

11119902 (120579)

(44)

From (43) (46) and the definition of 119860 we can get

20(120579) = 2119894120591

11989612059611989611988220(120579) + 119892

20119902 (120579) + 119892

02119902 (120579) (45)

Noticing that 119902(120579) = 119902(0)119890119894120591119896120596119896120579 we have

11988220(120579) =

11989411989220

120591119896120596119896

119902 (0) 119890119894120591119896120596119896120579+

11989411989202

3120591119896120596119896

119902 (0) 119890minus119894120591119896120596119896120579

+ 11986411198902119894120591119896120596119896120579

(46)

6 Abstract and Applied Analysis

051

152

253

Max(119910)

0 5 10 15 20120591

minus1

0minus05

Figure 2 The bifurcation graph with 120591 when 119870 = minus1

where 1198641= (119864(1)

1 119864(2)

1 119864(3)

1) isin 1198773 is a constant vector In the

same way we can also obtain

11988211 (120579) = minus

11989411989211

120591119896120596119896

119902 (0) 119890119894120591119896120596119896120579+

11989411989211

120591119896120596119896

119902 (0) 119890minus119894120591119896120596119896120579+ 1198642 (47)

where 1198642= (119864(1)

2 119864(2)

2 119864(3)

2) isin 1198773 is also a constant vector

In what follows we will seek appropriate 1198641and 119864

2 From

the definition of 119860 and (42) we obtain

int

0

minus1

119889120578 (120579)11988220(120579) = 2119894120591

11989612059611989611988220(0) minus 119867

20(0) (48)

int

0

minus1

119889120578 (120579)11988211 (120579) = minus11986711 (0) (49)

where 120578(120579) = 120578(0 120579) From (39) and (40) we have

11986720(0) = minus119892

20119902 (0) minus 119892

02119902 (0) + 2120591

119896(

0

0

4120572120573

) (50)

11986711 (0) = minus119892

11119902 (0) minus 119892

11119902 (0) + 2120591

119896(

0

0

4Re 120572120573) (51)

Substituting (46) and (50) into (48) and noticing that

(119894120596119896120591119896119868 minus int

0

minus1

119890119894120596119896120591119896120579119889120578 (120579)) 119902 (0) = 0

(minus119894120596119896120591119896119868 minus int

0

minus1

119890minus119894120596119896120591119896120579119889120578 (120579)) 119902 (0) = 0

(52)

we obtain

(2119894120596119896120591119896119868 minus int

0

minus1

1198902119894120596119896120591119896120579119889120578 (120579))119864

1= 2120591119896(

0

0

4120572120573

) (53)

which leads to

(

2119894120596119896minus 119886 1 0

minus1 2119894120596119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896 1

minus119886 0 2119894120596119896+ 8

)1198641

= 2(

0

0

4120572120573

)

(54)

It follows that

119864(1)

1=

8

119860120572120573

119864(2)

1=

8

119860120572120573 (2119894120596

119896minus 119886)

119864(3)

1=

8

119860120572120573 [(2119894120596

119896minus 119886) (2119894120596

119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896) + 1]

(55)

where119860 = (2119894120596

119896+ 8)

times [(2119894120596119896minus 119886) (2119894120596

119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896) + 1] minus 119886

(56)

Similarly substituting (47) and (51) into (49) we can get

(

minus119886 1 0

minus1 119887 1

minus119886 0 8

)1198642= 2(

0

0

4Re 120572120573) (57)

Thus we have

119864(1)

2=

8

119861Re 120572120573

119864(2)

2=

8

119861Re 120572120573

119864(3)

2=

8

119861Re 120572120573 (1 minus 119886119887)

(58)

where

119861 = 8 minus 119886 minus 8119886119887 (59)

Thus we can determine11988220(120579) and119882

11(120579) Furthermore we

can determine each 119892119894119895 Therefore each 119892

119894119895is determined by

the parameters and delay in (2) Thus we can compute thefollowing values

1198881(0) =

119894

2120596119896120591119896

(1198922011989211minus 2

1003816100381610038161003816119892111003816100381610038161003816

2minus1

3

1003816100381610038161003816119892021003816100381610038161003816

2) +

11989221

2

1205832= minus

Re 1198881(0)

Re 1205821015840 (120591119896)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(120591119896)

120596119896120591119896

1205732= 2Re 119888

1(0)

(60)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

119896 that is 120583

2

determines the directions of theHopf bifurcation if1205832gt 0 (lt

0) then theHopf bifurcation is supercritical (subcritical) andthe bifurcation exists for 120591 gt 120591

0(lt 1205910) 1205732determines the

stability of the bifurcation periodic solutions the bifurcatingperiodic solutions are stable (unstable) if 120573

2lt 0 (gt 0) and119879

2

determines the period of the bifurcating periodic solutionsthe period increases (decreases) if 119879

2gt 0 (lt 0)

Abstract and Applied Analysis 7

0 24

4

0

0

051

152

2

25

0 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

119909

119909

minus5

05

minus5

5

minus5

119910

119910

119911

119911

minus05

minus2 minus2minus4 minus4

119905

119905

119905

Figure 3 The trajectories and phase graphs of system (61) with 120591 = 07362 when 119870 = minus1

02

40

24

002

0406

080 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

0

119909

minus5

5

minus5

1

minus1

119910

119910

119911

119911

minus02

minus2 minus2minus4 minus4

119905

119905

119905

119909

Figure 4 The trajectories and phase graphs of system (61) with 120591 = 13497 when 119870 = minus1

4 Numerical Simulations andApplication to Control of Chaos

In this section we apply the results we get in the previoussections to system (61) for the purpose of control of chaosFrom Section 2 we know that under certain conditions afamily of periodic solutions bifurcates from the steady statesof system (61) at some critical values of 120591 and the stability ofthe steady state may be changed along with the increase of 120591If the bifurcating periodic solution is orbitally asymptoticallystable or some steady state becomes local stable then chaosmay vanish Following this idea we consider the followingdelayed feedback control system for an autonomy system

= 02119909 minus 119910

119910 = 119909 minus 002119910 minus 119911 + 119870 (119910 (119905) minus 119910 (119905 minus 120591))

= 02119909 + 4119911 (119910 minus 2)

(61)

which has two equilibriums 1198640(0 0 0) and 119864lowast(9749 19498

9710) When 120591 = 0 or 119870 = 0 1198640is unstable and system

(61) is chaotic (see Figure 1)The corresponding characteristicequation of system (61) at 119864

0is

1205823+ (782 minus 119870) 120582

2minus (0444 + 78119870) 120582 + 7768

+ 16119870 + (1205822+ 78120582 minus 16)119870119890

minus120582120591= 0

(62)

When 120591 = 0 (62) has one negative root and a pair of complexroots with positive parts By the discussion of Section 2we can get 119901 = 620404 minus 004119870 119902 = minus1212944 minus

25616119870 and 119903 = 603418 + 248576119870 and then we cancompute 998779 = 42129 + 27216119870 + 00016119870

2 and it is easyto know that 998779 gt 0 and when 119870 ge minus24275 119911lowast

1gt 0 always

holdsThen byTheorem 5 we have the following conclusion

Conclusion 1 If 119870 lt minus24275 or 119870 ge minus24275 and ℎ(119911lowast

1) lt 0

then there exist some 1205910gt 0 such that when 120591 isin [0 120591

0) 1198640is

always unstable

For example we fix119870 = minus1 Through computing we canget 119911lowast1= 10693 gt 0 and ℎ(119911lowast

1) = minus192713 lt 0 Furthermore

we can compute

8 Abstract and Applied Analysis

0 50 100 150 200

05

02

001 0 1 2

01

2

0

00501

0 50 100 150 200

0 50 100 150 200

119909

119909

minus5119910

119910

minus2

119911

119911

minus01

minus01

minus1 minus1

minus2minus2

minus3

minus005

119905

119905

119905

Figure 5 The trajectories and phase graphs of system (61) with 120591 = 26815 when 119870 = minus1

005

115

20 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

119911

minus05

4

02

119910 minus2

minus4

02

4

119909

minus2minus4

119905

119905

119905

Figure 6 The trajectories and phase graphs of system (61) with 120591 = 63804 when 119870 = minus1

1199111≐ 03714185849 120596

1≐ 06094

120591(119895)

1= 16243 +

2119895120587

1205961

ℎ1015840(1199111) = minus713282

1199112≐ 1494030027 120596

2≐ 12223

120591(119895)

2= 03768 +

2119895120587

1205962

ℎ1015840(1199112) = 725266

(63)

Thus from Lemma 4 we have

Re 120582 (120591(119895)1)

119889120591lt 0

Re 120582 (120591(119895)2)

119889120591gt 0

(64)

In addition notice that

120591(0)

2= 03768 lt 120591

(0)

1= 16243 lt 120591

(1)

2

= 55173 lt 120591(2)

2= 106577 lt 120591

(1)

1

= 119340 lt 120591(3)

2= 157982 lt sdot sdot sdot

(65)

Thus from (64) and (65) we have the following conclusionabout the stability of the zero equilibrium of system (61) andHopf bifurcations

Conclusion 2 Suppose that 120591(119895)119896 119896 = 1 2 119895 = 0 1 is

defined by (63)

(i) When 120591 isin [0 120591(0)

1) cup (120591(1)

2 +infin) the equilibrium 119864

0is

unstable (see Figures 3 4 6 7 and 8)(ii) When 120591 isin (120591

(0)

1 120591(1)

2) the equilibrium 119864

0is asymptot-

ically stable (see Figure 5)(iii) System (61) undergoes a Hopf bifurcation at the

equilibrium 1198640when 120591 = 120591

(119895)

119896(see Figures 3 4

and 6)

Thus if we take 120591 isin (120591(0)

1 120591(1)

2) the stability of zero

equilibrium of the systemwill change from unstable to stableFor this example there are more complicated dynamicalphenomena occurs Bifurcation diagram is plotted with theparameter 120591 when 119870 = minus1 (Figure 2) In Figure 2 we can seethat chaos vanishes via a period-doubling bifurcation when120591 varying from 0 to 03768 Figure 3 shows that when 120591 =

07362 a period-2 orbits bifurcating from zero equilibriumFigure 4 shows when 120591 = 13497 a period-1 periodicalorbits bifurcating from zero equilibrium In Figure 2 a stablewindow is obtained from 120591 = 16243 to 120591 = 55173 which isdemonstrated by Figure 5 with 120591 = 26815 When 120591 gt 55173Figure 2 shows that periodic orbits and strange attractors also

Abstract and Applied Analysis 9

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

005

115

2

119911

minus05

4

02

119910 minus2

minus4

02

4

119909minus2

minus4

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

119905

119905

119905

Figure 7 The trajectories and phase graphs of system (61) with 120591 = 73006 when 119870 = minus1

05

119909

minus5

05

minus5

05

minus5

119910119911

119911

4

02

119910 minus2

minus4

02

4

119909minus2minus4

minus1

012345

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

119905

119905

119905

Figure 8 The trajectories and phase graphs of system (61) with 120591 = 137423 when 119870 = minus1

occur and system (61) through Hopf bifurcation route stillgoes to chaos and it is demonstrated by Figures 6ndash8

5 Discussion

In this paper an autonomy system with time-delayed feed-back is studied by using the theory of functional differentialequation and Hassardrsquos method Illustrating with numericalsimulations we show that delayed feedback controller (DFC)is an effective method for chaos control

In fact system (2) has more complicated dynamicalbehaves From Figure 2 we can see that chaos degenerationprocess the orbits of system (2) continuously degenerate toa periodic solution region through reverse period-doublingbifurcation With the increasing of 120591 chaos occurs againthrough Hopf bifurcation route However the numericalsimulations indicate that there were points of similaritybetween this route to chaos and quasiperiodicity route tochaos [18] in our future research we will consider this topicAnd in this paper we only give detailed analysis on thetrivial equilibrium 119864

0 the detailed analysis of the interior

equilibrium 119864lowast is the second interested topic in our future

research

Acknowledgments

The authors would like to thank the anonymous refereefor the very helpful suggestions and comments which ledto the improvement of the original paper And this workis supported by 2013 Scientific Research Project of BeifangUniversity of Nationalities (2013XYZ021) Institute of Infor-mation and System Computation Science of Beifang Uni-versity (13xyb01) Science and Technology Department ofHenan Province (122300410417) and Education Departmentof Henan Province (13A110108)

References

[1] G Corso and F B Rizzato ldquoHamiltonian bifurcations leading tochaos in low-energy relativistic wave- particle systemrdquo PhysicaD vol 80 pp 296ndash306 1995

[2] M Saleem and T Agrawal ldquoChaos in a tumor growth modelwith delayed responses of the immune systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 891095 16 pages2012

[3] U Kumar ldquoA retrospection of chaotic phenomene in electricalsystemsrdquo Active and Passive Electronic Components vol 21 pp1ndash15 1998

10 Abstract and Applied Analysis

[4] Z Chen X Zhang and Q Bi ldquoBifurcations and chaos ofcoupled electrical circuitsrdquo Nonlinear Analysis vol 9 no 3 pp1158ndash1168 2008

[5] S Hallegatte M Ghil P Dumas and J Hourcade ldquoBusinesscycles bifurcations and chaos in a neo-classical model withinvestment dynamicsrdquo Journal of Economic Behavior and Orga-nization vol 67 pp 57ndash77 2008

[6] DGhoshA R Chowdhury andP Saha ldquoBifurcation continua-tion chaos and chaos control in nonlinear Bloch systemrdquo Com-munications inNonlinear Science andNumerical Simulation vol13 no 8 pp 1461ndash1471 2008

[7] A El-Gohary andA S Al-Ruzaiza ldquoChaos and adaptive controlin two prey one predator system with nonlinear feedbackrdquoChaos Solitons and Fractals vol 34 no 2 pp 443ndash453 2007

[8] H Salarieh and A Alasty ldquoControl of stochastic chaos usingsliding mode methodrdquo Journal of Computational and AppliedMathematics vol 225 no 1 pp 135ndash145 2009

[9] J Guan ldquoBifurcation analysis and chaos control in genesiosystem with delayed feedbackrdquo ISRNMathematical Physics vol2012 Article ID 843962 12 pages 2012

[10] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquoNonlinear Analysis vol 9 no 4 pp 1800ndash18102008

[11] N Vasegh and A Sedigh ldquoDelayed feedback control of time-delayed chaotic systems analytical approach at Hopf bifurca-tionrdquo Physics Letters A vol 372 pp 5110ndash5114 2008

[12] J Lu Z Ma and L Li ldquoDouble delayed feedback control for thestabilization of unstable steady states in chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 7 pp 3037ndash3045 2009

[13] M Bleich and J Socolar ldquoStability of periodic orbits controlledby time-delay feedbackrdquo Physics Letters A vol 210 pp 87ndash941996

[14] Y Ding W Jiang and H Wang ldquoDelayed feedback controland bifurcation analysis of Rossler chaotic systemrdquo NonlinearDynamics vol 61 no 4 pp 707ndash715 2010

[15] Y Song and J Wei ldquoBifurcation analysis for Chenrsquos system withdelayed feedback and its application to control of chaosrdquoChaosSolitons and Fractals vol 22 no 1 pp 75ndash91 2004

[16] S Ruan and J Wei ldquoOn the zeros of transcendental functionswith applications to stability of delay differential equations withtwo delaysrdquo Dynamics of Continuous Discrete and ImpulsiveSystems A vol 10 no 6 pp 863ndash874 2003

[17] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge UK 1981

[18] I Fuente L Martinez J Veguillas and J AguirregabirialdquoQuasiperiodicity route to chaos in a biochemical systemrdquoBiophysical Journal vol 71 pp 2375ndash2379 1998

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

Abstract and Applied Analysis 5

We consider only real solutions For the solution 119906119905isin 1198620of

(18) since 120583 = 0 we have

= 119894120596119896120591119896119911

+ ⟨119902lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579) + 2Re 119911 (119905) 119902 (120579))⟩

= 119894120596119896120591119896119911

+ 119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) + 2Re 119911 (119905) 119902 (0))

≜ 119894120596119896120591119896119911 + 119902lowast(0) 1198910(119911 119911) = 119894120596

119896120591119896119911 + 119892 (119911 119911)

(34)

where

119892 (119911 119911) = 119902lowast(0) 1198910 (119911 119911)

= 11989220(120579)

1199112

2+ 11989211(120579) 119911119911 + 119892

02(120579)

1199112

2+ sdot sdot sdot

(35)

By (32) we have 119906119905(120579) = (119906

1119905(120579) 1199062119905(120579) 1199063119905(120579))119879= 119882(119905 120579) +

119911119902(120579) + 119911 119902(120579) and then

1199061119905 (0) = 119911 + 119911 +119882

(1)

20(0)

1199112

2+119882(1)

11(0) 119911119911

+119882(1)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

1199062119905 (0) = 119911120572 + 119911 120572 +119882

(2)

20(0)

1199112

2+119882(2)

11(0) 119911119911

+119882(2)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

1199063119905 (0) = 119911120573 + 119911120573 +119882

(3)

20(0)

1199112

2+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)

(36)

It follows together with (20) that

119892 (119911 119911) = 119902lowast(0) 1198910 (119911 119911)

= 119863120591119896(1 120572lowast 120573lowast

)(

0

0

41199062119905(0) 1199063119905(0)

)

= 4119863120591119896120573lowast

[119911120572 + 119911 120572 +119882(2)

20(0)

1199112

2+119882(2)

11(0) 119911119911

+119882(2)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)]

times [119911120573 + 119911120573 +119882(3)

20(0)

1199112

2+119882(3)

11(0) 119911119911

+119882(3)

02(0)

1199112

2+ 119900 (|(119911 119911)|

3)]

(37)

Comparing the coefficients with (35) we have

11989220

= 8119863120591119896120573lowast

120572120573

11989211

= 8119863120591119896120573lowast

(120572120573 + 120572120573)

11989202

= 8119863120591119896120573lowast

120572120573

11989221

= 4119863120591119896120573lowast

[2120572119882(3)

11(0) + 120572119882

(3)

20(0)

+120573119882(2)

20(0) + 2120573119882

(2)

11(0)]

(38)

In order to determine 11989221 in the sequel we need to compute

11988220(120579) and119882

11(120579) From (24) and (32) we have

= 119905minus 119902 minus 119902

=

119860119882 minus 2Re 119902lowast (0) 1198910119902 (120579) 120579 isin [0 1)

119860119882 minus 2Re 119902lowast (0) 1198910119902 (120579) + 1198910 120579 = 0

≜ 119860119882 +119867 (119911 119911 120579)

(39)

where

119867(119911 119911 120579) = 11986720 (120579)

1199112

2+ 11986711 (120579) 119911119911 + 119867

02 (120579)1199112

2+ sdot sdot sdot

(40)

Notice that near the origin on the centermanifold1198620 we have

= 119882119911 + 119882

119911 (41)

Thus we have

(119860 minus 2119894120591119896120596119896119868)11988220 (120579) = minus119867

20 (120579)

11986011988211(120579) = minus119867

11(120579)

(42)

Since (39) for 120579 isin [minus1 0) we have

119867(119911 119911 120579) = minus 119902lowast(0) 1198910119902 (120579) minus 119902

lowast(0) 1198910119902 (120579)

= minus 119892119902 (120579) minus 119892 119902 (120579)

(43)

Comparing the coefficients with (40) gives that

11986720(120579) = minus119892

20119902 (120579) minus 119892

02119902 (120579)

11986711 (120579) = minus119892

11119902 (120579) minus 119892

11119902 (120579)

(44)

From (43) (46) and the definition of 119860 we can get

20(120579) = 2119894120591

11989612059611989611988220(120579) + 119892

20119902 (120579) + 119892

02119902 (120579) (45)

Noticing that 119902(120579) = 119902(0)119890119894120591119896120596119896120579 we have

11988220(120579) =

11989411989220

120591119896120596119896

119902 (0) 119890119894120591119896120596119896120579+

11989411989202

3120591119896120596119896

119902 (0) 119890minus119894120591119896120596119896120579

+ 11986411198902119894120591119896120596119896120579

(46)

6 Abstract and Applied Analysis

051

152

253

Max(119910)

0 5 10 15 20120591

minus1

0minus05

Figure 2 The bifurcation graph with 120591 when 119870 = minus1

where 1198641= (119864(1)

1 119864(2)

1 119864(3)

1) isin 1198773 is a constant vector In the

same way we can also obtain

11988211 (120579) = minus

11989411989211

120591119896120596119896

119902 (0) 119890119894120591119896120596119896120579+

11989411989211

120591119896120596119896

119902 (0) 119890minus119894120591119896120596119896120579+ 1198642 (47)

where 1198642= (119864(1)

2 119864(2)

2 119864(3)

2) isin 1198773 is also a constant vector

In what follows we will seek appropriate 1198641and 119864

2 From

the definition of 119860 and (42) we obtain

int

0

minus1

119889120578 (120579)11988220(120579) = 2119894120591

11989612059611989611988220(0) minus 119867

20(0) (48)

int

0

minus1

119889120578 (120579)11988211 (120579) = minus11986711 (0) (49)

where 120578(120579) = 120578(0 120579) From (39) and (40) we have

11986720(0) = minus119892

20119902 (0) minus 119892

02119902 (0) + 2120591

119896(

0

0

4120572120573

) (50)

11986711 (0) = minus119892

11119902 (0) minus 119892

11119902 (0) + 2120591

119896(

0

0

4Re 120572120573) (51)

Substituting (46) and (50) into (48) and noticing that

(119894120596119896120591119896119868 minus int

0

minus1

119890119894120596119896120591119896120579119889120578 (120579)) 119902 (0) = 0

(minus119894120596119896120591119896119868 minus int

0

minus1

119890minus119894120596119896120591119896120579119889120578 (120579)) 119902 (0) = 0

(52)

we obtain

(2119894120596119896120591119896119868 minus int

0

minus1

1198902119894120596119896120591119896120579119889120578 (120579))119864

1= 2120591119896(

0

0

4120572120573

) (53)

which leads to

(

2119894120596119896minus 119886 1 0

minus1 2119894120596119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896 1

minus119886 0 2119894120596119896+ 8

)1198641

= 2(

0

0

4120572120573

)

(54)

It follows that

119864(1)

1=

8

119860120572120573

119864(2)

1=

8

119860120572120573 (2119894120596

119896minus 119886)

119864(3)

1=

8

119860120572120573 [(2119894120596

119896minus 119886) (2119894120596

119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896) + 1]

(55)

where119860 = (2119894120596

119896+ 8)

times [(2119894120596119896minus 119886) (2119894120596

119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896) + 1] minus 119886

(56)

Similarly substituting (47) and (51) into (49) we can get

(

minus119886 1 0

minus1 119887 1

minus119886 0 8

)1198642= 2(

0

0

4Re 120572120573) (57)

Thus we have

119864(1)

2=

8

119861Re 120572120573

119864(2)

2=

8

119861Re 120572120573

119864(3)

2=

8

119861Re 120572120573 (1 minus 119886119887)

(58)

where

119861 = 8 minus 119886 minus 8119886119887 (59)

Thus we can determine11988220(120579) and119882

11(120579) Furthermore we

can determine each 119892119894119895 Therefore each 119892

119894119895is determined by

the parameters and delay in (2) Thus we can compute thefollowing values

1198881(0) =

119894

2120596119896120591119896

(1198922011989211minus 2

1003816100381610038161003816119892111003816100381610038161003816

2minus1

3

1003816100381610038161003816119892021003816100381610038161003816

2) +

11989221

2

1205832= minus

Re 1198881(0)

Re 1205821015840 (120591119896)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(120591119896)

120596119896120591119896

1205732= 2Re 119888

1(0)

(60)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

119896 that is 120583

2

determines the directions of theHopf bifurcation if1205832gt 0 (lt

0) then theHopf bifurcation is supercritical (subcritical) andthe bifurcation exists for 120591 gt 120591

0(lt 1205910) 1205732determines the

stability of the bifurcation periodic solutions the bifurcatingperiodic solutions are stable (unstable) if 120573

2lt 0 (gt 0) and119879

2

determines the period of the bifurcating periodic solutionsthe period increases (decreases) if 119879

2gt 0 (lt 0)

Abstract and Applied Analysis 7

0 24

4

0

0

051

152

2

25

0 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

119909

119909

minus5

05

minus5

5

minus5

119910

119910

119911

119911

minus05

minus2 minus2minus4 minus4

119905

119905

119905

Figure 3 The trajectories and phase graphs of system (61) with 120591 = 07362 when 119870 = minus1

02

40

24

002

0406

080 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

0

119909

minus5

5

minus5

1

minus1

119910

119910

119911

119911

minus02

minus2 minus2minus4 minus4

119905

119905

119905

119909

Figure 4 The trajectories and phase graphs of system (61) with 120591 = 13497 when 119870 = minus1

4 Numerical Simulations andApplication to Control of Chaos

In this section we apply the results we get in the previoussections to system (61) for the purpose of control of chaosFrom Section 2 we know that under certain conditions afamily of periodic solutions bifurcates from the steady statesof system (61) at some critical values of 120591 and the stability ofthe steady state may be changed along with the increase of 120591If the bifurcating periodic solution is orbitally asymptoticallystable or some steady state becomes local stable then chaosmay vanish Following this idea we consider the followingdelayed feedback control system for an autonomy system

= 02119909 minus 119910

119910 = 119909 minus 002119910 minus 119911 + 119870 (119910 (119905) minus 119910 (119905 minus 120591))

= 02119909 + 4119911 (119910 minus 2)

(61)

which has two equilibriums 1198640(0 0 0) and 119864lowast(9749 19498

9710) When 120591 = 0 or 119870 = 0 1198640is unstable and system

(61) is chaotic (see Figure 1)The corresponding characteristicequation of system (61) at 119864

0is

1205823+ (782 minus 119870) 120582

2minus (0444 + 78119870) 120582 + 7768

+ 16119870 + (1205822+ 78120582 minus 16)119870119890

minus120582120591= 0

(62)

When 120591 = 0 (62) has one negative root and a pair of complexroots with positive parts By the discussion of Section 2we can get 119901 = 620404 minus 004119870 119902 = minus1212944 minus

25616119870 and 119903 = 603418 + 248576119870 and then we cancompute 998779 = 42129 + 27216119870 + 00016119870

2 and it is easyto know that 998779 gt 0 and when 119870 ge minus24275 119911lowast

1gt 0 always

holdsThen byTheorem 5 we have the following conclusion

Conclusion 1 If 119870 lt minus24275 or 119870 ge minus24275 and ℎ(119911lowast

1) lt 0

then there exist some 1205910gt 0 such that when 120591 isin [0 120591

0) 1198640is

always unstable

For example we fix119870 = minus1 Through computing we canget 119911lowast1= 10693 gt 0 and ℎ(119911lowast

1) = minus192713 lt 0 Furthermore

we can compute

8 Abstract and Applied Analysis

0 50 100 150 200

05

02

001 0 1 2

01

2

0

00501

0 50 100 150 200

0 50 100 150 200

119909

119909

minus5119910

119910

minus2

119911

119911

minus01

minus01

minus1 minus1

minus2minus2

minus3

minus005

119905

119905

119905

Figure 5 The trajectories and phase graphs of system (61) with 120591 = 26815 when 119870 = minus1

005

115

20 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

119911

minus05

4

02

119910 minus2

minus4

02

4

119909

minus2minus4

119905

119905

119905

Figure 6 The trajectories and phase graphs of system (61) with 120591 = 63804 when 119870 = minus1

1199111≐ 03714185849 120596

1≐ 06094

120591(119895)

1= 16243 +

2119895120587

1205961

ℎ1015840(1199111) = minus713282

1199112≐ 1494030027 120596

2≐ 12223

120591(119895)

2= 03768 +

2119895120587

1205962

ℎ1015840(1199112) = 725266

(63)

Thus from Lemma 4 we have

Re 120582 (120591(119895)1)

119889120591lt 0

Re 120582 (120591(119895)2)

119889120591gt 0

(64)

In addition notice that

120591(0)

2= 03768 lt 120591

(0)

1= 16243 lt 120591

(1)

2

= 55173 lt 120591(2)

2= 106577 lt 120591

(1)

1

= 119340 lt 120591(3)

2= 157982 lt sdot sdot sdot

(65)

Thus from (64) and (65) we have the following conclusionabout the stability of the zero equilibrium of system (61) andHopf bifurcations

Conclusion 2 Suppose that 120591(119895)119896 119896 = 1 2 119895 = 0 1 is

defined by (63)

(i) When 120591 isin [0 120591(0)

1) cup (120591(1)

2 +infin) the equilibrium 119864

0is

unstable (see Figures 3 4 6 7 and 8)(ii) When 120591 isin (120591

(0)

1 120591(1)

2) the equilibrium 119864

0is asymptot-

ically stable (see Figure 5)(iii) System (61) undergoes a Hopf bifurcation at the

equilibrium 1198640when 120591 = 120591

(119895)

119896(see Figures 3 4

and 6)

Thus if we take 120591 isin (120591(0)

1 120591(1)

2) the stability of zero

equilibrium of the systemwill change from unstable to stableFor this example there are more complicated dynamicalphenomena occurs Bifurcation diagram is plotted with theparameter 120591 when 119870 = minus1 (Figure 2) In Figure 2 we can seethat chaos vanishes via a period-doubling bifurcation when120591 varying from 0 to 03768 Figure 3 shows that when 120591 =

07362 a period-2 orbits bifurcating from zero equilibriumFigure 4 shows when 120591 = 13497 a period-1 periodicalorbits bifurcating from zero equilibrium In Figure 2 a stablewindow is obtained from 120591 = 16243 to 120591 = 55173 which isdemonstrated by Figure 5 with 120591 = 26815 When 120591 gt 55173Figure 2 shows that periodic orbits and strange attractors also

Abstract and Applied Analysis 9

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

005

115

2

119911

minus05

4

02

119910 minus2

minus4

02

4

119909minus2

minus4

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

119905

119905

119905

Figure 7 The trajectories and phase graphs of system (61) with 120591 = 73006 when 119870 = minus1

05

119909

minus5

05

minus5

05

minus5

119910119911

119911

4

02

119910 minus2

minus4

02

4

119909minus2minus4

minus1

012345

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

119905

119905

119905

Figure 8 The trajectories and phase graphs of system (61) with 120591 = 137423 when 119870 = minus1

occur and system (61) through Hopf bifurcation route stillgoes to chaos and it is demonstrated by Figures 6ndash8

5 Discussion

In this paper an autonomy system with time-delayed feed-back is studied by using the theory of functional differentialequation and Hassardrsquos method Illustrating with numericalsimulations we show that delayed feedback controller (DFC)is an effective method for chaos control

In fact system (2) has more complicated dynamicalbehaves From Figure 2 we can see that chaos degenerationprocess the orbits of system (2) continuously degenerate toa periodic solution region through reverse period-doublingbifurcation With the increasing of 120591 chaos occurs againthrough Hopf bifurcation route However the numericalsimulations indicate that there were points of similaritybetween this route to chaos and quasiperiodicity route tochaos [18] in our future research we will consider this topicAnd in this paper we only give detailed analysis on thetrivial equilibrium 119864

0 the detailed analysis of the interior

equilibrium 119864lowast is the second interested topic in our future

research

Acknowledgments

The authors would like to thank the anonymous refereefor the very helpful suggestions and comments which ledto the improvement of the original paper And this workis supported by 2013 Scientific Research Project of BeifangUniversity of Nationalities (2013XYZ021) Institute of Infor-mation and System Computation Science of Beifang Uni-versity (13xyb01) Science and Technology Department ofHenan Province (122300410417) and Education Departmentof Henan Province (13A110108)

References

[1] G Corso and F B Rizzato ldquoHamiltonian bifurcations leading tochaos in low-energy relativistic wave- particle systemrdquo PhysicaD vol 80 pp 296ndash306 1995

[2] M Saleem and T Agrawal ldquoChaos in a tumor growth modelwith delayed responses of the immune systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 891095 16 pages2012

[3] U Kumar ldquoA retrospection of chaotic phenomene in electricalsystemsrdquo Active and Passive Electronic Components vol 21 pp1ndash15 1998

10 Abstract and Applied Analysis

[4] Z Chen X Zhang and Q Bi ldquoBifurcations and chaos ofcoupled electrical circuitsrdquo Nonlinear Analysis vol 9 no 3 pp1158ndash1168 2008

[5] S Hallegatte M Ghil P Dumas and J Hourcade ldquoBusinesscycles bifurcations and chaos in a neo-classical model withinvestment dynamicsrdquo Journal of Economic Behavior and Orga-nization vol 67 pp 57ndash77 2008

[6] DGhoshA R Chowdhury andP Saha ldquoBifurcation continua-tion chaos and chaos control in nonlinear Bloch systemrdquo Com-munications inNonlinear Science andNumerical Simulation vol13 no 8 pp 1461ndash1471 2008

[7] A El-Gohary andA S Al-Ruzaiza ldquoChaos and adaptive controlin two prey one predator system with nonlinear feedbackrdquoChaos Solitons and Fractals vol 34 no 2 pp 443ndash453 2007

[8] H Salarieh and A Alasty ldquoControl of stochastic chaos usingsliding mode methodrdquo Journal of Computational and AppliedMathematics vol 225 no 1 pp 135ndash145 2009

[9] J Guan ldquoBifurcation analysis and chaos control in genesiosystem with delayed feedbackrdquo ISRNMathematical Physics vol2012 Article ID 843962 12 pages 2012

[10] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquoNonlinear Analysis vol 9 no 4 pp 1800ndash18102008

[11] N Vasegh and A Sedigh ldquoDelayed feedback control of time-delayed chaotic systems analytical approach at Hopf bifurca-tionrdquo Physics Letters A vol 372 pp 5110ndash5114 2008

[12] J Lu Z Ma and L Li ldquoDouble delayed feedback control for thestabilization of unstable steady states in chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 7 pp 3037ndash3045 2009

[13] M Bleich and J Socolar ldquoStability of periodic orbits controlledby time-delay feedbackrdquo Physics Letters A vol 210 pp 87ndash941996

[14] Y Ding W Jiang and H Wang ldquoDelayed feedback controland bifurcation analysis of Rossler chaotic systemrdquo NonlinearDynamics vol 61 no 4 pp 707ndash715 2010

[15] Y Song and J Wei ldquoBifurcation analysis for Chenrsquos system withdelayed feedback and its application to control of chaosrdquoChaosSolitons and Fractals vol 22 no 1 pp 75ndash91 2004

[16] S Ruan and J Wei ldquoOn the zeros of transcendental functionswith applications to stability of delay differential equations withtwo delaysrdquo Dynamics of Continuous Discrete and ImpulsiveSystems A vol 10 no 6 pp 863ndash874 2003

[17] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge UK 1981

[18] I Fuente L Martinez J Veguillas and J AguirregabirialdquoQuasiperiodicity route to chaos in a biochemical systemrdquoBiophysical Journal vol 71 pp 2375ndash2379 1998

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

6 Abstract and Applied Analysis

051

152

253

Max(119910)

0 5 10 15 20120591

minus1

0minus05

Figure 2 The bifurcation graph with 120591 when 119870 = minus1

where 1198641= (119864(1)

1 119864(2)

1 119864(3)

1) isin 1198773 is a constant vector In the

same way we can also obtain

11988211 (120579) = minus

11989411989211

120591119896120596119896

119902 (0) 119890119894120591119896120596119896120579+

11989411989211

120591119896120596119896

119902 (0) 119890minus119894120591119896120596119896120579+ 1198642 (47)

where 1198642= (119864(1)

2 119864(2)

2 119864(3)

2) isin 1198773 is also a constant vector

In what follows we will seek appropriate 1198641and 119864

2 From

the definition of 119860 and (42) we obtain

int

0

minus1

119889120578 (120579)11988220(120579) = 2119894120591

11989612059611989611988220(0) minus 119867

20(0) (48)

int

0

minus1

119889120578 (120579)11988211 (120579) = minus11986711 (0) (49)

where 120578(120579) = 120578(0 120579) From (39) and (40) we have

11986720(0) = minus119892

20119902 (0) minus 119892

02119902 (0) + 2120591

119896(

0

0

4120572120573

) (50)

11986711 (0) = minus119892

11119902 (0) minus 119892

11119902 (0) + 2120591

119896(

0

0

4Re 120572120573) (51)

Substituting (46) and (50) into (48) and noticing that

(119894120596119896120591119896119868 minus int

0

minus1

119890119894120596119896120591119896120579119889120578 (120579)) 119902 (0) = 0

(minus119894120596119896120591119896119868 minus int

0

minus1

119890minus119894120596119896120591119896120579119889120578 (120579)) 119902 (0) = 0

(52)

we obtain

(2119894120596119896120591119896119868 minus int

0

minus1

1198902119894120596119896120591119896120579119889120578 (120579))119864

1= 2120591119896(

0

0

4120572120573

) (53)

which leads to

(

2119894120596119896minus 119886 1 0

minus1 2119894120596119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896 1

minus119886 0 2119894120596119896+ 8

)1198641

= 2(

0

0

4120572120573

)

(54)

It follows that

119864(1)

1=

8

119860120572120573

119864(2)

1=

8

119860120572120573 (2119894120596

119896minus 119886)

119864(3)

1=

8

119860120572120573 [(2119894120596

119896minus 119886) (2119894120596

119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896) + 1]

(55)

where119860 = (2119894120596

119896+ 8)

times [(2119894120596119896minus 119886) (2119894120596

119896minus 119870 + 119887 + 119870119890

minus2119894120596119896120591119896) + 1] minus 119886

(56)

Similarly substituting (47) and (51) into (49) we can get

(

minus119886 1 0

minus1 119887 1

minus119886 0 8

)1198642= 2(

0

0

4Re 120572120573) (57)

Thus we have

119864(1)

2=

8

119861Re 120572120573

119864(2)

2=

8

119861Re 120572120573

119864(3)

2=

8

119861Re 120572120573 (1 minus 119886119887)

(58)

where

119861 = 8 minus 119886 minus 8119886119887 (59)

Thus we can determine11988220(120579) and119882

11(120579) Furthermore we

can determine each 119892119894119895 Therefore each 119892

119894119895is determined by

the parameters and delay in (2) Thus we can compute thefollowing values

1198881(0) =

119894

2120596119896120591119896

(1198922011989211minus 2

1003816100381610038161003816119892111003816100381610038161003816

2minus1

3

1003816100381610038161003816119892021003816100381610038161003816

2) +

11989221

2

1205832= minus

Re 1198881(0)

Re 1205821015840 (120591119896)

1198792= minus

Im 1198881(0) + 120583

2Im 120582

1015840(120591119896)

120596119896120591119896

1205732= 2Re 119888

1(0)

(60)

which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591

119896 that is 120583

2

determines the directions of theHopf bifurcation if1205832gt 0 (lt

0) then theHopf bifurcation is supercritical (subcritical) andthe bifurcation exists for 120591 gt 120591

0(lt 1205910) 1205732determines the

stability of the bifurcation periodic solutions the bifurcatingperiodic solutions are stable (unstable) if 120573

2lt 0 (gt 0) and119879

2

determines the period of the bifurcating periodic solutionsthe period increases (decreases) if 119879

2gt 0 (lt 0)

Abstract and Applied Analysis 7

0 24

4

0

0

051

152

2

25

0 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

119909

119909

minus5

05

minus5

5

minus5

119910

119910

119911

119911

minus05

minus2 minus2minus4 minus4

119905

119905

119905

Figure 3 The trajectories and phase graphs of system (61) with 120591 = 07362 when 119870 = minus1

02

40

24

002

0406

080 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

0

119909

minus5

5

minus5

1

minus1

119910

119910

119911

119911

minus02

minus2 minus2minus4 minus4

119905

119905

119905

119909

Figure 4 The trajectories and phase graphs of system (61) with 120591 = 13497 when 119870 = minus1

4 Numerical Simulations andApplication to Control of Chaos

In this section we apply the results we get in the previoussections to system (61) for the purpose of control of chaosFrom Section 2 we know that under certain conditions afamily of periodic solutions bifurcates from the steady statesof system (61) at some critical values of 120591 and the stability ofthe steady state may be changed along with the increase of 120591If the bifurcating periodic solution is orbitally asymptoticallystable or some steady state becomes local stable then chaosmay vanish Following this idea we consider the followingdelayed feedback control system for an autonomy system

= 02119909 minus 119910

119910 = 119909 minus 002119910 minus 119911 + 119870 (119910 (119905) minus 119910 (119905 minus 120591))

= 02119909 + 4119911 (119910 minus 2)

(61)

which has two equilibriums 1198640(0 0 0) and 119864lowast(9749 19498

9710) When 120591 = 0 or 119870 = 0 1198640is unstable and system

(61) is chaotic (see Figure 1)The corresponding characteristicequation of system (61) at 119864

0is

1205823+ (782 minus 119870) 120582

2minus (0444 + 78119870) 120582 + 7768

+ 16119870 + (1205822+ 78120582 minus 16)119870119890

minus120582120591= 0

(62)

When 120591 = 0 (62) has one negative root and a pair of complexroots with positive parts By the discussion of Section 2we can get 119901 = 620404 minus 004119870 119902 = minus1212944 minus

25616119870 and 119903 = 603418 + 248576119870 and then we cancompute 998779 = 42129 + 27216119870 + 00016119870

2 and it is easyto know that 998779 gt 0 and when 119870 ge minus24275 119911lowast

1gt 0 always

holdsThen byTheorem 5 we have the following conclusion

Conclusion 1 If 119870 lt minus24275 or 119870 ge minus24275 and ℎ(119911lowast

1) lt 0

then there exist some 1205910gt 0 such that when 120591 isin [0 120591

0) 1198640is

always unstable

For example we fix119870 = minus1 Through computing we canget 119911lowast1= 10693 gt 0 and ℎ(119911lowast

1) = minus192713 lt 0 Furthermore

we can compute

8 Abstract and Applied Analysis

0 50 100 150 200

05

02

001 0 1 2

01

2

0

00501

0 50 100 150 200

0 50 100 150 200

119909

119909

minus5119910

119910

minus2

119911

119911

minus01

minus01

minus1 minus1

minus2minus2

minus3

minus005

119905

119905

119905

Figure 5 The trajectories and phase graphs of system (61) with 120591 = 26815 when 119870 = minus1

005

115

20 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

119911

minus05

4

02

119910 minus2

minus4

02

4

119909

minus2minus4

119905

119905

119905

Figure 6 The trajectories and phase graphs of system (61) with 120591 = 63804 when 119870 = minus1

1199111≐ 03714185849 120596

1≐ 06094

120591(119895)

1= 16243 +

2119895120587

1205961

ℎ1015840(1199111) = minus713282

1199112≐ 1494030027 120596

2≐ 12223

120591(119895)

2= 03768 +

2119895120587

1205962

ℎ1015840(1199112) = 725266

(63)

Thus from Lemma 4 we have

Re 120582 (120591(119895)1)

119889120591lt 0

Re 120582 (120591(119895)2)

119889120591gt 0

(64)

In addition notice that

120591(0)

2= 03768 lt 120591

(0)

1= 16243 lt 120591

(1)

2

= 55173 lt 120591(2)

2= 106577 lt 120591

(1)

1

= 119340 lt 120591(3)

2= 157982 lt sdot sdot sdot

(65)

Thus from (64) and (65) we have the following conclusionabout the stability of the zero equilibrium of system (61) andHopf bifurcations

Conclusion 2 Suppose that 120591(119895)119896 119896 = 1 2 119895 = 0 1 is

defined by (63)

(i) When 120591 isin [0 120591(0)

1) cup (120591(1)

2 +infin) the equilibrium 119864

0is

unstable (see Figures 3 4 6 7 and 8)(ii) When 120591 isin (120591

(0)

1 120591(1)

2) the equilibrium 119864

0is asymptot-

ically stable (see Figure 5)(iii) System (61) undergoes a Hopf bifurcation at the

equilibrium 1198640when 120591 = 120591

(119895)

119896(see Figures 3 4

and 6)

Thus if we take 120591 isin (120591(0)

1 120591(1)

2) the stability of zero

equilibrium of the systemwill change from unstable to stableFor this example there are more complicated dynamicalphenomena occurs Bifurcation diagram is plotted with theparameter 120591 when 119870 = minus1 (Figure 2) In Figure 2 we can seethat chaos vanishes via a period-doubling bifurcation when120591 varying from 0 to 03768 Figure 3 shows that when 120591 =

07362 a period-2 orbits bifurcating from zero equilibriumFigure 4 shows when 120591 = 13497 a period-1 periodicalorbits bifurcating from zero equilibrium In Figure 2 a stablewindow is obtained from 120591 = 16243 to 120591 = 55173 which isdemonstrated by Figure 5 with 120591 = 26815 When 120591 gt 55173Figure 2 shows that periodic orbits and strange attractors also

Abstract and Applied Analysis 9

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

005

115

2

119911

minus05

4

02

119910 minus2

minus4

02

4

119909minus2

minus4

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

119905

119905

119905

Figure 7 The trajectories and phase graphs of system (61) with 120591 = 73006 when 119870 = minus1

05

119909

minus5

05

minus5

05

minus5

119910119911

119911

4

02

119910 minus2

minus4

02

4

119909minus2minus4

minus1

012345

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

119905

119905

119905

Figure 8 The trajectories and phase graphs of system (61) with 120591 = 137423 when 119870 = minus1

occur and system (61) through Hopf bifurcation route stillgoes to chaos and it is demonstrated by Figures 6ndash8

5 Discussion

In this paper an autonomy system with time-delayed feed-back is studied by using the theory of functional differentialequation and Hassardrsquos method Illustrating with numericalsimulations we show that delayed feedback controller (DFC)is an effective method for chaos control

In fact system (2) has more complicated dynamicalbehaves From Figure 2 we can see that chaos degenerationprocess the orbits of system (2) continuously degenerate toa periodic solution region through reverse period-doublingbifurcation With the increasing of 120591 chaos occurs againthrough Hopf bifurcation route However the numericalsimulations indicate that there were points of similaritybetween this route to chaos and quasiperiodicity route tochaos [18] in our future research we will consider this topicAnd in this paper we only give detailed analysis on thetrivial equilibrium 119864

0 the detailed analysis of the interior

equilibrium 119864lowast is the second interested topic in our future

research

Acknowledgments

The authors would like to thank the anonymous refereefor the very helpful suggestions and comments which ledto the improvement of the original paper And this workis supported by 2013 Scientific Research Project of BeifangUniversity of Nationalities (2013XYZ021) Institute of Infor-mation and System Computation Science of Beifang Uni-versity (13xyb01) Science and Technology Department ofHenan Province (122300410417) and Education Departmentof Henan Province (13A110108)

References

[1] G Corso and F B Rizzato ldquoHamiltonian bifurcations leading tochaos in low-energy relativistic wave- particle systemrdquo PhysicaD vol 80 pp 296ndash306 1995

[2] M Saleem and T Agrawal ldquoChaos in a tumor growth modelwith delayed responses of the immune systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 891095 16 pages2012

[3] U Kumar ldquoA retrospection of chaotic phenomene in electricalsystemsrdquo Active and Passive Electronic Components vol 21 pp1ndash15 1998

10 Abstract and Applied Analysis

[4] Z Chen X Zhang and Q Bi ldquoBifurcations and chaos ofcoupled electrical circuitsrdquo Nonlinear Analysis vol 9 no 3 pp1158ndash1168 2008

[5] S Hallegatte M Ghil P Dumas and J Hourcade ldquoBusinesscycles bifurcations and chaos in a neo-classical model withinvestment dynamicsrdquo Journal of Economic Behavior and Orga-nization vol 67 pp 57ndash77 2008

[6] DGhoshA R Chowdhury andP Saha ldquoBifurcation continua-tion chaos and chaos control in nonlinear Bloch systemrdquo Com-munications inNonlinear Science andNumerical Simulation vol13 no 8 pp 1461ndash1471 2008

[7] A El-Gohary andA S Al-Ruzaiza ldquoChaos and adaptive controlin two prey one predator system with nonlinear feedbackrdquoChaos Solitons and Fractals vol 34 no 2 pp 443ndash453 2007

[8] H Salarieh and A Alasty ldquoControl of stochastic chaos usingsliding mode methodrdquo Journal of Computational and AppliedMathematics vol 225 no 1 pp 135ndash145 2009

[9] J Guan ldquoBifurcation analysis and chaos control in genesiosystem with delayed feedbackrdquo ISRNMathematical Physics vol2012 Article ID 843962 12 pages 2012

[10] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquoNonlinear Analysis vol 9 no 4 pp 1800ndash18102008

[11] N Vasegh and A Sedigh ldquoDelayed feedback control of time-delayed chaotic systems analytical approach at Hopf bifurca-tionrdquo Physics Letters A vol 372 pp 5110ndash5114 2008

[12] J Lu Z Ma and L Li ldquoDouble delayed feedback control for thestabilization of unstable steady states in chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 7 pp 3037ndash3045 2009

[13] M Bleich and J Socolar ldquoStability of periodic orbits controlledby time-delay feedbackrdquo Physics Letters A vol 210 pp 87ndash941996

[14] Y Ding W Jiang and H Wang ldquoDelayed feedback controland bifurcation analysis of Rossler chaotic systemrdquo NonlinearDynamics vol 61 no 4 pp 707ndash715 2010

[15] Y Song and J Wei ldquoBifurcation analysis for Chenrsquos system withdelayed feedback and its application to control of chaosrdquoChaosSolitons and Fractals vol 22 no 1 pp 75ndash91 2004

[16] S Ruan and J Wei ldquoOn the zeros of transcendental functionswith applications to stability of delay differential equations withtwo delaysrdquo Dynamics of Continuous Discrete and ImpulsiveSystems A vol 10 no 6 pp 863ndash874 2003

[17] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge UK 1981

[18] I Fuente L Martinez J Veguillas and J AguirregabirialdquoQuasiperiodicity route to chaos in a biochemical systemrdquoBiophysical Journal vol 71 pp 2375ndash2379 1998

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

Abstract and Applied Analysis 7

0 24

4

0

0

051

152

2

25

0 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

119909

119909

minus5

05

minus5

5

minus5

119910

119910

119911

119911

minus05

minus2 minus2minus4 minus4

119905

119905

119905

Figure 3 The trajectories and phase graphs of system (61) with 120591 = 07362 when 119870 = minus1

02

40

24

002

0406

080 50 100 150 200

0 50 100 150 200

0 50 100 150 200

05

0

0

119909

minus5

5

minus5

1

minus1

119910

119910

119911

119911

minus02

minus2 minus2minus4 minus4

119905

119905

119905

119909

Figure 4 The trajectories and phase graphs of system (61) with 120591 = 13497 when 119870 = minus1

4 Numerical Simulations andApplication to Control of Chaos

In this section we apply the results we get in the previoussections to system (61) for the purpose of control of chaosFrom Section 2 we know that under certain conditions afamily of periodic solutions bifurcates from the steady statesof system (61) at some critical values of 120591 and the stability ofthe steady state may be changed along with the increase of 120591If the bifurcating periodic solution is orbitally asymptoticallystable or some steady state becomes local stable then chaosmay vanish Following this idea we consider the followingdelayed feedback control system for an autonomy system

= 02119909 minus 119910

119910 = 119909 minus 002119910 minus 119911 + 119870 (119910 (119905) minus 119910 (119905 minus 120591))

= 02119909 + 4119911 (119910 minus 2)

(61)

which has two equilibriums 1198640(0 0 0) and 119864lowast(9749 19498

9710) When 120591 = 0 or 119870 = 0 1198640is unstable and system

(61) is chaotic (see Figure 1)The corresponding characteristicequation of system (61) at 119864

0is

1205823+ (782 minus 119870) 120582

2minus (0444 + 78119870) 120582 + 7768

+ 16119870 + (1205822+ 78120582 minus 16)119870119890

minus120582120591= 0

(62)

When 120591 = 0 (62) has one negative root and a pair of complexroots with positive parts By the discussion of Section 2we can get 119901 = 620404 minus 004119870 119902 = minus1212944 minus

25616119870 and 119903 = 603418 + 248576119870 and then we cancompute 998779 = 42129 + 27216119870 + 00016119870

2 and it is easyto know that 998779 gt 0 and when 119870 ge minus24275 119911lowast

1gt 0 always

holdsThen byTheorem 5 we have the following conclusion

Conclusion 1 If 119870 lt minus24275 or 119870 ge minus24275 and ℎ(119911lowast

1) lt 0

then there exist some 1205910gt 0 such that when 120591 isin [0 120591

0) 1198640is

always unstable

For example we fix119870 = minus1 Through computing we canget 119911lowast1= 10693 gt 0 and ℎ(119911lowast

1) = minus192713 lt 0 Furthermore

we can compute

8 Abstract and Applied Analysis

0 50 100 150 200

05

02

001 0 1 2

01

2

0

00501

0 50 100 150 200

0 50 100 150 200

119909

119909

minus5119910

119910

minus2

119911

119911

minus01

minus01

minus1 minus1

minus2minus2

minus3

minus005

119905

119905

119905

Figure 5 The trajectories and phase graphs of system (61) with 120591 = 26815 when 119870 = minus1

005

115

20 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

119911

minus05

4

02

119910 minus2

minus4

02

4

119909

minus2minus4

119905

119905

119905

Figure 6 The trajectories and phase graphs of system (61) with 120591 = 63804 when 119870 = minus1

1199111≐ 03714185849 120596

1≐ 06094

120591(119895)

1= 16243 +

2119895120587

1205961

ℎ1015840(1199111) = minus713282

1199112≐ 1494030027 120596

2≐ 12223

120591(119895)

2= 03768 +

2119895120587

1205962

ℎ1015840(1199112) = 725266

(63)

Thus from Lemma 4 we have

Re 120582 (120591(119895)1)

119889120591lt 0

Re 120582 (120591(119895)2)

119889120591gt 0

(64)

In addition notice that

120591(0)

2= 03768 lt 120591

(0)

1= 16243 lt 120591

(1)

2

= 55173 lt 120591(2)

2= 106577 lt 120591

(1)

1

= 119340 lt 120591(3)

2= 157982 lt sdot sdot sdot

(65)

Thus from (64) and (65) we have the following conclusionabout the stability of the zero equilibrium of system (61) andHopf bifurcations

Conclusion 2 Suppose that 120591(119895)119896 119896 = 1 2 119895 = 0 1 is

defined by (63)

(i) When 120591 isin [0 120591(0)

1) cup (120591(1)

2 +infin) the equilibrium 119864

0is

unstable (see Figures 3 4 6 7 and 8)(ii) When 120591 isin (120591

(0)

1 120591(1)

2) the equilibrium 119864

0is asymptot-

ically stable (see Figure 5)(iii) System (61) undergoes a Hopf bifurcation at the

equilibrium 1198640when 120591 = 120591

(119895)

119896(see Figures 3 4

and 6)

Thus if we take 120591 isin (120591(0)

1 120591(1)

2) the stability of zero

equilibrium of the systemwill change from unstable to stableFor this example there are more complicated dynamicalphenomena occurs Bifurcation diagram is plotted with theparameter 120591 when 119870 = minus1 (Figure 2) In Figure 2 we can seethat chaos vanishes via a period-doubling bifurcation when120591 varying from 0 to 03768 Figure 3 shows that when 120591 =

07362 a period-2 orbits bifurcating from zero equilibriumFigure 4 shows when 120591 = 13497 a period-1 periodicalorbits bifurcating from zero equilibrium In Figure 2 a stablewindow is obtained from 120591 = 16243 to 120591 = 55173 which isdemonstrated by Figure 5 with 120591 = 26815 When 120591 gt 55173Figure 2 shows that periodic orbits and strange attractors also

Abstract and Applied Analysis 9

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

005

115

2

119911

minus05

4

02

119910 minus2

minus4

02

4

119909minus2

minus4

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

119905

119905

119905

Figure 7 The trajectories and phase graphs of system (61) with 120591 = 73006 when 119870 = minus1

05

119909

minus5

05

minus5

05

minus5

119910119911

119911

4

02

119910 minus2

minus4

02

4

119909minus2minus4

minus1

012345

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

119905

119905

119905

Figure 8 The trajectories and phase graphs of system (61) with 120591 = 137423 when 119870 = minus1

occur and system (61) through Hopf bifurcation route stillgoes to chaos and it is demonstrated by Figures 6ndash8

5 Discussion

In this paper an autonomy system with time-delayed feed-back is studied by using the theory of functional differentialequation and Hassardrsquos method Illustrating with numericalsimulations we show that delayed feedback controller (DFC)is an effective method for chaos control

In fact system (2) has more complicated dynamicalbehaves From Figure 2 we can see that chaos degenerationprocess the orbits of system (2) continuously degenerate toa periodic solution region through reverse period-doublingbifurcation With the increasing of 120591 chaos occurs againthrough Hopf bifurcation route However the numericalsimulations indicate that there were points of similaritybetween this route to chaos and quasiperiodicity route tochaos [18] in our future research we will consider this topicAnd in this paper we only give detailed analysis on thetrivial equilibrium 119864

0 the detailed analysis of the interior

equilibrium 119864lowast is the second interested topic in our future

research

Acknowledgments

The authors would like to thank the anonymous refereefor the very helpful suggestions and comments which ledto the improvement of the original paper And this workis supported by 2013 Scientific Research Project of BeifangUniversity of Nationalities (2013XYZ021) Institute of Infor-mation and System Computation Science of Beifang Uni-versity (13xyb01) Science and Technology Department ofHenan Province (122300410417) and Education Departmentof Henan Province (13A110108)

References

[1] G Corso and F B Rizzato ldquoHamiltonian bifurcations leading tochaos in low-energy relativistic wave- particle systemrdquo PhysicaD vol 80 pp 296ndash306 1995

[2] M Saleem and T Agrawal ldquoChaos in a tumor growth modelwith delayed responses of the immune systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 891095 16 pages2012

[3] U Kumar ldquoA retrospection of chaotic phenomene in electricalsystemsrdquo Active and Passive Electronic Components vol 21 pp1ndash15 1998

10 Abstract and Applied Analysis

[4] Z Chen X Zhang and Q Bi ldquoBifurcations and chaos ofcoupled electrical circuitsrdquo Nonlinear Analysis vol 9 no 3 pp1158ndash1168 2008

[5] S Hallegatte M Ghil P Dumas and J Hourcade ldquoBusinesscycles bifurcations and chaos in a neo-classical model withinvestment dynamicsrdquo Journal of Economic Behavior and Orga-nization vol 67 pp 57ndash77 2008

[6] DGhoshA R Chowdhury andP Saha ldquoBifurcation continua-tion chaos and chaos control in nonlinear Bloch systemrdquo Com-munications inNonlinear Science andNumerical Simulation vol13 no 8 pp 1461ndash1471 2008

[7] A El-Gohary andA S Al-Ruzaiza ldquoChaos and adaptive controlin two prey one predator system with nonlinear feedbackrdquoChaos Solitons and Fractals vol 34 no 2 pp 443ndash453 2007

[8] H Salarieh and A Alasty ldquoControl of stochastic chaos usingsliding mode methodrdquo Journal of Computational and AppliedMathematics vol 225 no 1 pp 135ndash145 2009

[9] J Guan ldquoBifurcation analysis and chaos control in genesiosystem with delayed feedbackrdquo ISRNMathematical Physics vol2012 Article ID 843962 12 pages 2012

[10] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquoNonlinear Analysis vol 9 no 4 pp 1800ndash18102008

[11] N Vasegh and A Sedigh ldquoDelayed feedback control of time-delayed chaotic systems analytical approach at Hopf bifurca-tionrdquo Physics Letters A vol 372 pp 5110ndash5114 2008

[12] J Lu Z Ma and L Li ldquoDouble delayed feedback control for thestabilization of unstable steady states in chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 7 pp 3037ndash3045 2009

[13] M Bleich and J Socolar ldquoStability of periodic orbits controlledby time-delay feedbackrdquo Physics Letters A vol 210 pp 87ndash941996

[14] Y Ding W Jiang and H Wang ldquoDelayed feedback controland bifurcation analysis of Rossler chaotic systemrdquo NonlinearDynamics vol 61 no 4 pp 707ndash715 2010

[15] Y Song and J Wei ldquoBifurcation analysis for Chenrsquos system withdelayed feedback and its application to control of chaosrdquoChaosSolitons and Fractals vol 22 no 1 pp 75ndash91 2004

[16] S Ruan and J Wei ldquoOn the zeros of transcendental functionswith applications to stability of delay differential equations withtwo delaysrdquo Dynamics of Continuous Discrete and ImpulsiveSystems A vol 10 no 6 pp 863ndash874 2003

[17] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge UK 1981

[18] I Fuente L Martinez J Veguillas and J AguirregabirialdquoQuasiperiodicity route to chaos in a biochemical systemrdquoBiophysical Journal vol 71 pp 2375ndash2379 1998

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 Abstract and Applied Analysis

0 50 100 150 200

05

02

001 0 1 2

01

2

0

00501

0 50 100 150 200

0 50 100 150 200

119909

119909

minus5119910

119910

minus2

119911

119911

minus01

minus01

minus1 minus1

minus2minus2

minus3

minus005

119905

119905

119905

Figure 5 The trajectories and phase graphs of system (61) with 120591 = 26815 when 119870 = minus1

005

115

20 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

119911

minus05

4

02

119910 minus2

minus4

02

4

119909

minus2minus4

119905

119905

119905

Figure 6 The trajectories and phase graphs of system (61) with 120591 = 63804 when 119870 = minus1

1199111≐ 03714185849 120596

1≐ 06094

120591(119895)

1= 16243 +

2119895120587

1205961

ℎ1015840(1199111) = minus713282

1199112≐ 1494030027 120596

2≐ 12223

120591(119895)

2= 03768 +

2119895120587

1205962

ℎ1015840(1199112) = 725266

(63)

Thus from Lemma 4 we have

Re 120582 (120591(119895)1)

119889120591lt 0

Re 120582 (120591(119895)2)

119889120591gt 0

(64)

In addition notice that

120591(0)

2= 03768 lt 120591

(0)

1= 16243 lt 120591

(1)

2

= 55173 lt 120591(2)

2= 106577 lt 120591

(1)

1

= 119340 lt 120591(3)

2= 157982 lt sdot sdot sdot

(65)

Thus from (64) and (65) we have the following conclusionabout the stability of the zero equilibrium of system (61) andHopf bifurcations

Conclusion 2 Suppose that 120591(119895)119896 119896 = 1 2 119895 = 0 1 is

defined by (63)

(i) When 120591 isin [0 120591(0)

1) cup (120591(1)

2 +infin) the equilibrium 119864

0is

unstable (see Figures 3 4 6 7 and 8)(ii) When 120591 isin (120591

(0)

1 120591(1)

2) the equilibrium 119864

0is asymptot-

ically stable (see Figure 5)(iii) System (61) undergoes a Hopf bifurcation at the

equilibrium 1198640when 120591 = 120591

(119895)

119896(see Figures 3 4

and 6)

Thus if we take 120591 isin (120591(0)

1 120591(1)

2) the stability of zero

equilibrium of the systemwill change from unstable to stableFor this example there are more complicated dynamicalphenomena occurs Bifurcation diagram is plotted with theparameter 120591 when 119870 = minus1 (Figure 2) In Figure 2 we can seethat chaos vanishes via a period-doubling bifurcation when120591 varying from 0 to 03768 Figure 3 shows that when 120591 =

07362 a period-2 orbits bifurcating from zero equilibriumFigure 4 shows when 120591 = 13497 a period-1 periodicalorbits bifurcating from zero equilibrium In Figure 2 a stablewindow is obtained from 120591 = 16243 to 120591 = 55173 which isdemonstrated by Figure 5 with 120591 = 26815 When 120591 gt 55173Figure 2 shows that periodic orbits and strange attractors also

Abstract and Applied Analysis 9

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

005

115

2

119911

minus05

4

02

119910 minus2

minus4

02

4

119909minus2

minus4

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

119905

119905

119905

Figure 7 The trajectories and phase graphs of system (61) with 120591 = 73006 when 119870 = minus1

05

119909

minus5

05

minus5

05

minus5

119910119911

119911

4

02

119910 minus2

minus4

02

4

119909minus2minus4

minus1

012345

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

119905

119905

119905

Figure 8 The trajectories and phase graphs of system (61) with 120591 = 137423 when 119870 = minus1

occur and system (61) through Hopf bifurcation route stillgoes to chaos and it is demonstrated by Figures 6ndash8

5 Discussion

In this paper an autonomy system with time-delayed feed-back is studied by using the theory of functional differentialequation and Hassardrsquos method Illustrating with numericalsimulations we show that delayed feedback controller (DFC)is an effective method for chaos control

In fact system (2) has more complicated dynamicalbehaves From Figure 2 we can see that chaos degenerationprocess the orbits of system (2) continuously degenerate toa periodic solution region through reverse period-doublingbifurcation With the increasing of 120591 chaos occurs againthrough Hopf bifurcation route However the numericalsimulations indicate that there were points of similaritybetween this route to chaos and quasiperiodicity route tochaos [18] in our future research we will consider this topicAnd in this paper we only give detailed analysis on thetrivial equilibrium 119864

0 the detailed analysis of the interior

equilibrium 119864lowast is the second interested topic in our future

research

Acknowledgments

The authors would like to thank the anonymous refereefor the very helpful suggestions and comments which ledto the improvement of the original paper And this workis supported by 2013 Scientific Research Project of BeifangUniversity of Nationalities (2013XYZ021) Institute of Infor-mation and System Computation Science of Beifang Uni-versity (13xyb01) Science and Technology Department ofHenan Province (122300410417) and Education Departmentof Henan Province (13A110108)

References

[1] G Corso and F B Rizzato ldquoHamiltonian bifurcations leading tochaos in low-energy relativistic wave- particle systemrdquo PhysicaD vol 80 pp 296ndash306 1995

[2] M Saleem and T Agrawal ldquoChaos in a tumor growth modelwith delayed responses of the immune systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 891095 16 pages2012

[3] U Kumar ldquoA retrospection of chaotic phenomene in electricalsystemsrdquo Active and Passive Electronic Components vol 21 pp1ndash15 1998

10 Abstract and Applied Analysis

[4] Z Chen X Zhang and Q Bi ldquoBifurcations and chaos ofcoupled electrical circuitsrdquo Nonlinear Analysis vol 9 no 3 pp1158ndash1168 2008

[5] S Hallegatte M Ghil P Dumas and J Hourcade ldquoBusinesscycles bifurcations and chaos in a neo-classical model withinvestment dynamicsrdquo Journal of Economic Behavior and Orga-nization vol 67 pp 57ndash77 2008

[6] DGhoshA R Chowdhury andP Saha ldquoBifurcation continua-tion chaos and chaos control in nonlinear Bloch systemrdquo Com-munications inNonlinear Science andNumerical Simulation vol13 no 8 pp 1461ndash1471 2008

[7] A El-Gohary andA S Al-Ruzaiza ldquoChaos and adaptive controlin two prey one predator system with nonlinear feedbackrdquoChaos Solitons and Fractals vol 34 no 2 pp 443ndash453 2007

[8] H Salarieh and A Alasty ldquoControl of stochastic chaos usingsliding mode methodrdquo Journal of Computational and AppliedMathematics vol 225 no 1 pp 135ndash145 2009

[9] J Guan ldquoBifurcation analysis and chaos control in genesiosystem with delayed feedbackrdquo ISRNMathematical Physics vol2012 Article ID 843962 12 pages 2012

[10] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquoNonlinear Analysis vol 9 no 4 pp 1800ndash18102008

[11] N Vasegh and A Sedigh ldquoDelayed feedback control of time-delayed chaotic systems analytical approach at Hopf bifurca-tionrdquo Physics Letters A vol 372 pp 5110ndash5114 2008

[12] J Lu Z Ma and L Li ldquoDouble delayed feedback control for thestabilization of unstable steady states in chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 7 pp 3037ndash3045 2009

[13] M Bleich and J Socolar ldquoStability of periodic orbits controlledby time-delay feedbackrdquo Physics Letters A vol 210 pp 87ndash941996

[14] Y Ding W Jiang and H Wang ldquoDelayed feedback controland bifurcation analysis of Rossler chaotic systemrdquo NonlinearDynamics vol 61 no 4 pp 707ndash715 2010

[15] Y Song and J Wei ldquoBifurcation analysis for Chenrsquos system withdelayed feedback and its application to control of chaosrdquoChaosSolitons and Fractals vol 22 no 1 pp 75ndash91 2004

[16] S Ruan and J Wei ldquoOn the zeros of transcendental functionswith applications to stability of delay differential equations withtwo delaysrdquo Dynamics of Continuous Discrete and ImpulsiveSystems A vol 10 no 6 pp 863ndash874 2003

[17] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge UK 1981

[18] I Fuente L Martinez J Veguillas and J AguirregabirialdquoQuasiperiodicity route to chaos in a biochemical systemrdquoBiophysical Journal vol 71 pp 2375ndash2379 1998

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

Abstract and Applied Analysis 9

05

119909

minus5

0

0

5

2

minus5

minus2

119910119911

005

115

2

119911

minus05

4

02

119910 minus2

minus4

02

4

119909minus2

minus4

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

0 50 100 300 400150 250 350200

119905

119905

119905

Figure 7 The trajectories and phase graphs of system (61) with 120591 = 73006 when 119870 = minus1

05

119909

minus5

05

minus5

05

minus5

119910119911

119911

4

02

119910 minus2

minus4

02

4

119909minus2minus4

minus1

012345

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

0 100 200 300 400 500 600 700 800

119905

119905

119905

Figure 8 The trajectories and phase graphs of system (61) with 120591 = 137423 when 119870 = minus1

occur and system (61) through Hopf bifurcation route stillgoes to chaos and it is demonstrated by Figures 6ndash8

5 Discussion

In this paper an autonomy system with time-delayed feed-back is studied by using the theory of functional differentialequation and Hassardrsquos method Illustrating with numericalsimulations we show that delayed feedback controller (DFC)is an effective method for chaos control

In fact system (2) has more complicated dynamicalbehaves From Figure 2 we can see that chaos degenerationprocess the orbits of system (2) continuously degenerate toa periodic solution region through reverse period-doublingbifurcation With the increasing of 120591 chaos occurs againthrough Hopf bifurcation route However the numericalsimulations indicate that there were points of similaritybetween this route to chaos and quasiperiodicity route tochaos [18] in our future research we will consider this topicAnd in this paper we only give detailed analysis on thetrivial equilibrium 119864

0 the detailed analysis of the interior

equilibrium 119864lowast is the second interested topic in our future

research

Acknowledgments

The authors would like to thank the anonymous refereefor the very helpful suggestions and comments which ledto the improvement of the original paper And this workis supported by 2013 Scientific Research Project of BeifangUniversity of Nationalities (2013XYZ021) Institute of Infor-mation and System Computation Science of Beifang Uni-versity (13xyb01) Science and Technology Department ofHenan Province (122300410417) and Education Departmentof Henan Province (13A110108)

References

[1] G Corso and F B Rizzato ldquoHamiltonian bifurcations leading tochaos in low-energy relativistic wave- particle systemrdquo PhysicaD vol 80 pp 296ndash306 1995

[2] M Saleem and T Agrawal ldquoChaos in a tumor growth modelwith delayed responses of the immune systemrdquo Journal ofApplied Mathematics vol 2012 Article ID 891095 16 pages2012

[3] U Kumar ldquoA retrospection of chaotic phenomene in electricalsystemsrdquo Active and Passive Electronic Components vol 21 pp1ndash15 1998

10 Abstract and Applied Analysis

[4] Z Chen X Zhang and Q Bi ldquoBifurcations and chaos ofcoupled electrical circuitsrdquo Nonlinear Analysis vol 9 no 3 pp1158ndash1168 2008

[5] S Hallegatte M Ghil P Dumas and J Hourcade ldquoBusinesscycles bifurcations and chaos in a neo-classical model withinvestment dynamicsrdquo Journal of Economic Behavior and Orga-nization vol 67 pp 57ndash77 2008

[6] DGhoshA R Chowdhury andP Saha ldquoBifurcation continua-tion chaos and chaos control in nonlinear Bloch systemrdquo Com-munications inNonlinear Science andNumerical Simulation vol13 no 8 pp 1461ndash1471 2008

[7] A El-Gohary andA S Al-Ruzaiza ldquoChaos and adaptive controlin two prey one predator system with nonlinear feedbackrdquoChaos Solitons and Fractals vol 34 no 2 pp 443ndash453 2007

[8] H Salarieh and A Alasty ldquoControl of stochastic chaos usingsliding mode methodrdquo Journal of Computational and AppliedMathematics vol 225 no 1 pp 135ndash145 2009

[9] J Guan ldquoBifurcation analysis and chaos control in genesiosystem with delayed feedbackrdquo ISRNMathematical Physics vol2012 Article ID 843962 12 pages 2012

[10] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquoNonlinear Analysis vol 9 no 4 pp 1800ndash18102008

[11] N Vasegh and A Sedigh ldquoDelayed feedback control of time-delayed chaotic systems analytical approach at Hopf bifurca-tionrdquo Physics Letters A vol 372 pp 5110ndash5114 2008

[12] J Lu Z Ma and L Li ldquoDouble delayed feedback control for thestabilization of unstable steady states in chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 7 pp 3037ndash3045 2009

[13] M Bleich and J Socolar ldquoStability of periodic orbits controlledby time-delay feedbackrdquo Physics Letters A vol 210 pp 87ndash941996

[14] Y Ding W Jiang and H Wang ldquoDelayed feedback controland bifurcation analysis of Rossler chaotic systemrdquo NonlinearDynamics vol 61 no 4 pp 707ndash715 2010

[15] Y Song and J Wei ldquoBifurcation analysis for Chenrsquos system withdelayed feedback and its application to control of chaosrdquoChaosSolitons and Fractals vol 22 no 1 pp 75ndash91 2004

[16] S Ruan and J Wei ldquoOn the zeros of transcendental functionswith applications to stability of delay differential equations withtwo delaysrdquo Dynamics of Continuous Discrete and ImpulsiveSystems A vol 10 no 6 pp 863ndash874 2003

[17] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge UK 1981

[18] I Fuente L Martinez J Veguillas and J AguirregabirialdquoQuasiperiodicity route to chaos in a biochemical systemrdquoBiophysical Journal vol 71 pp 2375ndash2379 1998

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 Abstract and Applied Analysis

[4] Z Chen X Zhang and Q Bi ldquoBifurcations and chaos ofcoupled electrical circuitsrdquo Nonlinear Analysis vol 9 no 3 pp1158ndash1168 2008

[5] S Hallegatte M Ghil P Dumas and J Hourcade ldquoBusinesscycles bifurcations and chaos in a neo-classical model withinvestment dynamicsrdquo Journal of Economic Behavior and Orga-nization vol 67 pp 57ndash77 2008

[6] DGhoshA R Chowdhury andP Saha ldquoBifurcation continua-tion chaos and chaos control in nonlinear Bloch systemrdquo Com-munications inNonlinear Science andNumerical Simulation vol13 no 8 pp 1461ndash1471 2008

[7] A El-Gohary andA S Al-Ruzaiza ldquoChaos and adaptive controlin two prey one predator system with nonlinear feedbackrdquoChaos Solitons and Fractals vol 34 no 2 pp 443ndash453 2007

[8] H Salarieh and A Alasty ldquoControl of stochastic chaos usingsliding mode methodrdquo Journal of Computational and AppliedMathematics vol 225 no 1 pp 135ndash145 2009

[9] J Guan ldquoBifurcation analysis and chaos control in genesiosystem with delayed feedbackrdquo ISRNMathematical Physics vol2012 Article ID 843962 12 pages 2012

[10] H-T Yau and C-S Shieh ldquoChaos synchronization using fuzzylogic controllerrdquoNonlinear Analysis vol 9 no 4 pp 1800ndash18102008

[11] N Vasegh and A Sedigh ldquoDelayed feedback control of time-delayed chaotic systems analytical approach at Hopf bifurca-tionrdquo Physics Letters A vol 372 pp 5110ndash5114 2008

[12] J Lu Z Ma and L Li ldquoDouble delayed feedback control for thestabilization of unstable steady states in chaotic systemsrdquo Com-munications inNonlinear Science andNumerical Simulation vol14 no 7 pp 3037ndash3045 2009

[13] M Bleich and J Socolar ldquoStability of periodic orbits controlledby time-delay feedbackrdquo Physics Letters A vol 210 pp 87ndash941996

[14] Y Ding W Jiang and H Wang ldquoDelayed feedback controland bifurcation analysis of Rossler chaotic systemrdquo NonlinearDynamics vol 61 no 4 pp 707ndash715 2010

[15] Y Song and J Wei ldquoBifurcation analysis for Chenrsquos system withdelayed feedback and its application to control of chaosrdquoChaosSolitons and Fractals vol 22 no 1 pp 75ndash91 2004

[16] S Ruan and J Wei ldquoOn the zeros of transcendental functionswith applications to stability of delay differential equations withtwo delaysrdquo Dynamics of Continuous Discrete and ImpulsiveSystems A vol 10 no 6 pp 863ndash874 2003

[17] B D Hassard N D Kazarinoff and Y H Wan Theory andApplications of Hopf Bifurcation vol 41 Cambridge UniversityPress Cambridge UK 1981

[18] I Fuente L Martinez J Veguillas and J AguirregabirialdquoQuasiperiodicity route to chaos in a biochemical systemrdquoBiophysical Journal vol 71 pp 2375ndash2379 1998

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