# Two Types of Synchronization Problems in a New 5D Hyperchaotic System

Authors: Bin Li; Xue Yang; Qixing Liang; Zhi Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper investigates the synchronization problem in a new 5D hyperchaotic system. Firstly, the existence of two types of synchronization problems in the new 5D hyperchaotic system is proved. Then, by the dynamic feedback control method, one complete synchronization problem and three coexistence of complete synchronization and antisynchronization problems in such system are realized. Finally, numerical simulations are used to verify the validity and effectiveness of the theoretical results.


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## Body

## 1. Introduction

Since Lorenz proposed the first chaotic system in 1963, many researchers are stimulated to investigate this chaotic phenomena. Then, lots of chaotic systems and hyperchaotic systems are proposed. Since the OGY method [1] and PC method [2] were first observed for chaos control and chaos synchronization in 1990, respectively, the control problems and synchronization problems of those chaotic systems have become hot topics, see References [13–15] and the references therein. However, there are still some important questions need to be solved completely. For example, the existence of the synchronization problem in a given chaotic or hyperchaotic system, which is a fundamental theoretical base to design a physical controller. Furthermore, how to design a not only simple but also physical controller to realize the synchronization problem is also a question which needs to be solved. For the realization of such synchronization problems, there are many methods to use, linear feedback control method, dynamic feedback control method, sliding mode control method, and so on. Among those methods, the dynamic feedback control method has been applied often in applications, which is also used in this paper.The new 5D hyperchaotic system was firstly presented in [16], and it has complex dynamics. There are eight parameters to control with only one equilibrium point. This new dynamics can generate four-wing hyperchaotic and chaotic attractors for some specific parameters and initial conditions. One remarkable feature of the new system is that it can generate double-wing and four-wing smooth chaotic attractors with special appearance. However, the synchronization problem in this system has not been solved. Moreover, for the new 5D hyperchaotic system, the existence of the synchronization problems in such system is a fundamental question, but it still not be solved. Therefore, investigating the synchronization problems in the new 5D hyperchaotic system is very important in both theory and applications, which motivates our work in this paper.Motivated by the abovementioned conclusions, the synchronization problem in the new 5D chaotic system is studied by the dynamic feedback control method. The main contributions of this paper are given as follows:(1)
The existence of two types of synchronization problems in the new 5D system is proved(2)
These two types of synchronization problems in the new 5D system are realized by the dynamic feedback control method


## 2. Problem Formulation

According to [16], the new 5D hyperchaotic system is given as(1)x˙=fx,where x∈R5 is the state and(2)fx=f1xf2xf3xf4xf5x=−10x1+x2x3−60x2+x5−20x3+50x4+x1x215x4−10x340x5−x12x2.Let system (1) be the master system, then the controlled slave system is described as follows:(3)y˙=fy+Bu,where y∈R5 is the state, (4)B=000000100010001,and u∈R3 is the controller to be designed.This paper investigates the synchronization problem of the master system (1) and the slave system (3) and presents some new results.For the development of this paper, the dynamic feedback control method is introduced in the next.


## 3. Preliminary

Consider the following controlled chaotic system:(5)z˙=Fz+Bu,where z∈Rn is the state, Fz∈Rn is continuous vector function with F0=0, B∈Rn×r is a constant matrix, and u is the designed controller.Lemma 1 (see [17]).
Consider system (5), if Fz,B can be stabilized; then, the controller u is designed as follows:(6)u=Ktz,where K=ktBT, and the feedback gain kt is updated by the following equation:(7)k˙t=−zTz=−z2.


## 4. Main Results

### 4.1. The Existence of Synchronization Problems in the New 5D Hyperchaotic System

According to the results in [17], the existence of synchronization problems in the 5D hyperchaotic system (1) is proved by the following condition:(8)fαx≡αfx,where α=Diagα1,α2,…,α5, αi≠0, i=1,2,…,5,(9)f1αx−α1f1x=α2α3−α1x2x3≡0,f2αx−α2f2x=α2−α5x5≡0,f3αx−α3f3x=50α3−α4x4+α1α2−α3x1x2≡0,f4αx−α4f4x=−10α3−α4x3≡0,f5αx−α4f4x=−α12α2−α5x1≡0.It results in(10)α2α3=α1,α2=α5,α1α2=α3,α3=α4,α12α2=α5.Solving equation (10), we obtain(I)
α1=α2=α3=α4=α5=1, which implies the complete synchronization problem exists(II)
α1=α2=α5=−1 and α3=α4=1, which implies the coexistence of complete synchronization and antisynchronization problem exists(III)
α2=α3=α4=α5=−1 and α1=1, which implies the coexistence of complete synchronization and antisynchronization problem exists(IV)
α1=α3=α4=−1 and α2=α5=1, which implies the coexistence of complete synchronization and antisynchronization problem exists


### 4.2. Complete Synchronization in the New 5D Hyperchaotic System

For the master system (1) and the slave system (3), let e=y−x, and the error system is given as(11)e˙=fy−fx+Bu=gx,e+Bu,where(12)gx,e=g1x,eg2x,eg3x,eg4x,eg5x,e=−10e1+x2e3+x3e2+e2e3−60e2+e5−20e3+50e4+x1e2+x2e1+e1e215e4−10e340e5−2x1x2e1−2x1e2e1−x2e12−e12e2,and B is given in (4).According to Lemma1, we propose the following conclusion.Theorem 1.
Consider the error system (11). If the controller u is designed as follows(13)u=Ke,where K=ktBT, B is given in (4), and kt is updated by the following update law:(14)k˙=−e2.Then, the master system (1) and the slave system (3) reach complete synchronization.Proof.
For the error system (11) with u=0, if e3=e4=e5=0, then the following subsystem(15)e˙1=−10e1+x3e2,e˙2=−60e2,is asymptotically stable.
Thus,gx,e,B can be stabilized. According to Lemma 1, the error system (11) is stabilized by the abovementioned controller u given in (13), which completes the proof.In the following, numerical simulation is carried out with the initial conditions:x10=1,x20=−2,x30=3,x40=−4, y10=5,y20=−6,y30=7,y40=−8, and k0=−1. Figure 1 shows e1,e2,ande3 are asymptotically stable, Figure 2 shows e4ande5 are asymptotically stable, and Figure 3 shows that the feedback gain kt tends to constant.Figure 1
e
1
,
e
2
,
and
e
3 are asymptotically stable.Figure 2
e
4
and
e
5 are asymptotically stable.Figure 3
The feedback gainkt tends to constant.


### 4.3. Coexistence of Synchronization and Antisynchronization in the New 5D Hyperchaotic System

Case 1.
α
1
=
α
2
=
α
5
=
−
1 and α3=α4=1.
LetEi=xi+yi,i=1,2,5, and ei=yi−xi,i=3,4; then, the sum and error system is described as follows:(16)E˙e˙=E˙1E˙2E˙5e˙3e˙4=f1y+f1xf2y+f2xf5y+f5xf3y−f3xf4y−f4x+Bu=Gx,E,e+Bu,where(17)Gx,E,e=G1x,E,eG2x,E,eG5x,E,eG3x,E,eG4x,E,e=−10E1−x2e3−x3E2+E2e3−60E2+E540e5−2x1x2E1+2x1E2E1+x2E12−E12E2−20e3+50e4−x1E2−x2E1+E1E215e4−10e3,and B is given in (4).
According to Lemma1, we propose the following conclusion.Theorem 2.
Consider the sum and error system (16). If controller u is designed as follows(18)u=KEe,where K=ktBT, B is given as (4), and kt is updated by the following update law:(19)k˙=−e2+E2.Then, the master system (1) and the slave system (3) realize the coexistence of complete synchronization and antisynchronization.Proof.
For the sum and error system (16) with u=0, if e3=e4=E5=0, then the following subsystem(20)E˙1=−10E1−x3E2,E˙2=−60E2,is asymptotically stable.
Thus,Gx,E,e,B can be stabilized. According to Lemma 1, the sum and the error system (16) is stabilized by the abovementioned controller u given in (18), which completes the proof.In the following, numerical simulation is carried out with the initial conditions:x10=1,x20=−2,x30=3,x40=−4, y10=5,y20=−6,y30=7,y40=−8, and k0=−1. Figure 4 shows E1,E2,ande3 are asymptotically stable, Figure 5 shows e4andE5 are asymptotically stable, and Figure 6 shows that the feedback gain kt tends to constant.Figure 4
E
1
,
E
2
,
and
e
3 are asymptotically stable.Figure 5
e
4
,
and
E
5 are asymptotically stable.Figure 6
The feedback gainkt tends to constant.Case 2.
α
2
=
α
3
=
α
4
=
α
5
=
−
1 and α1=1.
LetEi=xi+yi,i=2,3,4,5, and e1=y1−x1, and then the sum and error system is described as follows:(21)E˙e˙=E˙2E˙3E˙4E˙5e˙1=f2y+f2xf3y+f3xf4y+f4xf5y+f5xf1y−f1x+B∗u=G∗x,E,e+B∗u,where(22)G∗x,E,e=G2∗x,E,eG3∗x,E,eG4∗x,E,eG5∗x,E,eG1∗x,E,e=−60E2+E5−20E3+50E4+x1E2−x2e1+e1E215E4−10E340E5−2x1x2e1−2x1E2e1+x2e12−e12E2−10e1+x2E3+x3E2+E2E3,and(23)B∗=000100010001000.According to Lemma 1, we propose the following conclusion.Theorem 3.
Consider the controlled sum and error system (21). If controller u is designed as follows(24)u=KEe,where K=ktB∗T, B∗ is given in (23), and kt is updated by the following update law:(25)k˙=−e2+E2.Then, the master system (1) and the slave system (3) realize the coexistence of complete synchronization and antisynchronization.Proof.
For the sum and error system (21) with u=0, if E3=E4=E5=0, then the following subsystem(26)e˙1=−10e1+x3E2,E˙2=−60E2,is asymptotically stable.
Thus,Gx,E,e,B can be stabilized. According to Lemma 1, the sum and the error system (21) is stabilized by the abovementioned controller u given in (24), which completes the proof.In the following, numerical simulation is carried out with the initial conditions:x10=1,x20=−2,x30=3,x40=−4, y10=5,y20=−6,y30=7,y40=−8, and k0=−1. Figure 7 shows e1,E2,andE3 are asymptotically stable, Figure 8 shows E4andE5 are asymptotically stable, and Figure 9 shows that the feedback gain kt tends to constant.Figure 7
e
1
,
E
2
,
and
E
3 are asymptotically stable.Figure 8
E
4
and
E
5 are asymptotically stable.Figure 9
The feedback gainkt tends to constant.Case 3.
α
1
=
α
3
=
α
4
=
−
1 and α2=α5=1.
LetEi=xi+yi,i=1,3,4, and ei=yi−xi,i=2,5, and then the sum and error system is described as follows:(27)E˙e˙=E˙1E˙3E˙4e˙2e˙5=f1y+f1xf3y+f3xf4y+f4xf2y−f2xf5y−f5x+B∗∗u=G∗∗x,E,e+B∗∗u,where(28)G∗∗x,E,e=G1∗∗x,E,eG3∗∗x,E,eG4∗∗x,E,eG2∗∗x,E,eG5∗∗x,E,e=−10E1+x2E3−x3e2+e2E3−20E3+50E4−x1e2+x2E1+E1e215E4−10E3−60e2+e540e5+2x1x2E1+2x1e2E1−x2E12−E12e2and(29)B∗∗=000100010000001.According to Lemma 1, we propose the following conclusion.Theorem 4.
Consider the controlled sum and error system (27). If controller u is designed as follows:(30)u=KEe,where K=ktB∗∗T, B∗∗ is given in (29), and kt is updated by the following update law:(31)k˙=−e2+E2.Then, the master system (1) and the slave system (3) realize the coexistence of complete synchronization and antisynchronization.Proof.
For the sum and error system (27) with u=0, if E3=E4=E5=0, then the following subsystem(32)E˙1=−10e1−x3e2,e˙2=−60e2,is asymptotically stable.
Thus,G∗∗x,E,e,B∗∗ can be stabilized. According to Lemma 1, the sum and the error system (27) is stabilized by the abovementioned controller u given in (30), which completes the proof.In the following, numerical simulation is carried out with the initial conditions:x10=1,x20=−2,x30=3,x40=−4, y10=5,y20=−6,y30=7,y40=−8, and k0=−1. Figure 10 shows E1,e2,andE3 are asymptotically stable, Figure 11 E4andE5 are asymptotically stable, Figure 12 shows that the feedback gain kt tends to constant.Figure 10
E
1
,
e
2
,
and
E
3 are asymptotically stable.Figure 11
E
4
and
E
5 are asymptotically stable.Figure 12
The feedback gainkt tends to constant.


## 4.1. The Existence of Synchronization Problems in the New 5D Hyperchaotic System

According to the results in [17], the existence of synchronization problems in the 5D hyperchaotic system (1) is proved by the following condition:(8)fαx≡αfx,where α=Diagα1,α2,…,α5, αi≠0, i=1,2,…,5,(9)f1αx−α1f1x=α2α3−α1x2x3≡0,f2αx−α2f2x=α2−α5x5≡0,f3αx−α3f3x=50α3−α4x4+α1α2−α3x1x2≡0,f4αx−α4f4x=−10α3−α4x3≡0,f5αx−α4f4x=−α12α2−α5x1≡0.It results in(10)α2α3=α1,α2=α5,α1α2=α3,α3=α4,α12α2=α5.Solving equation (10), we obtain(I)
α1=α2=α3=α4=α5=1, which implies the complete synchronization problem exists(II)
α1=α2=α5=−1 and α3=α4=1, which implies the coexistence of complete synchronization and antisynchronization problem exists(III)
α2=α3=α4=α5=−1 and α1=1, which implies the coexistence of complete synchronization and antisynchronization problem exists(IV)
α1=α3=α4=−1 and α2=α5=1, which implies the coexistence of complete synchronization and antisynchronization problem exists


## 4.2. Complete Synchronization in the New 5D Hyperchaotic System

For the master system (1) and the slave system (3), let e=y−x, and the error system is given as(11)e˙=fy−fx+Bu=gx,e+Bu,where(12)gx,e=g1x,eg2x,eg3x,eg4x,eg5x,e=−10e1+x2e3+x3e2+e2e3−60e2+e5−20e3+50e4+x1e2+x2e1+e1e215e4−10e340e5−2x1x2e1−2x1e2e1−x2e12−e12e2,and B is given in (4).According to Lemma1, we propose the following conclusion.Theorem 1.
Consider the error system (11). If the controller u is designed as follows(13)u=Ke,where K=ktBT, B is given in (4), and kt is updated by the following update law:(14)k˙=−e2.Then, the master system (1) and the slave system (3) reach complete synchronization.Proof.
For the error system (11) with u=0, if e3=e4=e5=0, then the following subsystem(15)e˙1=−10e1+x3e2,e˙2=−60e2,is asymptotically stable.
Thus,gx,e,B can be stabilized. According to Lemma 1, the error system (11) is stabilized by the abovementioned controller u given in (13), which completes the proof.In the following, numerical simulation is carried out with the initial conditions:x10=1,x20=−2,x30=3,x40=−4, y10=5,y20=−6,y30=7,y40=−8, and k0=−1. Figure 1 shows e1,e2,ande3 are asymptotically stable, Figure 2 shows e4ande5 are asymptotically stable, and Figure 3 shows that the feedback gain kt tends to constant.Figure 1
e
1
,
e
2
,
and
e
3 are asymptotically stable.Figure 2
e
4
and
e
5 are asymptotically stable.Figure 3
The feedback gainkt tends to constant.


## 4.3. Coexistence of Synchronization and Antisynchronization in the New 5D Hyperchaotic System

Case 1.
α
1
=
α
2
=
α
5
=
−
1 and α3=α4=1.
LetEi=xi+yi,i=1,2,5, and ei=yi−xi,i=3,4; then, the sum and error system is described as follows:(16)E˙e˙=E˙1E˙2E˙5e˙3e˙4=f1y+f1xf2y+f2xf5y+f5xf3y−f3xf4y−f4x+Bu=Gx,E,e+Bu,where(17)Gx,E,e=G1x,E,eG2x,E,eG5x,E,eG3x,E,eG4x,E,e=−10E1−x2e3−x3E2+E2e3−60E2+E540e5−2x1x2E1+2x1E2E1+x2E12−E12E2−20e3+50e4−x1E2−x2E1+E1E215e4−10e3,and B is given in (4).
According to Lemma1, we propose the following conclusion.Theorem 2.
Consider the sum and error system (16). If controller u is designed as follows(18)u=KEe,where K=ktBT, B is given as (4), and kt is updated by the following update law:(19)k˙=−e2+E2.Then, the master system (1) and the slave system (3) realize the coexistence of complete synchronization and antisynchronization.Proof.
For the sum and error system (16) with u=0, if e3=e4=E5=0, then the following subsystem(20)E˙1=−10E1−x3E2,E˙2=−60E2,is asymptotically stable.
Thus,Gx,E,e,B can be stabilized. According to Lemma 1, the sum and the error system (16) is stabilized by the abovementioned controller u given in (18), which completes the proof.In the following, numerical simulation is carried out with the initial conditions:x10=1,x20=−2,x30=3,x40=−4, y10=5,y20=−6,y30=7,y40=−8, and k0=−1. Figure 4 shows E1,E2,ande3 are asymptotically stable, Figure 5 shows e4andE5 are asymptotically stable, and Figure 6 shows that the feedback gain kt tends to constant.Figure 4
E
1
,
E
2
,
and
e
3 are asymptotically stable.Figure 5
e
4
,
and
E
5 are asymptotically stable.Figure 6
The feedback gainkt tends to constant.Case 2.
α
2
=
α
3
=
α
4
=
α
5
=
−
1 and α1=1.
LetEi=xi+yi,i=2,3,4,5, and e1=y1−x1, and then the sum and error system is described as follows:(21)E˙e˙=E˙2E˙3E˙4E˙5e˙1=f2y+f2xf3y+f3xf4y+f4xf5y+f5xf1y−f1x+B∗u=G∗x,E,e+B∗u,where(22)G∗x,E,e=G2∗x,E,eG3∗x,E,eG4∗x,E,eG5∗x,E,eG1∗x,E,e=−60E2+E5−20E3+50E4+x1E2−x2e1+e1E215E4−10E340E5−2x1x2e1−2x1E2e1+x2e12−e12E2−10e1+x2E3+x3E2+E2E3,and(23)B∗=000100010001000.According to Lemma 1, we propose the following conclusion.Theorem 3.
Consider the controlled sum and error system (21). If controller u is designed as follows(24)u=KEe,where K=ktB∗T, B∗ is given in (23), and kt is updated by the following update law:(25)k˙=−e2+E2.Then, the master system (1) and the slave system (3) realize the coexistence of complete synchronization and antisynchronization.Proof.
For the sum and error system (21) with u=0, if E3=E4=E5=0, then the following subsystem(26)e˙1=−10e1+x3E2,E˙2=−60E2,is asymptotically stable.
Thus,Gx,E,e,B can be stabilized. According to Lemma 1, the sum and the error system (21) is stabilized by the abovementioned controller u given in (24), which completes the proof.In the following, numerical simulation is carried out with the initial conditions:x10=1,x20=−2,x30=3,x40=−4, y10=5,y20=−6,y30=7,y40=−8, and k0=−1. Figure 7 shows e1,E2,andE3 are asymptotically stable, Figure 8 shows E4andE5 are asymptotically stable, and Figure 9 shows that the feedback gain kt tends to constant.Figure 7
e
1
,
E
2
,
and
E
3 are asymptotically stable.Figure 8
E
4
and
E
5 are asymptotically stable.Figure 9
The feedback gainkt tends to constant.Case 3.
α
1
=
α
3
=
α
4
=
−
1 and α2=α5=1.
LetEi=xi+yi,i=1,3,4, and ei=yi−xi,i=2,5, and then the sum and error system is described as follows:(27)E˙e˙=E˙1E˙3E˙4e˙2e˙5=f1y+f1xf3y+f3xf4y+f4xf2y−f2xf5y−f5x+B∗∗u=G∗∗x,E,e+B∗∗u,where(28)G∗∗x,E,e=G1∗∗x,E,eG3∗∗x,E,eG4∗∗x,E,eG2∗∗x,E,eG5∗∗x,E,e=−10E1+x2E3−x3e2+e2E3−20E3+50E4−x1e2+x2E1+E1e215E4−10E3−60e2+e540e5+2x1x2E1+2x1e2E1−x2E12−E12e2and(29)B∗∗=000100010000001.According to Lemma 1, we propose the following conclusion.Theorem 4.
Consider the controlled sum and error system (27). If controller u is designed as follows:(30)u=KEe,where K=ktB∗∗T, B∗∗ is given in (29), and kt is updated by the following update law:(31)k˙=−e2+E2.Then, the master system (1) and the slave system (3) realize the coexistence of complete synchronization and antisynchronization.Proof.
For the sum and error system (27) with u=0, if E3=E4=E5=0, then the following subsystem(32)E˙1=−10e1−x3e2,e˙2=−60e2,is asymptotically stable.
Thus,G∗∗x,E,e,B∗∗ can be stabilized. According to Lemma 1, the sum and the error system (27) is stabilized by the abovementioned controller u given in (30), which completes the proof.In the following, numerical simulation is carried out with the initial conditions:x10=1,x20=−2,x30=3,x40=−4, y10=5,y20=−6,y30=7,y40=−8, and k0=−1. Figure 10 shows E1,e2,andE3 are asymptotically stable, Figure 11 E4andE5 are asymptotically stable, Figure 12 shows that the feedback gain kt tends to constant.Figure 10
E
1
,
e
2
,
and
E
3 are asymptotically stable.Figure 11
E
4
and
E
5 are asymptotically stable.Figure 12
The feedback gainkt tends to constant.


## 5. Conclusions

In conclusion, two types of synchronization problems in the new 5D hyperchaotic system have been investigated. Firstly, the existence of these synchronization problems in such system has been proven. Secondly, those synchronization problems have been realized by the dynamic feedback control method. Finally, numerical simulations have been used to verify the validity and effectiveness of the proposed results.



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