# Master-Slave Synchronization of Chaotic Φ6 Duffing Oscillators by Linear State Error Feedback Control

Authors: Ke Ding
Journal: Complexity
 (2019)

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## Abstract

This paper is concerned with master-slave synchronization of chaoticΦ6 Duffing oscillators by using linear state error feedback control. Compared with some existing methods and results, this paper estimates the bound of the first trajectory (variable) of the controlled slave system and uses this bound to derive synchronization criteria for two chaotic Φ6 Duffing oscillators. The effectiveness of synchronization criteria is illustrated by three simulation examples.


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## Body

## 1. Introduction

Synchronization of chaotic systems has received considerable attention due to its theoretical importance and practical applications in secure communication and signal processing (see for example, [1–31] and references therein).As is well known, some models for damped and driven oscillators, such as stiffening springs, beam bulking, and superconducting Josephson parametric amplifiers, can be described asΦ6 Duffing oscillators which have been widely used in mechanical and electrical systems [1, 9–11, 32–36]. With proper parameters, Duffing oscillators have exhibited chaotic behaviors. For chaotic Φ6 Duffing oscillators, Njah [10, 11] used the active control to achieve master-slave synchronization, in which the active control removed all nonlinear terms of the error system. For chaotic Φ4 Duffing oscillators which is the special case of Φ6 Duffing oscillators, synchronization criteria were derived by the active control in [32–34, 37] and [35] in which the linear error system and synchronization criteria were derived. It should be pointed out that chaotic Φ6 Duffing oscillators are nonlinear systems in which the nonlinear terms play a key role in the generation of chaotic attractors. Thus, how to use the nonlinear properties of the error system and how to use linear state error feedback control to derive synchronization criteria for chaotic Φ6 Duffing oscillators is one motivation of this paper.The bounds of trajectories of the master system and slave system have been widely used to derive the synchronization criteria for chaotic systems (see for example, [36, 38–40]). But it was difficult to estimate the bounds of slave systems. Therefore, how to derive the bound of some (not all) trajectories of the controlled slave system before the master system and the slave system achieve synchronization and how to use the derived bound to achieve synchronization criteria for the chaotic Φ6 Duffing oscillators is another motivation of this paper.In this paper, we will construct a master-slave synchronization scheme for chaoticΦ6 Duffing oscillators by using linear state error feedback control. We will use the linear state error feedback control to derive the bound of the first trajectory of the slave system before the master system and the slave system achieve synchronization and use this bound to obtain synchronization criteria. Moreover, we will use three examples to illustrate the effectiveness of our synchronization criteria.The rest of this paper is as follows. In Section2, the related problems and concepts will be introduced. In Section 3, the synchronization results for chaotic Φ6 Duffing oscillators will be given. As applications, the synchronization results for classic Φ4 Duffing oscillators and parametrically excited Φ4 Duffing oscillators will be provided in Section 4. In Section 5, three simulation results will be given. Conclusions and future works will be presented in Section 6.


## 2. Problem Statement

The mathematical model ofΦ6 Duffing oscillator is(1)x¨t=−cx˙t−dxt−lx3t−ax5t+qcosωt,where xt is the displacement of rotation angle; x˙t=dxt/dt; x¨t=d2xt/dt2; a, c, d, l, q, and ω are constants; qcosωt is the excitation; cx˙t is the linear damping term; dxt+lx3t is a nonlinear force; and the initial condition is x0=x0 and x˙0=x0′. The potential of (1) is W6x=1/2dx2t+1/4lx4t+1/6ax6t, which is the reason why system (1) is called Φ6 Duffing oscillator.Remark 1.
Ifa=0, system (1) reduces to the following Φ4 Duffing oscillator:(2)x¨t=−cx˙t−dxt−lx3t+qcosωt,with the potential W4x=1/2dx2t+1/4lx4t.Lety1t=xt and y2t=y˙1t. The nonautonomous system (1) can be written as the following dimensionless system:(3)y˙1t=y2t,y˙2t=−dy1t−cy2t+gy1t+pt,where(4)pt=qcosωt,gy1t=−ly13t−ay15t.The initial condition of system (3) is given by y10=y10,y20=y20.Letyt=y1ty2t∈ℝ2. Write the system described by (3) as(5)y˙t=Ayt+φyt+rt,where(6)A=01−d−c,rt=0pt,φyt=0gy1t.Letzt=z1tz2t∈ℝ2. We can construct the following synchronization scheme for the system described by (5):(7)ℳ:y˙t=Ayt+φyt+rt,(8)S:z˙t=Azt+φzt+rt+ut,(9)C:ut=Kyt−zt,with the master system described by ℳ and the slave system described by S, where ut=u1tu2t∈ℝ2 is the controller and K=k11k2k3 in which k1>0, k2, and k3 are gains which can be determined later. The initial condition of system (8) is given by z10=z10andz20=z20.Defining a signalet=yt−zt=e1te2t∈ℝ2, one can obtain the error system(10)e˙1t=−k1e1t,e˙2t=−k2+de1t−k3+ce2t+gy1t−gz1t.In view of differential mean theorem, one can have(11)gy1t−gz1t=g′ξty1t−z1t,where(12)g′ξt=dgρdρρ=ξt=−3lξ2t+5aξ4t,for ξt∈miny1t,z1t,maxy1t,z1t, which results in(13)e˙t=K^tet,where(14)K^t=−k10−k2+d+g′ξt−k3+c.The initial condition of system (10) is e10=y10−z10,e20=y20−z20.Notice that the master system described by (7) is chaotic. Thus, there exist two scales m1>0 and m2>0 for any y10 and y20 in the attracting area such that(15)yit≤mi,i=1,2,∀t>0.From the first equation of system described by (10), we have(16)e1t=y10−z10exp−k1t,which indicates that(17)z1t=y1t−y10−z10exp−k1t.From the equation described by (17), we have(18)z1t≤y1t+y10−z10≤m1+y10−z10,∀t>0,k1>0.From inequalities (15) and (18) and equation (11), there exists a scale λ>0 such that(19)g′ξt≤λ,where(20)λ=3l+5aσσ,σ=maxζ2t,∀ζt∈miny1t,z1t,maxy1t,z1t.Remark 2.
Since the bounds ofy1t and z1t for any t>0 are given by (15) and (18), respectively, and g⋅ and g′⋅ are defined and differentiable, the bound of g′ξt for ξt∈miny1t,z1t,maxy1t,z1t can be estimated by (19).Remark 3.
For the cascaded system described by(21)ζ˙1t=ϕ1t,ζ1t+χt,ζ1t,ζ2t,(22)ζ˙2t=ϕ2t,ζ2t,where ϕ1,ϕ2,andχ are smooth, the origin 0,0 is uniformly globally asymptotically stable if ζ˙1t=ϕ1t,ζ1t and ζ˙2t=ϕ2t,ζ2t are uniformly globally asymptotically stable and the solutions of (21) and (22) are uniformly globally bounded (Lemma 2, [41]). The system described by (10) can be regarded as a cascaded system. Although, it is easy to obtain the conditions to ensure that ζ˙1t=−k1e1t and ζ˙2t=−k3+ce2t are uniformly globally asymptotically stable, we cannot directly claim that the solutions of (10) are uniformly globally bounded. Thus, we cannot directly use Lemma 2 of [41] to study the stability of the error system (10).
The purpose of this paper is to investigate the master-slave synchronization for the system described by (1) and to find the controller gain K, such that the system described by (10) is globally asymptotically stable, which indicates that the system described by (7)–(9) synchronizes.


## 3. Main Results: Master-Slave Synchronization Criteria

In this section, we give some stability criteria for the error system described by (10), which ensures that the system described by (7)–(9) synchronizes.Choosing the following Lyapunov function:(23)Vt=eTtPet,where P=p11p12p12p22∈ℝ2×2 is a real positive matrix, we state and establish the following result.Proposition 1.
The error system described by (10), (15), and (19) is globally asymptotically stable, i.e., the master system described by (7) and the slave system described by (8) achieve synchronization if(24)Θ1=−k2+dp12−p11k1+λp12<0,Θ2=−p22k3+c<0,−k1+k3+cp12−k2+dp22+λp222≤4Θ1Θ2.Proof.
Taking the derivative ofVt with respect to t along the trajectory of (10) yields(25)V˙t=eTtLtet,where(26)Lt=K^TtP+PK^t=l11tl12tl12tl22t,with(27)l11t=2g′ξt−k2−dp12−p11k1,l12t=−k1+k3+cp12+g′ξt−k2−dp22,l22t=−2p22k3+c.
Conditions(28)l11t<0,l22t<0,l122t<l11tl22t,can ensure(29)V˙t<0,∀e1t,e2t≠0.
It follows from (19) and (27) that(30)l11t2=g′ξtp12−k2+dp12−p11k1≤−k2+dp12−p11k1+g′ξtp12≤−k2+dp12−p11k1+λp12=Θ1,l22t2=−p22k3+c=Θ2,l122t=−k1+k3+cp12+g′ξt−k2−dp222≤−k1+k3+cp12−k2+dp22+g′ξtp222≤−k1+k3+cp12−k2+dp22+λp222.
From (30), one can see that inequalities (24) can guarantee inequalities (28). Thus, it follows from inequalities (19), (24), and (29) that the error system described by (10) is globally asymptotically stable. This completes the proof. Q.E.D.
LetP=1001. One can derive the following synchronization result by Proposition 1.Proposition 2.
The error system described by (10), (15), and (19) is globally asymptotically stable, i.e., the master system described by (7) and the slave system described by (8) achieve synchronization if(31)k1>0,c+k3>0,2k1c+k3−d+k2>λ.Ifk1=k3=k, we have the following corollary.Corollary 1.
The error system described by (10), (15), and (19) is globally asymptotically stable, i.e., the master system described by (7) and the slave system described by (8) achieve synchronization if(32)k>0,c+k>0,2kc+k−d+k2>λ.Ifk1=k3=k and k2=0, the following result is obtained.Corollary 2.
The error system described by (10), (15), and (19) is globally asymptotically stable, i.e., the master system described by (7) and the slave system described by (8) achieve synchronization if(33)k>0,c+k>0,2kc+k−d>λ.In some applications, one can only measure the position variables for a chaotic system, which means that we only usey1t−z1t in the feedback control. In this situation, one can obtain k3=0. The corresponding result is given as follows.Corollary 3.
The error system described by (10), (15), and (19) is globally asymptotically stable, i.e., the master system described by (7) and the slave system described by (8) achieve synchronization if(34)k1>0,c>0,2k1c−d+k2>λ.Furthermore, in the case ofk2=k3=0, one can have the following result.Corollary 4.
The error system described by (10), (15), and (19) is globally asymptotically stable, i.e., the master system described by (7) and the slave system described by (8) achieve synchronization if(35)k1>0,c>0,2k1c−d>λ.Remark 4.
Njah [10, 11] constructed the master-slave synchronization scheme for the Φ6 Duffing equation and studied master-slave synchronization by the active control, in which the active controller ut=u1tu2t∈ℝ2 was(36)u1t=−a^e1t−b^e2t,u2t=−ay15t−z15t−ly13t−z13t−η1e1t−η2e2t,where a^,b^,η1,andη2 are parameters for control inputs. Then, one can have the error system(37)e˙1t=a^e1t+1+b^e2t,e˙2t=−d−η1e1t−c−η2e2t.
Obviously, a linear error system described by (37) can be perfectly derived by the control (36) in [10, 11] which removed all nonlinear terms of the error system, and the stability criterion for the linear error system described by (37) can be easily obtained. However, the original Duffing oscillator (1) was completely canceled. Compared with control (36) in [10, 11], control (9) in this paper has two advantages. The first advantage is that the nonlinear term y13t−z13t of the error system described by (10) is kept which means that the error system described by (10) is a nonlinear system, rather than a linear error system (37) in [10, 11]. The second advantage is that it is easy to estimate the bounds for z1t and g′ξt by using (18) and (19), respectively, which are necessary for deriving the stability criterion for the error system described by (10).Remark 5.
In this paper, we only use the bound ofz1t because it can be estimated by (18).


## 4. Applications to Master-Slave Synchronization of ChaoticΦ4 Duffing Oscillators

### 4.1. Master-Slave Synchronization of ClassicΦ4 Duffing Oscillators

Now, we can study the synchronization of classicΦ4 Duffing oscillator (2). Let y1t=xt and y2t=y˙1t. One can derive the dimensionless system (3), in which gy1t is replaced by −ly13t. From (19), the bound of g′⋅ can be estimated as(38)g′ξt≤λ˜,where λ˜=3lσ in which σ=maxζ2t,∀ζt∈miny1t,z1t,maxy1t,z1t. One can construct the synchronization scheme described by (7)–(9) for the system described by (2) and derive the error system described by (13), where gy1t is replaced by −ly13t. Employing the Lyapunov function described by (23), one can have the following synchronization criterion for the Φ4 Duffing oscillator described by (2).Proposition 3.
The error system described by (13), (15), and (38) is globally asymptotically stable, i.e., the master system described by (7) and the slave system described by (8) achieve synchronization if(39)Θ1=−k2+dp12−p11k1+λ˜p12<0,Θ2=−p22k3+c<0,−k1+k3+cp12−k2+dp22+λ˜p222≤4Θ1Θ2.
IfP=1001 in (23), one can obtain the following synchronization criterion for Φ4 Duffing oscillator (2).Proposition 4.
The error system described by (13), (15), and (38) is globally asymptotically stable, i.e., the master system described by (7) and the slave system described by (8) achieve synchronization if(40)k1>0,c+k3>0,2k1c+k3−d+k2>λ˜.Remark 6.
Jiang [3] and Nijmeijer and Berghuis [9] studied the tracking control for Duffing oscillators which can be equivalent to master-slave synchronization for Duffing oscillators. The stability criteria for the error system were derived by using the control ut=K˜et−ςt [3], where K˜=00k4k5 and ςt=03k6z12te1t in which k4,k5,andk6 are gains, and the control ut=K˜et−υt [9], where K˜=00k7k8 and υt=03y1tz1te1t in which k7 and k8 are gains. It should be pointed out that those controls in [3, 9] were nonlinear feedback controls. Our control (9) ut=Kyt−zt is a linear feedback control.Remark 7.
Han et al. [32] and Njah and Vincent [37] used the active control to derive synchronization criteria for chaotic Φ4 Duffing oscillators, in which the active controller ut=u1tu2t∈ℝ2 was(41)u1t=−z1t+y1t+k˘1e1t,u2t=dz1t+cz2t+lz13t−dy1t−cy2t−ly13t+k˘2e2t,where k˘1andk˘2 are feedback gains. Thus, the error system was(42)e˙1t=−k˘1e1t,e˙2t=−k˘2e2t.
As discussed in Remark 4, the error system described by (42) was a linear system. Compared with the control method in [32, 37], we keep the linear and nonlinear terms and fully use the term −dy1t−cy2t−ly13t to derive the synchronization criterion Proposition 3.


### 4.2. Master-Slave Synchronization of Parametrically ExcitedΦ4 Duffing Oscillators

Wu et al. [36] studied the following parametrically excited Φ4 Duffing oscillator:(43)y˙1t=y2t,y˙1t=1+μsinωty1t−y13t−cy2t,where μ and ω are constants, which can be rewritten as(44)y˙t=Aˇtyt+φˇyt,where(45)yt=y1ty2t∈ℝ2,Aˇt=011+μsinωt−c,φˇyt=0−y13t.In [36], the master-slave scheme was constructed as follows:(46)ℳ:y˙t=Aˇtyt+φˇyt,(47)S:z˙t=Aˇtzt+φˇzt+ut,(48)C:ut=Kˇyt−zt,where zt=z1tz2t∈ℝ2 and Kˇ=kˇ11kˇ12kˇ21kˇ22 is a gain matrix which can be determined. The initial conditions of the master and slave system were y10=y10,y20=y20, and z10=z10,z20=z20, respectively. The error system was(49)e˙t=Aˇt+Mt−Kˇet,where(50)et=yt−zt=e1te2t=y1t−y2tz1t−z2t,Mtet=φˇyt−φˇztwithMt=00−Ft0,Ft=y12t+z12t+y1tz1t.The initial conditions of the error system weree10=e10=y10−z10,e20=e20=y20−z20.Choosing our method, one can use the control as follows:(51)C:ut=K¯yt−zt,where K¯=k¯11k¯2k¯3 in which k¯1>0,k¯2,andk¯3 can be determined later. From master-slave schemes (46), (47), and (51), one can have the following error system:(52)e˙1t=−k¯1e1t,e˙2t=1+μsinωt−k¯2e1t+g¯y1t−g¯z1t−c+k¯3e2t,where(53)g¯y1t=−y13t,and the initial condition is e10=e10ande20=e20. From (19), one can obtain(54)g¯y1t−g¯z1t=−y13t+z13t=g¯′ξty1t−z1t,g¯′ξt=−3ξ2t,ξt∈miny1t,z1t,maxy1t,z1t.Then, the error system described by (52) can be rewritten as(55)e˙t=K´tet,where(56)K´t=−k¯101+μsinωt−k¯2+g¯′ξt−c−k¯3.It follows from (19) that there exists a scale λ¯>0 such that(57)g¯′ξt≤λ¯,where(58)λ¯=3σ,σ=maxζ2t,∀ζt∈miny1t,z1t,maxy1t,z1t. By using the similar proof of Proposition 1, one can have the synchronization result for (43).Proposition 5.
The error system described by (52) is globally asymptotically stable, i.e., the master system described by (46) and the slave system described by (47) achieve synchronization if(59)k¯1>0,c+k¯3>0,2k¯1c+k¯3−1+μ+k¯2>λ˜.


## 4.1. Master-Slave Synchronization of ClassicΦ4 Duffing Oscillators

Now, we can study the synchronization of classicΦ4 Duffing oscillator (2). Let y1t=xt and y2t=y˙1t. One can derive the dimensionless system (3), in which gy1t is replaced by −ly13t. From (19), the bound of g′⋅ can be estimated as(38)g′ξt≤λ˜,where λ˜=3lσ in which σ=maxζ2t,∀ζt∈miny1t,z1t,maxy1t,z1t. One can construct the synchronization scheme described by (7)–(9) for the system described by (2) and derive the error system described by (13), where gy1t is replaced by −ly13t. Employing the Lyapunov function described by (23), one can have the following synchronization criterion for the Φ4 Duffing oscillator described by (2).Proposition 3.
The error system described by (13), (15), and (38) is globally asymptotically stable, i.e., the master system described by (7) and the slave system described by (8) achieve synchronization if(39)Θ1=−k2+dp12−p11k1+λ˜p12<0,Θ2=−p22k3+c<0,−k1+k3+cp12−k2+dp22+λ˜p222≤4Θ1Θ2.
IfP=1001 in (23), one can obtain the following synchronization criterion for Φ4 Duffing oscillator (2).Proposition 4.
The error system described by (13), (15), and (38) is globally asymptotically stable, i.e., the master system described by (7) and the slave system described by (8) achieve synchronization if(40)k1>0,c+k3>0,2k1c+k3−d+k2>λ˜.Remark 6.
Jiang [3] and Nijmeijer and Berghuis [9] studied the tracking control for Duffing oscillators which can be equivalent to master-slave synchronization for Duffing oscillators. The stability criteria for the error system were derived by using the control ut=K˜et−ςt [3], where K˜=00k4k5 and ςt=03k6z12te1t in which k4,k5,andk6 are gains, and the control ut=K˜et−υt [9], where K˜=00k7k8 and υt=03y1tz1te1t in which k7 and k8 are gains. It should be pointed out that those controls in [3, 9] were nonlinear feedback controls. Our control (9) ut=Kyt−zt is a linear feedback control.Remark 7.
Han et al. [32] and Njah and Vincent [37] used the active control to derive synchronization criteria for chaotic Φ4 Duffing oscillators, in which the active controller ut=u1tu2t∈ℝ2 was(41)u1t=−z1t+y1t+k˘1e1t,u2t=dz1t+cz2t+lz13t−dy1t−cy2t−ly13t+k˘2e2t,where k˘1andk˘2 are feedback gains. Thus, the error system was(42)e˙1t=−k˘1e1t,e˙2t=−k˘2e2t.
As discussed in Remark 4, the error system described by (42) was a linear system. Compared with the control method in [32, 37], we keep the linear and nonlinear terms and fully use the term −dy1t−cy2t−ly13t to derive the synchronization criterion Proposition 3.


## 4.2. Master-Slave Synchronization of Parametrically ExcitedΦ4 Duffing Oscillators

Wu et al. [36] studied the following parametrically excited Φ4 Duffing oscillator:(43)y˙1t=y2t,y˙1t=1+μsinωty1t−y13t−cy2t,where μ and ω are constants, which can be rewritten as(44)y˙t=Aˇtyt+φˇyt,where(45)yt=y1ty2t∈ℝ2,Aˇt=011+μsinωt−c,φˇyt=0−y13t.In [36], the master-slave scheme was constructed as follows:(46)ℳ:y˙t=Aˇtyt+φˇyt,(47)S:z˙t=Aˇtzt+φˇzt+ut,(48)C:ut=Kˇyt−zt,where zt=z1tz2t∈ℝ2 and Kˇ=kˇ11kˇ12kˇ21kˇ22 is a gain matrix which can be determined. The initial conditions of the master and slave system were y10=y10,y20=y20, and z10=z10,z20=z20, respectively. The error system was(49)e˙t=Aˇt+Mt−Kˇet,where(50)et=yt−zt=e1te2t=y1t−y2tz1t−z2t,Mtet=φˇyt−φˇztwithMt=00−Ft0,Ft=y12t+z12t+y1tz1t.The initial conditions of the error system weree10=e10=y10−z10,e20=e20=y20−z20.Choosing our method, one can use the control as follows:(51)C:ut=K¯yt−zt,where K¯=k¯11k¯2k¯3 in which k¯1>0,k¯2,andk¯3 can be determined later. From master-slave schemes (46), (47), and (51), one can have the following error system:(52)e˙1t=−k¯1e1t,e˙2t=1+μsinωt−k¯2e1t+g¯y1t−g¯z1t−c+k¯3e2t,where(53)g¯y1t=−y13t,and the initial condition is e10=e10ande20=e20. From (19), one can obtain(54)g¯y1t−g¯z1t=−y13t+z13t=g¯′ξty1t−z1t,g¯′ξt=−3ξ2t,ξt∈miny1t,z1t,maxy1t,z1t.Then, the error system described by (52) can be rewritten as(55)e˙t=K´tet,where(56)K´t=−k¯101+μsinωt−k¯2+g¯′ξt−c−k¯3.It follows from (19) that there exists a scale λ¯>0 such that(57)g¯′ξt≤λ¯,where(58)λ¯=3σ,σ=maxζ2t,∀ζt∈miny1t,z1t,maxy1t,z1t. By using the similar proof of Proposition 1, one can have the synchronization result for (43).Proposition 5.
The error system described by (52) is globally asymptotically stable, i.e., the master system described by (46) and the slave system described by (47) achieve synchronization if(59)k¯1>0,c+k¯3>0,2k¯1c+k¯3−1+μ+k¯2>λ˜.


## 5. Simulation Study

### 5.1. Simulation for ChaoticΦ6 Duffing Oscillators

Consider the chaoticΦ6 Duffing oscillator (1) with a=0.1, c=0.4, d=1.1, l=0.4, q=1.8, and ω=2.1. If we choose the initial condition of (1) as y10=0 and z10=−1, there is an attractor which is demonstrated in Figure 1. From Figure 1, one can obtain that the bound of y1t is 1.4, i.e., m1=1.4.Figure 1
The phase diagram of the chaoticΦ6 Duffing oscillator.For the master-slave scheme (7)–(9) with (1), one can choose the initial condition of the slave system as z10=0.1 and z20=−1.5. By virtue of (18), one can obtain z1t≤1.4+0.1=1.5. It follows from (19) that g′ξt≤5.2313=λ. Let k2=0 and k1=k3. From Corollary 2, we have k1c+k1>λ+d/2, which implies that k1≥2.97. We choose k1=2.98.Figures2–4 give the simulation results for the master system, the slave system, and the error system with k1=2.98, k2=0, and k3=2.98, respectively, from which one can see that the error system (10) is globally asymptotically stable; i.e., the master-slave synchronization scheme described by (7)–(9) indeed achieves synchronization.Figure 2
The simulation result for the master system.Figure 3
The simulation result for the slave system withk1=2.98,k2=0,andk3=2.98.Figure 4
The simulation result for the error system withk1=2.98,k2=0,andk3=2.98.


### 5.2. Simulation for ClassicΦ4 Duffing Oscillators

For the classicΦ4 Duffing oscillator (2), parameters c,d,l,q,andω are the same as those defined in the abovementioned Φ6 Duffing oscillators. If we choose the initial condition of (1) as y10=0 and z10=−1, there is an attractor which is demonstrated in Figure 5. From Figure 5, we have that the bound of y1t is 1.4, i.e., m1=1.4.Figure 5
The phase diagram of the chaoticΦ4 Duffing oscillator.Consider the master-slave scheme (7)–(9) with (2), where the initial condition of the slave system is z10=0.1 and z20=1.5. From (18), we have z1t≤1.4+0.1=1.5. It follows from (38) that g′ξt≤2.7=λ˜. Let k2=0 and k1=k3. From Proposition 4, we have k1c+k1>λ+d/2, which implies that k1≥1.71. We choose k1=1.72.Figures6–8 give the simulation results for the master system, the slave system, and the error system with k1=1.72, k2=0, and k3=1.72, respectively. It follows from Figures 6–8 that the error system (10) is globally asymptotically stable; i.e., the master-slave synchronization scheme described by (7)–(9) indeed achieves synchronization.Figure 6
The simulation result for the master system.Figure 7
The simulation result for the slave system withk1=1.72,k2=0,andk3=1.72.Figure 8
The simulation result for error system withk1=1.72,k2=0,andk3=1.72.


### 5.3. Simulation for Parametrically ExcitedΦ4 Duffing Oscillators

Now, we study the synchronization of parametrically excitedΦ4 Duffing oscillators (43) where c=0.2, μ=0.5, and ω=1. The initial conditions of master system (46) and slave system (47) are y10=1.6,y20=0.2 and z10=2,z20=1.4, respectively. It follows from Figure 9 that the up bound of y1t is 1.66. However, we can see that the up bound of z1t is larger than 2 because z10=2.Figure 9
The simulation result for the master system.It follows from (57) that z1t≤1.66+0.4=2.06 and g¯′ξt≤3×2.062=12.7308=λ¯. Let k¯1=k¯3andk¯2=0. Using Proposition 5, one can have k¯1=k¯3>7.01. Let k¯1=k¯3=7.1. Figure 10 illustrates that the error system is globally asymptotically stable; i.e., the master-slave synchronization scheme described by (46), (47), and (51) indeed achieves synchronization.Figure 10
The simulation result for the error system withk1=7.1,k2=0,andk3=7.1.


## 5.1. Simulation for ChaoticΦ6 Duffing Oscillators

Consider the chaoticΦ6 Duffing oscillator (1) with a=0.1, c=0.4, d=1.1, l=0.4, q=1.8, and ω=2.1. If we choose the initial condition of (1) as y10=0 and z10=−1, there is an attractor which is demonstrated in Figure 1. From Figure 1, one can obtain that the bound of y1t is 1.4, i.e., m1=1.4.Figure 1
The phase diagram of the chaoticΦ6 Duffing oscillator.For the master-slave scheme (7)–(9) with (1), one can choose the initial condition of the slave system as z10=0.1 and z20=−1.5. By virtue of (18), one can obtain z1t≤1.4+0.1=1.5. It follows from (19) that g′ξt≤5.2313=λ. Let k2=0 and k1=k3. From Corollary 2, we have k1c+k1>λ+d/2, which implies that k1≥2.97. We choose k1=2.98.Figures2–4 give the simulation results for the master system, the slave system, and the error system with k1=2.98, k2=0, and k3=2.98, respectively, from which one can see that the error system (10) is globally asymptotically stable; i.e., the master-slave synchronization scheme described by (7)–(9) indeed achieves synchronization.Figure 2
The simulation result for the master system.Figure 3
The simulation result for the slave system withk1=2.98,k2=0,andk3=2.98.Figure 4
The simulation result for the error system withk1=2.98,k2=0,andk3=2.98.


## 5.2. Simulation for ClassicΦ4 Duffing Oscillators

For the classicΦ4 Duffing oscillator (2), parameters c,d,l,q,andω are the same as those defined in the abovementioned Φ6 Duffing oscillators. If we choose the initial condition of (1) as y10=0 and z10=−1, there is an attractor which is demonstrated in Figure 5. From Figure 5, we have that the bound of y1t is 1.4, i.e., m1=1.4.Figure 5
The phase diagram of the chaoticΦ4 Duffing oscillator.Consider the master-slave scheme (7)–(9) with (2), where the initial condition of the slave system is z10=0.1 and z20=1.5. From (18), we have z1t≤1.4+0.1=1.5. It follows from (38) that g′ξt≤2.7=λ˜. Let k2=0 and k1=k3. From Proposition 4, we have k1c+k1>λ+d/2, which implies that k1≥1.71. We choose k1=1.72.Figures6–8 give the simulation results for the master system, the slave system, and the error system with k1=1.72, k2=0, and k3=1.72, respectively. It follows from Figures 6–8 that the error system (10) is globally asymptotically stable; i.e., the master-slave synchronization scheme described by (7)–(9) indeed achieves synchronization.Figure 6
The simulation result for the master system.Figure 7
The simulation result for the slave system withk1=1.72,k2=0,andk3=1.72.Figure 8
The simulation result for error system withk1=1.72,k2=0,andk3=1.72.


## 5.3. Simulation for Parametrically ExcitedΦ4 Duffing Oscillators

Now, we study the synchronization of parametrically excitedΦ4 Duffing oscillators (43) where c=0.2, μ=0.5, and ω=1. The initial conditions of master system (46) and slave system (47) are y10=1.6,y20=0.2 and z10=2,z20=1.4, respectively. It follows from Figure 9 that the up bound of y1t is 1.66. However, we can see that the up bound of z1t is larger than 2 because z10=2.Figure 9
The simulation result for the master system.It follows from (57) that z1t≤1.66+0.4=2.06 and g¯′ξt≤3×2.062=12.7308=λ¯. Let k¯1=k¯3andk¯2=0. Using Proposition 5, one can have k¯1=k¯3>7.01. Let k¯1=k¯3=7.1. Figure 10 illustrates that the error system is globally asymptotically stable; i.e., the master-slave synchronization scheme described by (46), (47), and (51) indeed achieves synchronization.Figure 10
The simulation result for the error system withk1=7.1,k2=0,andk3=7.1.


## 6. Conclusion

We have constructed a master-slave synchronization scheme for chaoticΦ6 Duffing oscillators by using linear feedback control. By estimating the first trajectory of the controlled slave system and keeping the nonlinear property of the error system, we have derived some synchronization criteria. Then, we have used three examples to illustrate the effectiveness of synchronization criteria for Duffing oscillators. In this paper, master and slave systems are all Φ6 Duffing oscillators. The synchronization between Φ6 and Φ4 Duffing oscillators and the synchronization between Φ6 Duffing oscillators with different parameters are our future research interests. Moreover, how to design the time delayed feedback control to achieve synchronization between Φ6 Duffing oscillators with different parameters can be our future research interest as well.



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*Source: 3637902-2019-10-30.xml*