# Chaotic Control and Generalized Synchronization for a Hyperchaotic Lorenz-Stenflo System

Authors: Yin Li; Yulin Zhao; Zheng-an Yao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is devoted to investigate the tracking control and generalized
synchronization of the hyperchaotic Lorenz-Stenflo system using the tracking model
and the feedback control scheme. We suppress the chaos to unstable equilibrium via
three feedback methods, and we achieve three globally generalized synchronization
controls. Novel tracking controllers with corresponding parameter update laws are
designed such that the Lorenz-Stenflo systems can be synchronized asymptotically.
Moreover, numerical simulations are presented to demonstrate the effectiveness,
through the contrast between the orbits before being stabilized and the ones after
being stabilized.


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## Body

## 1. Introduction

Study of chaotic control and generalized synchronization has received great attention in the past several decades [1–10]; many hyperchaotic systems have been proposed and studied in the last decade, for example, a new hyperchaotic Rössler system [4], the hyperchaotic L system [5], Chua’s circuit [6], the hyperchaotic Chen system [7, 8], and so forth. Hyperchaotic system has been proposed for secure communication and the presence of more than one positive Lyapunov exponent clearly improves the security of the communication scheme [9–12]. Therefore, hyperchaotic system generates more complex dynamics than the low-dimensional chaotic system, which has much wider application than the low-dimensional chaotic system.Until now, a variety of approaches have been proposed for the synchronization of low-dimensional chaotic systems, including Q-S method [13, 14], active control [15, 16], adaptive control [17–24], and time-delay feedback control [25]. Recently, Stenflo [26] presented a new hyperchaotic Lorenz-Stenflo (LS) system:
(1)x˙=a(y-x)+dw,y˙=x(c-z)-y,z˙=yx-bz,w˙=-x-aw.
In (1), x, y, z, and w are the state variables of the system and a, b, c, and d are real constant parameters. System (1) is generated from the originally three-dimensional Lorenz chaotic system by introducing a new control parameter b and a state variable w.In this paper, we will consider chaos control and generalized synchronization related to hyperchaotic Lorenz-Stenflo system. we found that the feedback control achieved in the low-dimensional system like many other studies of dynamics in low-dimensional systems. We suppress the hyperchaotic Lorenz-Stenflo system to unstabilize equilibrium via three control methods: linear feedback control, speed feedback control, and doubly-periodic function feedback control. By designing a nonlinear controller, we achieve the generalized synchronization of two Lorenz-Stenflo systems up to a scaling factor. Moreover, numerical simulations are applied to verify the effectiveness of the obtained controllers.


## 2. The Hyperchaotic Lorenz-Stenflo System

In the following we would like to consider the hyperchaotic cases of system (1). When a=1.0, b=0.7, c=26, and d=1.5, system (1) exhibits hyperchaotic behavior. Simulated results are depicted in Figures 1 and 2. Figures 1(a)–1(d) depict the projection of the chaotic attractor in different spaces; Figures 2(a)–2(d) depict the states of system (1) before being stabilized, respectively. The volume of the elements in the phase space δX(t)=δxδyδzδw and the divergence of flow (1) are defined by
(2)∇X=∂X∂x+∂X∂y+∂X∂z+∂X∂w=-(2a+b+1),
where X=(x˙,y˙,z˙,w˙)=[a(y-x)+dw,x(c-z)-y,yx-bz,-x-aw].Chaotic attractor in different spaces: (a)(x,y,z), (b) (x,y,w), (c) (x,w,z), and (d) (y,w,z).
(a)
(b)
(c)
(d)The states of system (1) before being stabilized: (a) x-t, (b) y-t, (c) z-t, and (d) w-t.
(a)
(b)
(c)
(d)System (1) is dissipative when 2a+b+1>0. Moreover, an exponential contraction rate is given by
(3)dX(t)dt=-(2a+b+1)X(t).
It is clear that X(t)=X0e-(2a+b+1)t, which implies that the solutions of system (1) are bounded as t->+∞. It is easy to find the three equilibria E1(0,0,0,0),
(4)E2(((c-1)a2-d)b(d+a2)d+a2,((c-1)a2-d)b(d+a2)a2,(c-1)a2-da2,-(d+a2)b(a2c-a2-d)(d+a2)a),E3(-((c-1)a2-d)b(d+a2)d+a2,-((c-1)a2-d)b(d+a2)a2,(c-1)a2-da2,(d+a2)b(a2c-a2-d)(d+a2)a).To determine the stability of the equilibria pointE1(0,0,0,0), evaluating the Jacobian matrix of system (1) at E1 yields
(5)J|E1=(-aa0dc-10000-b0-100-a).
If a=1.0, b=0.7, c=26, and d=1.5, the four eigenvalues of the characteristic polynomial of Jacobian matrix (5) are
(6)λ1=3.94975,λ2=-5.9497,λ3=-1,λ4=-0.7.
Thus, the equilibria E1 is a saddle point of the hyperchaotic system (1). The Jacobian matrix of system (1) at E2 yields(7)J|E2=(-aa0dc-1((c-1)a2-d)b(d+a2)d+a20((c-1)a2-d)b(d+a2)a2((c-1)a2-d)b(d+a2)d+a2-b0-100-a). If a=1.0, b=0.7, c=26, and d=1.5, the four eigenvalues of the characteristic polynomial of Jacobian matrix (7) are
(8)λ1=4.8887,λ2=-6.3032,λ3=-1.1427+0.5438i,λ4=-1.1427-0.5438i.
Thus, the equilibria E2 are unstable, and E3 is similar.


## 3. The Hyperchaotic Control for Lorenz-Stenflo System

In this section, we control the hyperchaotic system (1) such that all trajections converge to the equilibrium point (0,0,0,0). The controlled hyperchaotic Lorenz-Stenflo system is given by
(9)x˙=a(y-x)+dw+u1,y˙=x(c-z)-y+u2,z˙=yx-bz+u3,w˙=-x-aw+u4,
where u1, u2, u3, and u4 are external control inputs which will be suitably derived from the trajectory of the chaotic system (1), specified by (x,y,z,w) to the equilibrium (0,0,0) of uncontrolled system ui=0(i=1,2,3,4).


### 3.1. Linear Function Feedback Control

For the modified hyperchaotic Lorenz-Stenflo system (9), if one of the following feedback controllers ui(i=1,2,3,4) is chosen for the system (9), then u1=-k1x1, u2=-k2y, u3=-k3z, u4=-k4w, and where ki’s are feedback coefficients. Therefore controlled system (9) is rewritten as
(10)x˙=a(y-x)+dw-k1x,y˙=x(c-z)-y-k2y,z˙=yx-bz-k3z,w˙=-x-aw-k4w
whose Jacobian matrix is
(11)J=(-a-k1a0dc-1-k20000-b-k30-100-a-k4).
The characteristic equation of J is
(12)λ4+(-B-A-D-C)λ3+(CD+AB+BD+BC-ca+AD+AC+d)λ2+(-ABD-ABC-BCD+caD+caC-ACD-dC-dB)λ+E=0,
where A=-a-k1, B=-1-k2, C=-b-k3, D=-a-k4, and E=ABCD-caCD+dBC.The abbreviated characteristic equation is(13)λ4+R1λ3+R2λ2+R3λ+E=0,
where R1=-B-A-D-C, R2=CD+AB+BD+BC-ca+AD+AC+d, and R3=-ABD-ABC-BCD+caD+caC-ACD-dC-dB, R4=E.According to the Routh-Hurwitz criterion, constraints are imposed as follows:(14)H1=R1=-B-A-D-C>0,H2=|R1R31R2|=R1R2-R3=(-A-C-B)D2+((-2A-2C)B-B2-A2-C2-d-2AC)D+(-A-C)B2-A2C+(-2AC+ca-A2-C2)B+(-d-C2+ca)A>0,H3=|R1R301R2E0R1R3|=R1R2R3-ER12-R32>0,H4=|R1R3001R2R400R1R3001R2R4|=R1R2R3R4-R12R42-R32R4>0.This characteristic polynomial has four roots, all with negative real roots, under the condition ofH1>0, H2>0, H3>0, and H4>0. Therefore, the equilibria (0,0,0,0) are the stable manifold Ws and the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable. The concrete dynamics of (9) can be demonstrated by Proposition 1, while the Maple program is demonstrated in Appendix A.Proposition 1.
If one chooses the control coefficientsk1=8, k2=4, k3=3, and k4=2 and the parameters a=1.0, b=0.7, c=26, and d=1.5, the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium (0,0,0,0).Proof.
When the parameters were selected by the above value, we obtain the Jacobian matrix(15)J=(-9.01.001.526-50000-3.70-100-3.0).
The characteristic equation of J is given by
(16)λ4+20710λ3+12525λ2+29534λ+2381320=0.
According to Appendix A, we easily obtain
(17)H1=20710>0,H2=2300>0,H3=578>0,H4=69500>0,
which yields the eigenvalues via the compute simulation
(18)λ1=-12.36844706,λ2=-1.931149367,λ3=-2.700403573,λ4=-3.70000.
Thus the zero solution of system (9) is exponentially stable, Proposition 1 is proved.Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (1) using the fourth-order Runge-Kutta scheme with time step 0.01. The parameters and the corresponding feedback coefficients are given by the above value. The initial values are taken as [x(0)=1,y(0)=0.7,z(0)=20,w(0)=0.1]. The behaviors of the states (x,y,z,w) of the controlled chaotic Lorenz-Stenflo system (1) with time are displayed in Figures 3(a)–3(d).The states of system (9): (a) x-t, (b) y-t, (c) z-t, (d) w-t.
(a)
(b)
(c)
(d)


### 3.2. Speed Function Feedback Control

Suppose thatu1=u3=0; u2 and u4 are of the speed forms u2=k2(-bz+xy) and u4=-k4[a(y-x)+dw], where k2 and k4 are speed feedback coefficients. Therefore the controlled chaotic system (9) is rewritten as
(19)x˙=a(y-x)+dw,y˙=x(c-z)-y+k2(-bz+xy),z˙=yx-bz,w˙=-x-aw-k4[a(y-x)+dw].
The Jacobian matrix is
(20)J=(-aa0dc-1-k2b000-b0-1+k4a-k4a0-a-k4d).
The characteristic equation of J is
(21)λ4+(1-k1+2a+b)λ3+(2ab-k1b-k1a+2a-k1+b+d+a2-ca)λ2+(-ca2-cab+2ab+db+a2b-k1a+a2-k1b-k1ba+d)λ-ca2b-k1ba+a2b+db=0.Proposition 2.
If one chooses the control coefficients:k2=9, k4=-1 and the parameters a=1.0, b=0.7, c=26, and d=1.5, the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium (0,0,0,0).Similar to Proposition1, the proof of Proposition 2 is straightforward and thus is omitted.In the following, we give the eigenvalues via the compute simulation. When the parameters were selected by the above value, we obtain the Jacobian matrix(22)J=(-1.01.001.526-1-6.3000-0.70-2.01.000.5).
The characteristic equation of J changes the following:
(23)λ4+2.2λ3-21.95λ2-39.6λ-16.45=0.
The eigenvalues of the above equation are
(24)λ1=-5.10412,λ2=-1,λ3=-0.70000,λ4=-4.60412.
Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (20) using the fourth-order Runge-Kutta scheme with time step 0.01. The initial values are taken as [x(0)=1,y(0)=0.7,z(0)=20,w(0)=0.1]. The behaviors of the states (x,y,z,w) of the controlled chaotic Lorenz-Stenflo system (20) with time are displayed in Figures 4(a)–4(d).The states of system (19): (a) x-t, (b) y-t, (c) z-t, and (d) w-t.
(a)
(b)
(c)
(d)


### 3.3. The Doubly Periodic Function Feedback Control

Suppose thatu2=0, u3=0, and u4=0; u1 is of the doubly periodic function u1=k1cn(x,m), where k1 is speed feedback coefficients and 0<m<1 is the modulus of Jacobi elliptic function. Therefore the controlled chaotic system (9) is rewritten as
(25)x˙=a(y-x)+dw+k1cn(x,m),y˙=x(c-z)-y,z˙=yx-bz,w˙=-x-aw.
The Jacobian matrix is
(26)J=(-a+k1a0dc-100-100-a).
The characteristic equation of J is
(27)λ4+(1-k1+2a+b)λ3+(a222ab-k1b-k1a+2a-k1+b+d+a2-caa22)λ2+(-ca2-cab+2ab+db+a2b-k1a+a2-k1b-k1ba+d)λ-ca2b-k1ba+a2b+db=0.Proposition 3.
If one chooses the control coefficientsk1=-30, m=0.3 and the parameters a=1.0, b=0.7, c=26, and d=1.5, the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium (0,0,0,0).The proof of Proposition3 is the same as Proposition 1, which is straightforward and thus is omitted.In the following, we give the eigenvalues via the compute simulation. When the parameters were selected by the above value, we obtain the Jacobian matrix(28)J=(-31.01.001.526-10000-0.70-1.000-1.0).
The characteristic equation of J changes the following:
(29)λ4+33.7λ3+61.60λ2+33.450λ+4.55=0.
The eigenvalues of the above equation are
(30)λ1=-31.795569,λ2=-0.20443,λ3=-1,λ4=-0.70000.
Thus the zero solution of system (9) is exponentially stable; Proposition 3 is proved.Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (25) using the fourth-order Runge-Kutta scheme with time step 0.01. The initial values are taken as [x(0)=1,y(0)=0.7,z(0)=20,w(0)=0.1]. The behaviors of the states (x,y,z,w) of the controlled chaotic Lorenz-Stenflo system (25) with time are displayed in Figures 5(a)–5(d).The states of system (25): (a) x-t, (b) y-t, (c) z-t, and (d) w-t.
(a)
(b)
(c)
(d)


## 3.1. Linear Function Feedback Control

For the modified hyperchaotic Lorenz-Stenflo system (9), if one of the following feedback controllers ui(i=1,2,3,4) is chosen for the system (9), then u1=-k1x1, u2=-k2y, u3=-k3z, u4=-k4w, and where ki’s are feedback coefficients. Therefore controlled system (9) is rewritten as
(10)x˙=a(y-x)+dw-k1x,y˙=x(c-z)-y-k2y,z˙=yx-bz-k3z,w˙=-x-aw-k4w
whose Jacobian matrix is
(11)J=(-a-k1a0dc-1-k20000-b-k30-100-a-k4).
The characteristic equation of J is
(12)λ4+(-B-A-D-C)λ3+(CD+AB+BD+BC-ca+AD+AC+d)λ2+(-ABD-ABC-BCD+caD+caC-ACD-dC-dB)λ+E=0,
where A=-a-k1, B=-1-k2, C=-b-k3, D=-a-k4, and E=ABCD-caCD+dBC.The abbreviated characteristic equation is(13)λ4+R1λ3+R2λ2+R3λ+E=0,
where R1=-B-A-D-C, R2=CD+AB+BD+BC-ca+AD+AC+d, and R3=-ABD-ABC-BCD+caD+caC-ACD-dC-dB, R4=E.According to the Routh-Hurwitz criterion, constraints are imposed as follows:(14)H1=R1=-B-A-D-C>0,H2=|R1R31R2|=R1R2-R3=(-A-C-B)D2+((-2A-2C)B-B2-A2-C2-d-2AC)D+(-A-C)B2-A2C+(-2AC+ca-A2-C2)B+(-d-C2+ca)A>0,H3=|R1R301R2E0R1R3|=R1R2R3-ER12-R32>0,H4=|R1R3001R2R400R1R3001R2R4|=R1R2R3R4-R12R42-R32R4>0.This characteristic polynomial has four roots, all with negative real roots, under the condition ofH1>0, H2>0, H3>0, and H4>0. Therefore, the equilibria (0,0,0,0) are the stable manifold Ws and the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable. The concrete dynamics of (9) can be demonstrated by Proposition 1, while the Maple program is demonstrated in Appendix A.Proposition 1.
If one chooses the control coefficientsk1=8, k2=4, k3=3, and k4=2 and the parameters a=1.0, b=0.7, c=26, and d=1.5, the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium (0,0,0,0).Proof.
When the parameters were selected by the above value, we obtain the Jacobian matrix(15)J=(-9.01.001.526-50000-3.70-100-3.0).
The characteristic equation of J is given by
(16)λ4+20710λ3+12525λ2+29534λ+2381320=0.
According to Appendix A, we easily obtain
(17)H1=20710>0,H2=2300>0,H3=578>0,H4=69500>0,
which yields the eigenvalues via the compute simulation
(18)λ1=-12.36844706,λ2=-1.931149367,λ3=-2.700403573,λ4=-3.70000.
Thus the zero solution of system (9) is exponentially stable, Proposition 1 is proved.Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (1) using the fourth-order Runge-Kutta scheme with time step 0.01. The parameters and the corresponding feedback coefficients are given by the above value. The initial values are taken as [x(0)=1,y(0)=0.7,z(0)=20,w(0)=0.1]. The behaviors of the states (x,y,z,w) of the controlled chaotic Lorenz-Stenflo system (1) with time are displayed in Figures 3(a)–3(d).The states of system (9): (a) x-t, (b) y-t, (c) z-t, (d) w-t.
(a)
(b)
(c)
(d)


## 3.2. Speed Function Feedback Control

Suppose thatu1=u3=0; u2 and u4 are of the speed forms u2=k2(-bz+xy) and u4=-k4[a(y-x)+dw], where k2 and k4 are speed feedback coefficients. Therefore the controlled chaotic system (9) is rewritten as
(19)x˙=a(y-x)+dw,y˙=x(c-z)-y+k2(-bz+xy),z˙=yx-bz,w˙=-x-aw-k4[a(y-x)+dw].
The Jacobian matrix is
(20)J=(-aa0dc-1-k2b000-b0-1+k4a-k4a0-a-k4d).
The characteristic equation of J is
(21)λ4+(1-k1+2a+b)λ3+(2ab-k1b-k1a+2a-k1+b+d+a2-ca)λ2+(-ca2-cab+2ab+db+a2b-k1a+a2-k1b-k1ba+d)λ-ca2b-k1ba+a2b+db=0.Proposition 2.
If one chooses the control coefficients:k2=9, k4=-1 and the parameters a=1.0, b=0.7, c=26, and d=1.5, the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium (0,0,0,0).Similar to Proposition1, the proof of Proposition 2 is straightforward and thus is omitted.In the following, we give the eigenvalues via the compute simulation. When the parameters were selected by the above value, we obtain the Jacobian matrix(22)J=(-1.01.001.526-1-6.3000-0.70-2.01.000.5).
The characteristic equation of J changes the following:
(23)λ4+2.2λ3-21.95λ2-39.6λ-16.45=0.
The eigenvalues of the above equation are
(24)λ1=-5.10412,λ2=-1,λ3=-0.70000,λ4=-4.60412.
Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (20) using the fourth-order Runge-Kutta scheme with time step 0.01. The initial values are taken as [x(0)=1,y(0)=0.7,z(0)=20,w(0)=0.1]. The behaviors of the states (x,y,z,w) of the controlled chaotic Lorenz-Stenflo system (20) with time are displayed in Figures 4(a)–4(d).The states of system (19): (a) x-t, (b) y-t, (c) z-t, and (d) w-t.
(a)
(b)
(c)
(d)


## 3.3. The Doubly Periodic Function Feedback Control

Suppose thatu2=0, u3=0, and u4=0; u1 is of the doubly periodic function u1=k1cn(x,m), where k1 is speed feedback coefficients and 0<m<1 is the modulus of Jacobi elliptic function. Therefore the controlled chaotic system (9) is rewritten as
(25)x˙=a(y-x)+dw+k1cn(x,m),y˙=x(c-z)-y,z˙=yx-bz,w˙=-x-aw.
The Jacobian matrix is
(26)J=(-a+k1a0dc-100-100-a).
The characteristic equation of J is
(27)λ4+(1-k1+2a+b)λ3+(a222ab-k1b-k1a+2a-k1+b+d+a2-caa22)λ2+(-ca2-cab+2ab+db+a2b-k1a+a2-k1b-k1ba+d)λ-ca2b-k1ba+a2b+db=0.Proposition 3.
If one chooses the control coefficientsk1=-30, m=0.3 and the parameters a=1.0, b=0.7, c=26, and d=1.5, the controlled chaotic Lorenz-Stenflo system (9) is asymptotically stable at the equilibrium (0,0,0,0).The proof of Proposition3 is the same as Proposition 1, which is straightforward and thus is omitted.In the following, we give the eigenvalues via the compute simulation. When the parameters were selected by the above value, we obtain the Jacobian matrix(28)J=(-31.01.001.526-10000-0.70-1.000-1.0).
The characteristic equation of J changes the following:
(29)λ4+33.7λ3+61.60λ2+33.450λ+4.55=0.
The eigenvalues of the above equation are
(30)λ1=-31.795569,λ2=-0.20443,λ3=-1,λ4=-0.70000.
Thus the zero solution of system (9) is exponentially stable; Proposition 3 is proved.Numerical simulations are used to investigate the controlled chaotic Lorenz-Stenflo system (25) using the fourth-order Runge-Kutta scheme with time step 0.01. The initial values are taken as [x(0)=1,y(0)=0.7,z(0)=20,w(0)=0.1]. The behaviors of the states (x,y,z,w) of the controlled chaotic Lorenz-Stenflo system (25) with time are displayed in Figures 5(a)–5(d).The states of system (25): (a) x-t, (b) y-t, (c) z-t, and (d) w-t.
(a)
(b)
(c)
(d)


## 4. Globally Exponential Hyperchaotic Projective Synchronization Control

Consider two chaotic systems given by(31)x˙m=f(xm,t),(32)y˙s=g(ys,t)+u(xm,ys,t),
where xm=(x1m,x2m,…xnm)T, ys=(y1s,y2s,…yns)T, f,g∈Cr[R+×Rn,Rn], u∈Cr[R+×Rn×Rn,Rn], and r≥1. R+ comprises the set of non negative real numbers. Assume that (31) is the master system, (32) is the slave system, and u(xm,ys,t) is the control vector. Let the error state be
(33)e(t)=[e1(t),e2(t)…,en(t)]T=[x1m-(a11y1s+a12)y1s,…,xnm-(an1yns+an2)y1s].
Then the error dynamics of e(t) are defined by
(34)e˙(t)=f(xm,t)-g(ys,t)-u(xm,ys,t).
The slave and master systems are said to be exponential, hyperchaotic, projective and synchronized if, for allxm(t0), ys(t0)∈Rn, and i∈Rn, ∥xm-(ai1ys+ai2)ys∥→0 as t→∞.Lemma 4 (see [27, 28]).
The zero solution of the error dynamical system (34) is globally and exponentially stable; the master-slave systems (31) and (32) are globally and exponentially projective, synchronized, if there exists a positive definite quadratic polynomial V=(e1,e2,…en)P(e1,e2,…en)T such that dV/dt=(e1,e2,…en)Q(e1,e2,…en)T. Moreover, the following negative Lyapunov exponent estimation for the error dynamical system (34) holds:
(35)∑i=1n‍ei2(t)≤λmax(P)λmin(P)∑i=1n‍ei2(0)exp[-λmax(Q)λmin(P)(t)],
where P=PT∈Rn×n and Q=QT∈Rn×n are both positive definite matrices, λmax(P) and λmin(P) stand for the minimum and maximum eigenvalues of the matrix P, respectively, and λmin(Q) denotes the minimum eigenvalue of the matrix Q.In the following, we consider the hyperchaotic system (1) as a master system:
(36)x˙m=a(ym-xm)+dwm,y˙m=xm(c-zm)-ym,z˙m=ymxm-bzm,w˙m=-xm-awm
and the system related to (36), given by
(37)x˙s=a(ys-xs)+dws+u1,y˙s=xs(c-zs)-ys+u2,z˙s=ysxs-bzs+u3,w˙s=-xs-aws+u4,
as a slave system, where the subscripts “m” and “s” stand for the master system and slave system, respectively. Let the error state be
(38)e(t)=(e1,e2,e3,e4)T=[xm-(a11xs+a12)xs,ym-(a21ys+a22)ys,zm-(a31zs+a32)zs,wm-(a41ws+a42)ws]T.
Then the derivative of e(t) along the trajectories of (36) and (37), we obtain the error system:
(39)e˙1=a(ym-xm)+dwm-2a11xs×[a(ys-xs)+bws+u1]-[a(ys-xs)+bws+u1]a12,e˙2=(c-zm)xm-ym-2a21ys×[(c-zs)xs-ys+u2]-[(c-zs)xs-ys+u2]a22,e˙3=-bzm+xmym-2a31zs×(-dzs+xsys+u3)-(-dzs+xsys+u3)a32,e˙4=-xm-awm-2a41ws(-xs-aws+u4)-(-xs-aws+u4)a42.
To demonstrate the synchronization control between systems (36) and (37), we have the following cases based on [29–44].Case  1 (modified projective synchronization control).Theorem I.
When[a11,a12,a21,a22,a31,a32,a41,a42]=[0,m,0,m,0,m,0,m], a>0 and b>0. For the hyperchaotic system (1), if one of the following families of feedback controllers ui(i=1,2,3,4) is given for the slave system (36):
(40)u1=(d-b)mw2-v1m,u2=(1-m)mx2z2-v2m,u3=(mx2y2-bz2+dz2-x2y2)m-v3m,u4=-x1+mx2-v4m,
there exist many possible choices for v1, v2, v3, and v4. Then the zero solution of the error dynamical system (39) is globally and exponentially stable, and thus globally exponential modified projective synchronization can be achieved.The concrete proof of Theorem I can be demonstrated by AppendixB. In the following, tracking numerical simulations are used to solve differential equations (36), (37), and (42) with Runge-Kutta integration method. The initial values of the drive system and response system are x1(0)=0.1, y1(0)=0.1, z1(0)=20, and w1(0)=0.1 and x2(0)=0.2, y2(0)=0.2, z2(0)=21, and w2(0)=0.2 respectively. The parameters are chosen to be m=-2, a=1.0, b=0.7, d=1.5, and c=26 so that the Lorenz-Stenflo hyperchaotic system exhibits a chaotic behavior if no control is applied. The numerical simulation of the master and the slave systems without active synchronization control law is shown in Figures 6(a)–6(d). The diagram of the solutions of the master and the slave systems with feedback control law is presented in Figures 7(a)–7(d). The synchronization errors are shown in Figures 8(a)–8(d). Figures 9(a)–9(d) depicts the projection of the synchronized attractors. Figures 10(a)–10(f) depicts the phase portraits of the synchronized attractors.The solutions of the master and slave systems without active control law. (a) Signalsx1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
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(d)The solutions of the drive and response systems with control law. (a) Signalsx1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
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(d)The dynamics of synchronization errors. (a) Signale1, (b) signal e2, (c) signal e3 and (d) signal e4.
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(d)The projection of the synchronized attractors in different spaces: (a)(x,y,z), (b) (x,y,w), (c) (x,w,z), and (d) (y,w,z). The solid line denotes and the drive system; the dashed line denotes respond system synchronized.
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(d)Simulated phase portraits of the chaotic system (1), projected on (a) the x-y plane, (b) the x-z plane, (c) the x-w plane, (d) the y-z plane, (e) the y-w plane, and (f) the z-w plane.
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(f)Case  2 (generalized projective synchronization control).Theorem II.
When[a11,a12,a21,a22,a31,a32,a41,a42]=[0,m1,0,m2,0,m3,0,m4], mi’s are different, a>0 and b>0. For the hyperchaotic system (1), if one of the following families of feedback controllers ui(i=1,2,3,4) is given for the slave system (36):
(41)u1=(m2-m1)ay2+(dm4-m1b)w2-v1m1,u2=(m1-m2)cx2+(m2-m1m3)cx2-v2m2,u3=(d-b)m3z2+(m1m2-m3)cx2-v3m3,u4=-x1+m4x2-v4m4,
there exist many possible choices for v1, v2, v3, and v4. Then the zero solution of the error dynamical system (39) is globally and exponentially stable, and thus globally exponential generalized projective synchronization.The concrete proof of Theorem II can be demonstrated by AppendixB. In the following, tracking numerical simulations are used to solve differential equations (36), (37), and (39) with Runge-Kutta integration method. The parameters are chosen to be m1=2, m2=3, m3=3, and m4=5. The initial values and others parameters are the same as the above cases so that the Lorenz-Stenflo hyperchaotic system exhibits a chaotic behavior if no control is applied. The diagram of the solutions of the master and the slave systems with feedback control law is presented in Figures 11(a)–11(d). The synchronization errors are shown in Figures 12(a)–12(d).The solutions of the master and slave systems with control law. (a) Signalsx1 (the dashed line) and x2 (the solid line). (b) Signals y1 (the dashed line) and y2 (the solid line). (c) Signals z1 (the dashed line) and z2 (the solid line). (d) Signals w1 (the dashed line) and w2 (the solid line).
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(d)The dynamics of synchronization errors. (a) Signale1, (b) signal e2, (c) signal e3, and (d) signal e4.
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(d)Case  3 (function synchronization control [45–58]).Theorem III.
Whenf1=a11x2+a12, f2=a21y2+a22, f3=a31z2+a32, f4=a41w2+a42, ai1≠0, a>0, b>0. For the hyperchaotic system (1), if one of the following families of feedback controllers ui(i=1,2,3,4) is given for the slave system (36):
(42)u1=(f2-f1)ay2+(df4-f1b)w2-k1e1f1,u2=(f1-f2)cx2+(f2-f1f3)cx2-k2e2f2,u3=(d-b)f3z2+(f1f2-f3)cx2-k3e3f3,u4=-x1+f4x2-k4e4f4,
where k1>0, k2>0, k3>0, and k4>0, then the zero solution of the error dynamical system (39) is globally stable, and thus global function, projective synchronization occurs between the master systems (36) and (37).The concrete Proof of Theorem III can be demonstrated by AppendixB. In the following, tracking numerical simulations are used to solve differential equations (36), (37), and (39) with Runge-Kutta integration method. The initial values and the parameters are the same as the above cases so that the Lorenz-Stenflo hyperchaotic system exhibits a chaotic behavior if no control is applied. The diagram of the solutions of the master and the slave systems with active control law is presented in Figures 13(a)–13(d). The synchronization errors are shown in Figures 14(a)–14(d).The function projection of the synchronized attractors in different spaces: (a)(x,y,z), (b) (x,y,w), (c) (x,w,z), and (d) (y,w,z). The solid line denotes the drive system; the dashed line denotes respond system synchronized.
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(d)The dynamics of synchronization errors. (a) Signale1, (b) signal e2, (c) signal e3 and (d) signal e4.
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## 5. Summary and Conclusions

In this paper, we have introduced the tracking control and generalized synchronization of the hyperchaotic system which is different from the Lorenz-Stenflo attractor. We suppress the chaos to unstabilize equilibrium via three feedback methods, and we achieve three globally generalized synchronization controls of two Lorenz-Stenflo systems. As a result, some powerful controllers are obtained. Then, we investigate the hyperchaotic system applying the complex system calculus technique. Moreover, numerical simulations are used to verify the effectiveness of our results, through the contrast between the orbits before being stabilized and the ones after being stabilized.



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*Source: 515106-2013-12-10.xml*