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Synchronization of hyperchaotic systems via linear control
Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
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Synchronization of hyperchaotic systems via linear control q Hua Wang a,*, Zheng-zhi Han b, Zhen Mo b a b
School of Mechatronics Engineering and Automation, Shanghai University, Shanghai 200072, China School of Electronic, Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e
i n f o
Article history: Received 6 November 2008 Received in revised form 27 July 2009 Accepted 28 July 2009 Available online 6 August 2009
a b s t r a c t In this paper, synchronization of hyperchaotic system is discussed. Based on the stability theory in the cascade system, a simple linear feedback law is presented to realize synchronization of hyperchaotic systems. Simulation results are given to illustrate the effectiveness of the proposed method. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Hyperchaotic systems Synchronization Linear feedback Input-to-state stability (ISS)
1. Introduction Chaotic phenomenon has received increasing attention in the past several decades. Compared with the ordinary chaotic systems, hyperchaotic systems hold more than one positive Lyapunov exponents. Hence, hyperchaotic systems possess more complicated attractors. Since hyperchaotic systems have the characteristics of high capacity, high security and high efficiency, they have been researched more and more in the fields of secure communication, information processing, biological engineering, chemical processing and other fields [1–9]. Many methods have been proposed to realize chaos synchronization in low dimensional attractors with one positive Lyapunov exponent. However, chaotic system with higher dimensional attractor have much wider application. The presence of more than one Lyapunov exponent clearly improves security of the communication scheme by generating more complex dynamics. So, hyperchaotic systems are being given more and more interest [10–13]. Several methods such as nonlinear control method [11], linear control method [12], adaptive control method [13,14], neural networks method [15] have been provided for hyperchaotic synchronization. Studies on the stabilization of cascade systems have attracted many researchers’ attention in nonlinear control field [16,17]. Among theses advances are several constructive design methods such as backstepping and forwarding, which are based on recursive applications of cascade designs. From some aspects, the hyperchaotic system can be seen as cascade system. In this paper, linear feedback law is designed based on the stability theory in cascade system. Taking some error states as virtual control inputs, input-to-state stability (ISS) theory [18,19] is technically implemented to attain global asymptotical stability of the overall error system. Linear controller is designed step by step and the feedback control obtained in this way is simple. It can be easily implemented in the practical process.
q
Supported by the National Natural Science Foundation of China (Grant No. 60674024). * Corresponding author. Tel.: +86 21 34203655; fax: +86 21 62932083. E-mail address: [email protected] (H. Wang).
1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.07.023
H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
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2. Preliminary results and lemmas Before giving the main results, let us introduce some necessary lemmas. Definition 1. A continuous function a : ½0; aÞ ! ½0; 1Þ is said to belong to class K if it is strictly increasing and að0Þ ¼ 0. It is said to belong to class K 1 if a ¼ 1 and aðrÞ ! 1 as r ! 1. Definition 2. A continuous function b : ½0; a ½0; 1Þ ! ½0; 1Þ is said to belong to class KL if, for each fixed s, the mapping bðr; sÞ belongs to class K with respect to r and, for each fixed r, the mapping bðr; sÞ is decreasing with respect to s and bðr; sÞ ! 0 as s ! 1. Lemma 1 [20]. Let x ¼ 0 be an equilibrium point for x_ ¼ f ðt; xÞ and D Rn be domain containing x ¼ 0. Let V : ½0; 1Þ D ! R be a continuously differentiable function such that
a1 kxkb 6 Vðt; xÞ 6 a2 kxkb ; ð1Þ
@V @V þ x_ 6 a3 kxkb ; @t @x
8t P 0 and 8x 2 D; where ai ði ¼ 1; 2; 3Þ and b are positive constants. Then, x ¼ 0 is exponentially stable. If the assumptions hold globally, then x ¼ 0 is globally exponentially stable. Lemma 2 [20]. Consider the system
x_ ¼ f ðt; x; uÞ:
ð2Þ n
m
n
Suppose f : ½0; 1Þ R R ! R is continuously differentiable and globally Lipschitz in ðx; uÞ, uniformly in t. If the unforced system x_ ¼ f ðt; x; 0Þ has a globally exponentially stable equilibrium point at the origin x ¼ 0, then the system (2) is input-to-state stable. Lemma 3 [20]. Consider the following system
x_ 1 ¼ f1 ðt; x1 ; x2 Þ;
ð3Þ
x_ 2 ¼ f2 ðt; x2 Þ;
ð4Þ n1
n2
n1
n2
n2
T
where f1 : ½0; 1Þ R R ! R and f2 : ½0; 1Þ R ! R are piecewise continuous in t and Locally Lipschitz in x ¼ ½x1 ; x2 . If the system (3), with x2 as input, is input-to-state stable and the origin of (4) is globally uniformly asymptotically stable, then the origin of the cascade system (3) and (4) is globally uniformly asymptotically stable. Proof. By the definition of asymptotical stability and the definition of ISS, the condition of Lemma 3 implies that there exist KL-class functions b1 and b2 , and a K-class function c1 such that for any t P s P t 0 ,
kx1 ðtÞk 6 b1 ðkx1 ðsÞk; t sÞ þ c1 sup kx2 ðsÞk ;
ð5Þ
kx2 ðtÞk 6 b2 ðkx2 ðsÞk; t sÞ;
ð6Þ
s6s6t
Let s ¼
tþt 0 . 2
It follows t P s ¼
tþt 0 . 2
Because t P s P t 0 , (5) becomes
0 1 t þ t0 t t0 @ þ c1 sup kx2 ðsÞkA: kx1 ðtÞk 6 b1 x1 ; 2 2 tþt 2
Using (5) again, but now t is replaced by
tþt 0 2
06
ð7Þ
s6t
and s ¼ t 0 , we have
0 1 t þ t t t 0 0 6 b1 kx1 ðt 0 Þk; x1 þ c1 @ sup kx2 ðsÞkA: 2 2 tþt t 6 s6 0 0
ð8Þ
2
By (6),
kx2 ðtÞk 6 b2 ðkx2 ðt0 Þk; tÞ 6 b2 ðkx2 ðt 0 Þk; 0Þ:
ð9Þ
Note that the right side is a constant, hence
sup kx2 ðsÞk 6 b2 ðkx2 ðt 0 Þk; 0Þ; t 0 6 s6
tþt 0 2
By (10), (8) leads to
ð10Þ
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x1 t þ t0 6 b1 kx1 ðt 0 Þk; t t 0 þ c ðb2 ðkx2 ðt 0 Þk; 0Þ: 1 2 2
ð11Þ
where c1 ðb2 ðkx2 ðt 0 Þk; 0Þ is a K – class function of kx2 ðt 0 Þk. The right side of (11) is a KL-class function of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 kx1 ðt 0 Þk2 þ kx2 ðt0 Þk2 and tt , i.e., there is a KL-class function b3 such that 2
kxðt0 Þk ¼
x1 t þ t0 6 b3 kxðt0 Þk; t t 0 : 2 2
ð12Þ
Substituting it into the first term of the right side of (7) leads to
t þ t0 t t0 t t0 t t0 ; b1 x b kxðt Þk; 6 b ; : 0 1 3 1 2 2 2 2 The right side is still a KL-class function of kxðt 0 Þk and
t þ t0 ; t t0 6 b4 ðkxðt0 Þk; t t0 Þ: b1 x 1 2 2
tt 0 , 2
ð13Þ
and we denote the KL-class function as b4 , i.e.,
ð14Þ
0 On the other hand, if s ¼ tþt , from (6) we have 2
t þ t0 ; t t 0 6 b2 kx2 ðt0 Þk; t t0 : kx2 ðtÞk 6 b2 x 2 2 2 2
ð15Þ
The last inequality is valid after t P T for some T, since x2 ðtÞ is asymptotically stable. Then
sup kx2 ðsÞk 6 b2 kx2 ðt 0 Þk;
tþt 0 2
tþt0 6 2
s6t
t0 2
! ¼ b2 ðkx2 ðt0 Þk;
t t0 : 4
ð16Þ
Furthermore,
0
1
c1 @ sup kx2 ðsÞkA 6 c1 b2 kx2 ðt0 Þk; tþt0 6 2
s6t
t t0 4
ð17Þ
:
Its right side is a KL-class function of kx2 ðt0 Þk and t t 0 . We denote the function by b5 , i.e.,
0
1
c1 @ sup kx2 ðsÞkA 6 b5 ðkx2 ðt0 Þk; t t0 Þ: tþt0 6 2
ð18Þ
s6t
By (14) and (18), (7) results in
kx1 ðtÞk 6 b4 ðkxðt 0 Þk; t t0 Þ þ b5 ðkx2 ðt 0 Þk; t t 0 Þ:
ð19Þ
At last, the lemma is verified by
kxðtÞk 6 kx1 ðtÞk þ kx2 ðtÞk 6 b4 ðkxðt 0 Þk; t t 0 Þ þ b5 ðkx2 ðt0 Þk; t t0 Þ þ b2 ðkx2 ðt0 Þk; t t0 Þ 6 b4 ðkxðt 0 Þk; t t 0 Þ þ b5 ðkxðt 0 Þk; t t 0 Þ þ b2 ðkxðt0 Þk; t t0 Þ
ð20Þ
Theorem 1. For the cascade system (3) and (4), if 1. f1 is continuously differentiable and globally Lipschitz in ðx1 ; x2 Þ, uniformly in t and f2 is piecewise continuous in t and Locally Lipschitz in x2 . 2. System x_ 1 ¼ f1 ðt; x1 ; 0Þ is globally exponentially stable and the origin of (4) is globally uniformly asymptotically stable. Then the origin of the cascade system (3) and (4) is globally uniformly asymptotically stable. Proof. This result is quite straightforward from Lemmas 2 and 3.
h
3. Main results Since the discovery of the hyperchaotic Rössler system [21], many hyperchaotic systems have been presented such as the hyperchaotic MCK circuit [22], the hyperchaotic Lü [23] and the hyperchaotic Chen [24] system, etc. In this section, we mainly research synchronization of the hyperchaotic Lü and the hyperchaotic Chen system via linear control. The method given below can also be extended to other hyperchaotic systems.
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3.1. Synchronization of hyperchaotic Lü system In 2006, Lü et al. [23] presented a new hyperchaotic system, called the hyperchaotic Lü system, which is described by
8 x_ 1 > > > < x_ 2 > _3 x > > : x_ 4
¼ a1 ðx2 x1 Þ þ x4 ; ¼ x1 x3 þ c1 x2 ;
ð21Þ
¼ x 1 x2 b 1 x3 ; ¼ x 1 x3 þ r 1 x4 ;
where xi ði ¼ 1; 2; 3; 4Þ are state variables. a1 ; b1 ; c1 and r 1 are real constants. Ref. [23] shows, when a1 ¼ 36; b1 ¼ 3; c1 ¼ 20; 1:03 6 r1 6 0:46, the system (21) has periodic orbit. If a1 ¼ 36; b1 ¼ 3; c1 ¼ 20; 0:46 < r 1 6 0:35, the system (21) has chaotic attractor. If a1 ¼ 36; b1 ¼ 3; c1 ¼ 20; 0:35 < r1 6 1:3, the system (21) has hyperchaotic attractor. Now, we consider how to realize synchronization of the hyperchaotic Lü system. The master (drive) system is system (21) and the slave(response) system is
8 y_ 1 > > > < y_ 2 > _3 y > > : y_ 4
¼ a1 ðy2 y1 Þ þ y4 þ u1 ; ¼ y1 y3 þ c1 y2 þ u2 ;
ð22Þ
¼ y1 y2 b1 y3 ¼ y1 y3 þ r 1 y4 þ u3 :
where u1 ; u2 and u3 are the control inputs to be designed. Denote the state errors as
e1 ¼ y1 x1 ;
e 2 ¼ y 2 x2 ;
e 3 ¼ y 3 x3 ;
e4 ¼ y4 x4 :
Subtracting Eq. (21) from Eq. (22), we can get the error system as follows
8 e_ 1 > > > < e_ 2 > _3 e > > : e_ 4
¼ a1 ðe2 e1 Þ þ e4 þ u1 ; ¼ e1 e3 e1 x3 x1 e3 þ c1 e2 þ u2 ; ¼ b1 e3 þ e1 e2 þ e1 x2 þ x1 e2 ;
ð23Þ
¼ e1 e3 þ e1 x3 þ e3 x1 þ r1 e4 þ u3 :
Now the synchronization problem of (21) and (22) is transformed into the stabilization problem of the error system (23), i.e., if system (23) is asymptotically stabilized by uj ðj ¼ 1; 2; 3Þ, synchronization of hyperchaotic Lü is realized. We present a linear feedback to achieve the stabilization. The design process of the linear controller is mainly divided into 3 steps. Step 1 Let u1 ¼ a1 e2 e4 , then the first equation of the error system (23) becomes
e_ 1 ¼ a1 e1 :
ð24Þ
Obviously, system (24) is globally uniformly asymptotically stable. Step 2 Consider the second and third equation of error system (23) and let u2 ¼ k1 e2 , where k1 > c1 is a constant. Then
e_ 2 ¼ e1 e3 e1 x3 x1 e3 þ c1 e2 k1 e2 ; e_ 3 ¼ b1 e3 þ e1 e2 þ e1 x2 þ x1 e2 :
ð25Þ
If we regard e1 as the virtual input of the system (25), then the systems (24) and (25) satisfy the first condition of Theorem 1. (The detailed proof is given in the appendix of this paper.) Now we verify that they satisfy the second condition of Theorem 1. Consider the following unforced system of system (25)
e_ 2 ¼ x1 e3 þ c1 e2 k1 e2 ; e_ 3 ¼ b1 e3 þ x1 e2 :
ð26Þ
Choose the following function as the candidate Lyapunov function for system (26)
V1 ¼
1 2 1 2 e þ e : 2 2 2 3
ð27Þ
The time derivative of V 1 along trajectories of system (26) is
V_1 ¼ e2 ðx1 e3 þ c1 e2 k1 e2 Þ þ e3 ðb1 e3 þ x1 e2 Þ ¼ ðk1 c1 Þe22 b1 e23 6 k1 ðe22 þ e23 Þ;
ð28Þ
where k1 ¼ minðk1 c1 ; b1 Þ. From Lemma 1, the system (26) is globally exponentially stable at e2 ¼ 0; e3 ¼ 0. This means system (25) satisfies the second condition in Theorem 1. So system (25) is globally uniformly asymptotically stable, i.e. limt!1 e2 ðtÞ ¼ 0; limt!1 e3 ðtÞ ¼ 0. Step 3 Finally, let us consider the last equation in error system (23). Let u3 ¼ k2 e4 , where k2 > r 1 is a constant. Then the last equation in error system (23) with such a u3 becomes
e_ 4 ¼ e1 e3 þ e1 x3 þ e3 x1 þ r1 e4 k2 e4 :
ð29Þ
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Take e1 ; e2 and e3 as the virtual inputs to this system, and we can see this system satisfy the first condition of Theorem 1. (The detailed proof is given in the appendix.) Then the unforced system becomes
e_ 4 ¼ ðk2 r 1 Þe4 :
ð30Þ
As k2 > r 1 , system (30) is globally exponentially stable at e4 ¼ 0. With the consideration that e1 ; e2 , and e3 are globally uniformly asymptotically stable, then by Theorem 1, system (29) is globally uniformly asymptotically stable, i.e. limt!1 e4 ¼ 0. From the above analysis, the error system (23) is globally uniformly asymptotically stable with the linear controller
u2 ¼ k1 e2 ðk1 > c1 Þ;
u3 ¼ k2 e4 ðk2 > r 1 Þ:
10
e1 e2 e3 e4
8 6 4 2 0 −2 −4 −6
0
2
4
6
time(s)
8
10
Fig. 1. Synchronization errors of hyperchaotic Lü system.
30
x1 y1
25 20
x1,y1
15 10 5 0 −5 −10 −15 −20
0
5
10
15
20
time(s)
25
30
35
Fig. 2. Synchronization of hyperchaotic Lü system x1 and y1 .
40
x2 y2
30 20
x2,y2
u1 ¼ a1 e2 e4 ;
10 0
−10 −20 −30 0
5
10
15
20
time(s)
25
30
Fig. 3. Synchronization of hyperchaotic Lü system x2 and y2 .
35
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H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
So the slave system (22) can asymptotically synchronize the master hyperchaotic Lü system (21). It can be seen that the controller designed here is more simpler than the nonlinear controller in Ref. [11].
3.2. Synchronization of hyperchaotic chen system This subsection turns to the consideration of the hyperchaotic Chen system [24]. The hyperchaotic Chen system [25] is described by
50
x3 y3
40
x3,y3
30 20 10 0 −10
0
5
10
15
20
time(s)
25
30
35
Fig. 4. Synchronization of hyperchaotic Lü system x3 and y3 .
100
x4 y4
50
x4,y4
0 −50 −100 −150 −200
0
5
10
15
20
time(s)
25
30
35
Fig. 5. Synchronization of hyperchaotic Lü system x4 and y4 .
10
hat(e)1 hat(e)2 hat(e)3 hat(e)4
5 0 −5 −10 −15 −20
0
5
10
time(s)
15
20
Fig. 6. Synchronization errors of hyperchaotic Chen system.
25
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H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
8 ^x_ 1 > > > > < ^_ x2 > > ^x_ 3 > > :_ ^x4
¼ að^x2 ^x1 Þ þ ^x4 ; ¼ d^x1 ^x1 ^x3 þ c^x2 ;
ð31Þ
¼ ^x1 ^x2 b^x3 ; ¼ ^x2 ^x3 þ r^x4 ;
where ^ xi ði ¼ 1; . . . ; 4Þ are system state variables. a; b; c; d, and r are real constants. When a ¼ 35; b ¼ 3; c ¼ 12; d ¼ 7; 0 6 r 6 0:085, system (31) is chaotic; when a ¼ 35; b ¼ 3; c ¼ 12; d ¼ 7; 0:085 < r 6 0:798, system (31) is hyperchaotic; when a ¼ 35; b ¼ 3; c ¼ 12; d ¼ 7; 0:798 < r 6 0:9, system (31) is periodic.
30
hat(x)1 hat(y)1
25 20
x1,y1
15 10 5 0 −5 −10 −15 −20
0
5
10
15
20
time(s)
25
30
35
^1 . Fig. 7. Synchronization of hyperchaotic Chen system ^ x1 and y
30
hat(x)2 hat(y)2
25 20 15
x2,y2
10 5 0 −5 −10 −15 −20 0
5
10
15
20
time(s)
25
30
35
^2 . Fig. 8. Synchronization of hyperchaotic Chen system ^ x2 and y
45
hat(x)3 hat(y)3
40 35
x3,y3
30 25 20 15 10 5 0 −5
0
5
10
15
20
time(s)
25
30
^3 . Fig. 9. Synchronization of hyperchaotic Chen system ^ x3 and y
35
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Let us consider system (31) as the master system, the slave system is
8 ^_ 1 y > > > > ^_ 3 >y > > :_ ^4 y where
^2 y ^1 Þ þ y ^4 þ v 1 ; ¼ aðy ^3 þ c y ^1 y ^1 y ^2 þ v 2 ; ¼ dy
ð32Þ
^1 y ^3 ; ^2 by ¼y ^3 þ r y ^2 y ^4 þ v 3 ; ¼y
v 1 ; v 2 and v 3 are the controllers to be determined to realize hyperchaotic synchronization. Denote ^1 ^x1 ; ^e1 ¼ y
^e2 ¼ y ^2 ^x2 ;
^e3 ¼ y ^3 ^x3 ;
^e4 ¼ y ^4 ^x4 :
From (31) and (32) we can get the following error system
8 ^e_ > > > 1 > < ^_ e2 > ^e_ 3 > > > :_ ^e4
¼ að^e2 ^e1 Þ þ ^e4 þ v 1 ; ¼ d^e1 ^e1 ^e3 ^e1 ^x3 ^x1 ^e3 þ c^e2 þ v 2 ;
ð33Þ
¼ b^e3 þ ^e1 ^e2 þ ^e1 ^x2 þ ^x1 ^e2 ; ¼ ^e2 ^e3 þ ^e2 ^x3 þ ^e3 ^x2 þ r^e4 þ v 3 :
Design the linear controller as
v 1 ¼ a^e2 ^e4 ; v 2 ¼ L1 ^e2 ðL1 > cÞ; v 3 ¼ L2 ^e4 ðL2 > rÞ: Then the error system (33) with such linear controller becomes
8 ^e_ 1 ¼ a^e1 ; > > > > < ^e_ ¼ d^e ^e ^e ^e ^x ^x ^e þ ðc L Þ^e ; 2 1 1 3 1 3 1 3 1 2 > > ^e_ 3 ¼ b^e3 þ ^e1 ^e2 þ ^e1 ^x2 þ ^x1 ^e2 ; > > :_ ^e4 ¼ ^e2 ^e3 þ ^e2 ^x3 þ ^e3 ^x2 þ ðr L2 Þ^e4 :
ð34Þ
We want to know whether system (34) is asymptotically stable. Now we analyze this stability problem of the error system step by step. Similarly as we have done for the hyperchaotic Lü system, this process is mainly divided into three steps. Step 1 The first equation of the system (34) is
^e_ 1 ¼ a^e1 : As a ¼ 35 > 0, it can be proved easily that this system is globally asymptotically stable. The detailed proof process is omitted here. Step 2 Consider the second and third equation in system (34)
(
^e_ 2 ¼ d^e1 ^e1 ^e3 ^e1 ^x3 ^x1 ^e3 þ ðc L1 Þ^e2 ; ^e_ 3 ¼ b^e3 þ ^e1 ^e2 þ ^e1 ^x2 þ ^x1 ^e2 :
ð35Þ
Take ^e1 as the virtual input of system (35). Then the next problem is whether the system is exponentially stable without the virtual input ^e1 . The unforced system is
100
hat(x)4 hat(y)4
x4,y4
50 0 −50 −100 −150
0
5
10
15
time(s)
20
25
30
^4 . Fig. 10. Synchronization of hyperchaotic Chen system ^ x4 and y
35
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H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
(
^e_ 2 ¼ ^x1 ^e3 þ ðc L1 Þ^e2 ; ^e_ 3 ¼ b^e3 þ ^x1 ^e2 :
ð36Þ
Chose the candidate Lyapunov function for system (36) as
V2 ¼
1 2 1 2 ^e þ ^e : 2 2 2 3
ð37Þ
Calculate the derivation of V 2 along the trajectories of system (36) and we can get
V_2 ¼ ^e2 ð^x1 ^e3 þ ðc L1 Þ^e2 Þ þ ^e3 ðb^e3 þ ^x1 ^e2 Þ ¼ ðL1 cÞ^e22 b^e23 6 k2 ^e22 þ ^e23
ð38Þ
where k2 ¼ minfL1 c; bg. From Lemma 1, system (36) is globally exponentially stable. Additional with Theorem 1, system (35) is globally asymptotically stable if we consider ^ e1 as virtual input. Step 3 Finally, let us analyze the stability problem of the last equation in system (34)
^e_ 4 ¼ ^e2 ^e3 þ ^e2 ^x3 þ ^e3 ^x2 þ ðr L2 Þ^e4 :
ð39Þ
Take ^ e2 ; ^ e3 as the virtual input, then the unforced system is
^e_ 4 ¼ ðL2 rÞ^e4 :
ð40Þ
Since L2 > r, this system is globally exponentially stable at e4 ¼ 0. As we can see from the above analysis process, e2 and e3 are globally asymptotically stable. From Theorem 1, the system (39) is globally asymptotically stable. So the error system (33) is globally asymptotically stable with the linear controller
v 1 ¼ a^e2 ^e4 ; v 2 ¼ L1 ^e2 ðL1 > cÞ; v 3 ¼ L2 ^e4 ðL2 > rÞ: Notation. In the above analysis, whether system (35) and (39) satisfy the global Lipschitz condition is not discussed. This is because we can analyze it in the same way as we do for the hyperchaotic Lü system in the appendix. So the discussion of this question is omitted here. 4. Simulation results In this section, simulation results for the hyperchaotic Lü and Chen systems are given, respectively. Fourth order Runge–Kutta integration method is used to solve the differential equations with time step length 0.0001. The parameters of hyperchaotic Lü system are selected as a1 ¼ 36; b1 ¼ 3; c1 ¼ 20; r1 ¼ 1:3. The Lü system exhibits hyperchaotic behavior. Controller gains are chosen as k1 ¼ c1 þ 1; k2 ¼ r 1 þ 1. The initial values of the master and slave systems are x1 ð0Þ ¼ 5; x2 ð0Þ ¼ 8; x3 ð0Þ ¼ 1; x4 ð0Þ ¼ 3 and y1 ð0Þ ¼ 3; y2 ð0Þ ¼ 4; y3 ð0Þ ¼ 5; y4 ð0Þ ¼ 5, respectively. Fig. 1 displays the time response of the controlled synchronization error system (23). Figs. 2–5 show the state response between the master system (21) and the slave system (22). From the figures we can see that the asymptotical synchronization of the hyperchotic Lü system is realized via the proposed linear feedback control. Chose the parameters of hyperchaotic Chen system as a ¼ 35; b ¼ 3; c ¼ 12; d ¼ 7; r ¼ 0:5 and the controller gain as L1 ¼ c þ 1; L2 ¼ r þ 1. The initial states of the master and slave systems are x1 ð0Þ ¼ 3; x2 ð0Þ ¼ 5; x3 ð0Þ ¼ 5; x4 ð0Þ ¼ 3 and y1 ð0Þ ¼ 1; y2 ð0Þ ¼ 2; y3 ð0Þ ¼ 2; y4 ð0Þ ¼ 3, respectively. Fig. 6 shows the time response of the synchronization error system (33). Figs. 7–10 provide the state trajectories of the master and slave system. From Figs. 6–10 we can see the slave system (32) asymptotically synchronize the master system (31) with the proposed simple linear controller. 5. Conclusion In this paper, linear controllers are proposed for hyperchaotic synchronization of the hyperchaotic Lü and Chen systems, respectively. Since the states of the hyperchaotic system are bounded, we can use the input-to-state theory in the cascade system to design the controller step by step. The linear feedback control presented is simple and easy to be implemented in real engineering process. Numerical simulations results illustrate the effectiveness of the proposed synchronization method. Appendix A Now we prove that the system (22) satisfies the first condition of Theorem 1, i.e. it meets the global Lipshitz condition. Let us introduce a lemma firstly. Lemma 4 [26]. Consider the nonlinear time-varying system x_ ¼ f ðx; tÞ, where x 2 U Rn ; t 2 J. If there exists a differentiable function VðÞ : U J ! R satisfying the following conditions: kðÞ : U ! R such that (1) There exist a positive constant k0 and a scalar function
k0 kxk2 6 Vðx; tÞ 6 kðxÞkxk2 ;
8ðx; tÞ 2 U J:
ð41Þ
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H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
(2) There exist positive constants k1 > 0 and 2 _ Vðx; tÞjx¼f _ ðx;tÞ 6 k1 kðxÞkxk þ e;
e P 0 such that
8ðx; tÞ 2 U J:
ð42Þ
then the solution of the system x_ ¼ f ðx; tÞ is bounded by
kxk2 6
1 e Vðx0 ; t 0 ÞekV ðtt0 Þ þ ð1 ek1 ðtt0 Þ Þ 6 M 2 ; k0 k0 k1
ð43Þ
where M 2 ¼ k10 Vðx0 ; t 0 Þ þ k0ek1 . Then we can get that kxk 6 M.
Proof. Consider the following differential equation:
y_ ¼ k1 y þ ;
y0 ¼ Vðx0 ; t 0 Þ:
ð44Þ
then
dy ¼ k1 y þ ¼ k1 y : dt k1
ð45Þ
Then, we can get the following differential equation:
dðy k1 Þ y k1
¼ k1 dt:
ð46Þ
Integrate this differential equation with the initial condition y0 ¼ Vðx0 ; t0 Þ and we can get
y ¼ y0 ek1 ðtt0 Þ þ
e k1
1 ek1 ðtt0 Þ :
ð47Þ
From the comparison Lemma [20] we know
Vðx; tÞ 6 Vðx0 ; t 0 Þek1 ðtt0 Þ þ
e k1
1 ek1 ðtt0 Þ :
ð48Þ
Then
kxk2 6
1 e Vðx0 ; t 0 Þek1 ðtt0 Þ þ 1 ek1 ðtt0 Þ 6 M 2 ; k0 k0 k1
ð49Þ
where M2 ¼ k10 Vðx0 ; t 0 Þ þ k0ek1 . h Lemma 5 [27]. Let x; y be two n-dimensional vectors and q be a positive real number. Then the following inequality holds
2xT y 6 qxT x þ q1 yT y: Now we turn to prove that the system (22) satisfies the global Lipshitz condition. Derivations of computations concerning Step 2 of Section 3.1. Consider the system (22) as follows:
e_ 2 ¼ e1 e3 e1 x3 x1 e3 þ c1 e2 k1 e2 ; e_ 3 ¼ b1 e3 þ e1 e2 þ e1 x2 þ x1 e2 :
ð50Þ
Since system (21) is a hyperchaotic system, its state trajectories are bounded. There exist positive constants ni such that jxi j < ni ði ¼ 1; 2; 3; 4Þ. Consider the following positive function
V 3 ðe2 ; e3 Þ ¼ e22 þ e23 :
ð51Þ
Its derivation along the trajectories of system (50) is
V_3 ðe2 ; e3 Þ ¼ 2e2 ðe1 e3 e1 x3 x1 e3 þ c1 e2 k1 e2 Þ þ 2e3 ðb1 e3 þ e1 e2 þ e1 x2 þ x1 e2 Þ ¼ 2ðk1 c1 Þe22 2b1 e23 2e1 e2 x3 þ 2e1 e3 x2 6 ð2k1 2c1 e1 Þe22 ð2b1 e2 Þe23 þ 6 ð2k1 2c1 e1 Þe22 ð2b1 e2 Þe23 þ M;
1
e1
e21 x23 þ
1
e2
e21 x22 ð52Þ
where e1 and e2 are two positive constants and selected such that k1 c1 e1 > 0; b1 e2 > 0. The positive constant M is M ¼ e11 e21 ð0Þn23 þ e12 e21 ð0Þn22 . From Lemma 4, states of system (50) are bounded. So there exist positive constants rj ðj ¼ 2; 3Þ such that jej j < rj ðj ¼ 2; 3Þ.
1920
H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
Denote f ¼ ðe2 ; e3 ÞT and system (50) can be rewritten as
f_ ¼ gðf; tÞ ¼
g 1 ðf; tÞ g 2 ðf; tÞ
¼
e1 e3 e1 x3 x1 e3 þ c1 e2 k1 e2 b1 e3 þ e1 e2 þ e1 x2 þ x1 e2
:
ð53Þ
Then
jg 1 ðf f0 Þ; tj ¼ j e1 e3 e1 x3 x1 e3 þ ðc1 k1 Þe2 þ e01 e03 þ e01 x3 þ x1 e03 ðc1 k1 Þe02 j 6 je03 þ x3 ke1 e01 j þ jc1 k1 ke2 e02 j þ je1 þ x1 ke3 e03 j 6 ðr3 þ n3 Þje1 e01 j þ jc1 k1 ke2 e02 j þ ðje1 ð0Þj þ n1 Þje3 e03 j:
ð54Þ
For g 2 ðf; tÞ, we have
jg 2 ðf f0 Þ; tj ¼ j b1 e3 þ e1 e2 þ e1 x2 þ x1 e2 ðb1 e03 þ e01 e02 þ e01 x2 þ x1 e02 Þj 6 je02 þ x2 ke1 e01 j þ jx1 þ e1 ke2 e02 j þ b1 je3 e03 j 6 ðr2 þ n2 Þje1 e01 j þ ðn1 þ je1 ð0ÞjÞje2 e02 j þ b1 je3 e03 j:
ð55Þ
Then
g ðf f0 Þ; t 2 6 ðjg 1 ðf f0 Þ; tj þ jg 2 ðf f0 Þ; tjÞ2 6 ðg je1 e0 j þ g je2 e0 j þ g je3 e0 jÞ2 ; kgðf f0 Þ; tk2 ¼ 1 1 2 3 1 2 3 0 g 2 ðf f Þ; t
ð56Þ
g1 ¼ r2 þ r3 þ n2 þ n3 ; g2 ¼ jc1 k1 j þ n1 þ je1 ð0Þj; g3 ¼ je1 ð0Þj þ n1 þ b1 :
ð57Þ
where
Inequality (56) implies that gðf; tÞ is global Lipschitz in ðe2 ; e3 ÞT . Derivations of computations concerning Step 3 of Section 3.1. Consider the following system
e_ 4 ¼ e1 e3 þ e1 x3 þ e3 x1 þ r 1 e4 k2 e4 :
ð58Þ
Being similar to the above proof process, we can prove that the right hand of system (58) is global Lipschitz in e4 . References [1] Udaltsov VS, Goedgebuer JP, Larger L, Cuenot JB, Levy P, Rhodes WT. Communicating with hyperchaos: the dynamics of a DNLF emitter and recovery of transmitted information. Opt Spectrosc 2003;95(1):114–8. [2] Cafagna D, Grassi G. New 3D-scroll attractors in hyperchaotic Chua’s circuits forming a ring. Int J Bifurcat Chaos 2003;13(10):2889–903. [3] Sugawara T, Tachikawa M, Tsukamoto T, et al. Observation of synchronization in laser chaos. Phys Rev Lett 1994;72(22):3502–5. [4] Wang H, Han ZZ, Zhang W, Xie QY. Synchronization of unified chaotic systems with uncertain parameters based on the CLF. Nonlinear Anal: Real World Appl 2009;10(5):715–22. [5] Wang H, Han ZZ, Xie QY, Zhang W. Sliding mode control for chaotic systems based on LMI. Commun Nonlinear Sci Numer Simul 2009;14(4):1410–7. [6] Wang H, Han ZZ, Xie QY, Zhang W. Finite-time chaos control of unified chaotic systems with uncertain parameters. Nonlinear Dyn 2009;55(4):323–8. [7] Wang H, Han ZZ, Zhang W, Xie QY. Chaotic synchronization and secure communication based on descriptor observer. Nonlinear Dyn 2009;57:69–73. [8] Wang H, Han ZZ, Xie QY, Zhang W. Finite-time synchronization of uncertain unified chaotic systems based on CLF. Nonlinear Anal: Real World Appl 2009;10(5):2842–9. [9] Wang H, Han ZZ, Xie QY, Zhang W. Finite-time chaos control via nonsingular terminal sliding mode control. Commun Nonlinear Sci Numer Simul 2009;14(6):2728–33. [10] Park JH. Adaptive synchronization of hyperchaotic Chen system with uncertain parameters. Chaos Solitons Fract 2005;26(3):959–64. [11] Yassen MT. Synchronization hyperchaos of hyperchaotic systems. Chaos Solitons Fract 2008;37(2):465–75. [12] Marat Rafikov, Jose Manoel Balthazar. On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commun Nonlinear Sci Numer Simul 2008;13:1246–55. [13] Wang B, Wen GJ, Xie K. On the synchronization of a hyperchaotic system based on adaptive method. Phys Lett A 2008;372:3015–20. [14] Lu JQ, Cao JD. Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters. Chaos 2005;15(4):043901. [15] Cao JD, Lu JQ. Adaptive synchronization of neural networks with or without time-varying delay. Chaos 2006;16(1):013133. [16] Jankovic M, Sepulchre R, Kokotovic PV. Constructive Lyapunov stabilization of nonlinear cascade systems. IEEE Trans Automat Control 1996;41:1723–35. [17] Arcak M, Angeli D, Sontag EA. Unifying integral ISS framework for stability of nonlinear cascades. SIAM J Control Optimization 2002;40:1888–904. [18] Sontag ED. On the input-to-state stability property. Eur J Control 1995;1:24–36. [19] Sontag ED, Wang Y. New characterizations of the input to state stability property. IEEE Trans Automat Control 1996;41:1283–94. [20] Khalil HK. Nonlinear systems. 3rd ed.[M]. NJ: Prentice-Hall; 2002. [21] Rössler OE. Continuous chaos-four prototype equations. Ann NY Acad Sci 1979;316:376–92. [22] Matsumoto T, Chua LO, Kobayashi K. Hyperchaos: laboratory experiment and numerical confirmation. IEEE Trans Circ Syst 1986;33:1143–7. [23] Chen A, Lu J, Lü J, Yu S. Generating hyperchaotic Lü attractor via state feedback control. Physica A 2006;364:103–10. [24] Chen G, Ueta T. Yet another chaotic attractor. Int J Bifurcat Chaos 1999;9:1465–6. [25] Yan Z. Controlling hyperchaos in the new hyperchaotic Chen system. Appl Math Comput 2005;168:1239–50. [26] Mei SW, Shen TL, Liu KZ. Modern robust control theory and application. Beijing: Qinghua University Press; 2003 [in Chinese]. [27] Cao Y, Sun Y, Cheng C. Delay-dependent robust stabilization of uncertain systems with multiple state delays. IEEE Trans Automat Control 1998;43:1608–12.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Synchronization of hyperchaotic systems via linear control q Hua Wang a,*, Zheng-zhi Han b, Zhen Mo b a b
School of Mechatronics Engineering and Automation, Shanghai University, Shanghai 200072, China School of Electronic, Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e
i n f o
Article history: Received 6 November 2008 Received in revised form 27 July 2009 Accepted 28 July 2009 Available online 6 August 2009
a b s t r a c t In this paper, synchronization of hyperchaotic system is discussed. Based on the stability theory in the cascade system, a simple linear feedback law is presented to realize synchronization of hyperchaotic systems. Simulation results are given to illustrate the effectiveness of the proposed method. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Hyperchaotic systems Synchronization Linear feedback Input-to-state stability (ISS)
1. Introduction Chaotic phenomenon has received increasing attention in the past several decades. Compared with the ordinary chaotic systems, hyperchaotic systems hold more than one positive Lyapunov exponents. Hence, hyperchaotic systems possess more complicated attractors. Since hyperchaotic systems have the characteristics of high capacity, high security and high efficiency, they have been researched more and more in the fields of secure communication, information processing, biological engineering, chemical processing and other fields [1–9]. Many methods have been proposed to realize chaos synchronization in low dimensional attractors with one positive Lyapunov exponent. However, chaotic system with higher dimensional attractor have much wider application. The presence of more than one Lyapunov exponent clearly improves security of the communication scheme by generating more complex dynamics. So, hyperchaotic systems are being given more and more interest [10–13]. Several methods such as nonlinear control method [11], linear control method [12], adaptive control method [13,14], neural networks method [15] have been provided for hyperchaotic synchronization. Studies on the stabilization of cascade systems have attracted many researchers’ attention in nonlinear control field [16,17]. Among theses advances are several constructive design methods such as backstepping and forwarding, which are based on recursive applications of cascade designs. From some aspects, the hyperchaotic system can be seen as cascade system. In this paper, linear feedback law is designed based on the stability theory in cascade system. Taking some error states as virtual control inputs, input-to-state stability (ISS) theory [18,19] is technically implemented to attain global asymptotical stability of the overall error system. Linear controller is designed step by step and the feedback control obtained in this way is simple. It can be easily implemented in the practical process.
q
Supported by the National Natural Science Foundation of China (Grant No. 60674024). * Corresponding author. Tel.: +86 21 34203655; fax: +86 21 62932083. E-mail address: [email protected] (H. Wang).
1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.07.023
H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
1911
2. Preliminary results and lemmas Before giving the main results, let us introduce some necessary lemmas. Definition 1. A continuous function a : ½0; aÞ ! ½0; 1Þ is said to belong to class K if it is strictly increasing and að0Þ ¼ 0. It is said to belong to class K 1 if a ¼ 1 and aðrÞ ! 1 as r ! 1. Definition 2. A continuous function b : ½0; a ½0; 1Þ ! ½0; 1Þ is said to belong to class KL if, for each fixed s, the mapping bðr; sÞ belongs to class K with respect to r and, for each fixed r, the mapping bðr; sÞ is decreasing with respect to s and bðr; sÞ ! 0 as s ! 1. Lemma 1 [20]. Let x ¼ 0 be an equilibrium point for x_ ¼ f ðt; xÞ and D Rn be domain containing x ¼ 0. Let V : ½0; 1Þ D ! R be a continuously differentiable function such that
a1 kxkb 6 Vðt; xÞ 6 a2 kxkb ; ð1Þ
@V @V þ x_ 6 a3 kxkb ; @t @x
8t P 0 and 8x 2 D; where ai ði ¼ 1; 2; 3Þ and b are positive constants. Then, x ¼ 0 is exponentially stable. If the assumptions hold globally, then x ¼ 0 is globally exponentially stable. Lemma 2 [20]. Consider the system
x_ ¼ f ðt; x; uÞ:
ð2Þ n
m
n
Suppose f : ½0; 1Þ R R ! R is continuously differentiable and globally Lipschitz in ðx; uÞ, uniformly in t. If the unforced system x_ ¼ f ðt; x; 0Þ has a globally exponentially stable equilibrium point at the origin x ¼ 0, then the system (2) is input-to-state stable. Lemma 3 [20]. Consider the following system
x_ 1 ¼ f1 ðt; x1 ; x2 Þ;
ð3Þ
x_ 2 ¼ f2 ðt; x2 Þ;
ð4Þ n1
n2
n1
n2
n2
T
where f1 : ½0; 1Þ R R ! R and f2 : ½0; 1Þ R ! R are piecewise continuous in t and Locally Lipschitz in x ¼ ½x1 ; x2 . If the system (3), with x2 as input, is input-to-state stable and the origin of (4) is globally uniformly asymptotically stable, then the origin of the cascade system (3) and (4) is globally uniformly asymptotically stable. Proof. By the definition of asymptotical stability and the definition of ISS, the condition of Lemma 3 implies that there exist KL-class functions b1 and b2 , and a K-class function c1 such that for any t P s P t 0 ,
kx1 ðtÞk 6 b1 ðkx1 ðsÞk; t sÞ þ c1 sup kx2 ðsÞk ;
ð5Þ
kx2 ðtÞk 6 b2 ðkx2 ðsÞk; t sÞ;
ð6Þ
s6s6t
Let s ¼
tþt 0 . 2
It follows t P s ¼
tþt 0 . 2
Because t P s P t 0 , (5) becomes
0 1 t þ t0 t t0 @ þ c1 sup kx2 ðsÞkA: kx1 ðtÞk 6 b1 x1 ; 2 2 tþt 2
Using (5) again, but now t is replaced by
tþt 0 2
06
ð7Þ
s6t
and s ¼ t 0 , we have
0 1 t þ t t t 0 0 6 b1 kx1 ðt 0 Þk; x1 þ c1 @ sup kx2 ðsÞkA: 2 2 tþt t 6 s6 0 0
ð8Þ
2
By (6),
kx2 ðtÞk 6 b2 ðkx2 ðt0 Þk; tÞ 6 b2 ðkx2 ðt 0 Þk; 0Þ:
ð9Þ
Note that the right side is a constant, hence
sup kx2 ðsÞk 6 b2 ðkx2 ðt 0 Þk; 0Þ; t 0 6 s6
tþt 0 2
By (10), (8) leads to
ð10Þ
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H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
x1 t þ t0 6 b1 kx1 ðt 0 Þk; t t 0 þ c ðb2 ðkx2 ðt 0 Þk; 0Þ: 1 2 2
ð11Þ
where c1 ðb2 ðkx2 ðt 0 Þk; 0Þ is a K – class function of kx2 ðt 0 Þk. The right side of (11) is a KL-class function of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 kx1 ðt 0 Þk2 þ kx2 ðt0 Þk2 and tt , i.e., there is a KL-class function b3 such that 2
kxðt0 Þk ¼
x1 t þ t0 6 b3 kxðt0 Þk; t t 0 : 2 2
ð12Þ
Substituting it into the first term of the right side of (7) leads to
t þ t0 t t0 t t0 t t0 ; b1 x b kxðt Þk; 6 b ; : 0 1 3 1 2 2 2 2 The right side is still a KL-class function of kxðt 0 Þk and
t þ t0 ; t t0 6 b4 ðkxðt0 Þk; t t0 Þ: b1 x 1 2 2
tt 0 , 2
ð13Þ
and we denote the KL-class function as b4 , i.e.,
ð14Þ
0 On the other hand, if s ¼ tþt , from (6) we have 2
t þ t0 ; t t 0 6 b2 kx2 ðt0 Þk; t t0 : kx2 ðtÞk 6 b2 x 2 2 2 2
ð15Þ
The last inequality is valid after t P T for some T, since x2 ðtÞ is asymptotically stable. Then
sup kx2 ðsÞk 6 b2 kx2 ðt 0 Þk;
tþt 0 2
tþt0 6 2
s6t
t0 2
! ¼ b2 ðkx2 ðt0 Þk;
t t0 : 4
ð16Þ
Furthermore,
0
1
c1 @ sup kx2 ðsÞkA 6 c1 b2 kx2 ðt0 Þk; tþt0 6 2
s6t
t t0 4
ð17Þ
:
Its right side is a KL-class function of kx2 ðt0 Þk and t t 0 . We denote the function by b5 , i.e.,
0
1
c1 @ sup kx2 ðsÞkA 6 b5 ðkx2 ðt0 Þk; t t0 Þ: tþt0 6 2
ð18Þ
s6t
By (14) and (18), (7) results in
kx1 ðtÞk 6 b4 ðkxðt 0 Þk; t t0 Þ þ b5 ðkx2 ðt 0 Þk; t t 0 Þ:
ð19Þ
At last, the lemma is verified by
kxðtÞk 6 kx1 ðtÞk þ kx2 ðtÞk 6 b4 ðkxðt 0 Þk; t t 0 Þ þ b5 ðkx2 ðt0 Þk; t t0 Þ þ b2 ðkx2 ðt0 Þk; t t0 Þ 6 b4 ðkxðt 0 Þk; t t 0 Þ þ b5 ðkxðt 0 Þk; t t 0 Þ þ b2 ðkxðt0 Þk; t t0 Þ
ð20Þ
Theorem 1. For the cascade system (3) and (4), if 1. f1 is continuously differentiable and globally Lipschitz in ðx1 ; x2 Þ, uniformly in t and f2 is piecewise continuous in t and Locally Lipschitz in x2 . 2. System x_ 1 ¼ f1 ðt; x1 ; 0Þ is globally exponentially stable and the origin of (4) is globally uniformly asymptotically stable. Then the origin of the cascade system (3) and (4) is globally uniformly asymptotically stable. Proof. This result is quite straightforward from Lemmas 2 and 3.
h
3. Main results Since the discovery of the hyperchaotic Rössler system [21], many hyperchaotic systems have been presented such as the hyperchaotic MCK circuit [22], the hyperchaotic Lü [23] and the hyperchaotic Chen [24] system, etc. In this section, we mainly research synchronization of the hyperchaotic Lü and the hyperchaotic Chen system via linear control. The method given below can also be extended to other hyperchaotic systems.
H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
1913
3.1. Synchronization of hyperchaotic Lü system In 2006, Lü et al. [23] presented a new hyperchaotic system, called the hyperchaotic Lü system, which is described by
8 x_ 1 > > > < x_ 2 > _3 x > > : x_ 4
¼ a1 ðx2 x1 Þ þ x4 ; ¼ x1 x3 þ c1 x2 ;
ð21Þ
¼ x 1 x2 b 1 x3 ; ¼ x 1 x3 þ r 1 x4 ;
where xi ði ¼ 1; 2; 3; 4Þ are state variables. a1 ; b1 ; c1 and r 1 are real constants. Ref. [23] shows, when a1 ¼ 36; b1 ¼ 3; c1 ¼ 20; 1:03 6 r1 6 0:46, the system (21) has periodic orbit. If a1 ¼ 36; b1 ¼ 3; c1 ¼ 20; 0:46 < r 1 6 0:35, the system (21) has chaotic attractor. If a1 ¼ 36; b1 ¼ 3; c1 ¼ 20; 0:35 < r1 6 1:3, the system (21) has hyperchaotic attractor. Now, we consider how to realize synchronization of the hyperchaotic Lü system. The master (drive) system is system (21) and the slave(response) system is
8 y_ 1 > > > < y_ 2 > _3 y > > : y_ 4
¼ a1 ðy2 y1 Þ þ y4 þ u1 ; ¼ y1 y3 þ c1 y2 þ u2 ;
ð22Þ
¼ y1 y2 b1 y3 ¼ y1 y3 þ r 1 y4 þ u3 :
where u1 ; u2 and u3 are the control inputs to be designed. Denote the state errors as
e1 ¼ y1 x1 ;
e 2 ¼ y 2 x2 ;
e 3 ¼ y 3 x3 ;
e4 ¼ y4 x4 :
Subtracting Eq. (21) from Eq. (22), we can get the error system as follows
8 e_ 1 > > > < e_ 2 > _3 e > > : e_ 4
¼ a1 ðe2 e1 Þ þ e4 þ u1 ; ¼ e1 e3 e1 x3 x1 e3 þ c1 e2 þ u2 ; ¼ b1 e3 þ e1 e2 þ e1 x2 þ x1 e2 ;
ð23Þ
¼ e1 e3 þ e1 x3 þ e3 x1 þ r1 e4 þ u3 :
Now the synchronization problem of (21) and (22) is transformed into the stabilization problem of the error system (23), i.e., if system (23) is asymptotically stabilized by uj ðj ¼ 1; 2; 3Þ, synchronization of hyperchaotic Lü is realized. We present a linear feedback to achieve the stabilization. The design process of the linear controller is mainly divided into 3 steps. Step 1 Let u1 ¼ a1 e2 e4 , then the first equation of the error system (23) becomes
e_ 1 ¼ a1 e1 :
ð24Þ
Obviously, system (24) is globally uniformly asymptotically stable. Step 2 Consider the second and third equation of error system (23) and let u2 ¼ k1 e2 , where k1 > c1 is a constant. Then
e_ 2 ¼ e1 e3 e1 x3 x1 e3 þ c1 e2 k1 e2 ; e_ 3 ¼ b1 e3 þ e1 e2 þ e1 x2 þ x1 e2 :
ð25Þ
If we regard e1 as the virtual input of the system (25), then the systems (24) and (25) satisfy the first condition of Theorem 1. (The detailed proof is given in the appendix of this paper.) Now we verify that they satisfy the second condition of Theorem 1. Consider the following unforced system of system (25)
e_ 2 ¼ x1 e3 þ c1 e2 k1 e2 ; e_ 3 ¼ b1 e3 þ x1 e2 :
ð26Þ
Choose the following function as the candidate Lyapunov function for system (26)
V1 ¼
1 2 1 2 e þ e : 2 2 2 3
ð27Þ
The time derivative of V 1 along trajectories of system (26) is
V_1 ¼ e2 ðx1 e3 þ c1 e2 k1 e2 Þ þ e3 ðb1 e3 þ x1 e2 Þ ¼ ðk1 c1 Þe22 b1 e23 6 k1 ðe22 þ e23 Þ;
ð28Þ
where k1 ¼ minðk1 c1 ; b1 Þ. From Lemma 1, the system (26) is globally exponentially stable at e2 ¼ 0; e3 ¼ 0. This means system (25) satisfies the second condition in Theorem 1. So system (25) is globally uniformly asymptotically stable, i.e. limt!1 e2 ðtÞ ¼ 0; limt!1 e3 ðtÞ ¼ 0. Step 3 Finally, let us consider the last equation in error system (23). Let u3 ¼ k2 e4 , where k2 > r 1 is a constant. Then the last equation in error system (23) with such a u3 becomes
e_ 4 ¼ e1 e3 þ e1 x3 þ e3 x1 þ r1 e4 k2 e4 :
ð29Þ
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H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
Take e1 ; e2 and e3 as the virtual inputs to this system, and we can see this system satisfy the first condition of Theorem 1. (The detailed proof is given in the appendix.) Then the unforced system becomes
e_ 4 ¼ ðk2 r 1 Þe4 :
ð30Þ
As k2 > r 1 , system (30) is globally exponentially stable at e4 ¼ 0. With the consideration that e1 ; e2 , and e3 are globally uniformly asymptotically stable, then by Theorem 1, system (29) is globally uniformly asymptotically stable, i.e. limt!1 e4 ¼ 0. From the above analysis, the error system (23) is globally uniformly asymptotically stable with the linear controller
u2 ¼ k1 e2 ðk1 > c1 Þ;
u3 ¼ k2 e4 ðk2 > r 1 Þ:
10
e1 e2 e3 e4
8 6 4 2 0 −2 −4 −6
0
2
4
6
time(s)
8
10
Fig. 1. Synchronization errors of hyperchaotic Lü system.
30
x1 y1
25 20
x1,y1
15 10 5 0 −5 −10 −15 −20
0
5
10
15
20
time(s)
25
30
35
Fig. 2. Synchronization of hyperchaotic Lü system x1 and y1 .
40
x2 y2
30 20
x2,y2
u1 ¼ a1 e2 e4 ;
10 0
−10 −20 −30 0
5
10
15
20
time(s)
25
30
Fig. 3. Synchronization of hyperchaotic Lü system x2 and y2 .
35
1915
H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
So the slave system (22) can asymptotically synchronize the master hyperchaotic Lü system (21). It can be seen that the controller designed here is more simpler than the nonlinear controller in Ref. [11].
3.2. Synchronization of hyperchaotic chen system This subsection turns to the consideration of the hyperchaotic Chen system [24]. The hyperchaotic Chen system [25] is described by
50
x3 y3
40
x3,y3
30 20 10 0 −10
0
5
10
15
20
time(s)
25
30
35
Fig. 4. Synchronization of hyperchaotic Lü system x3 and y3 .
100
x4 y4
50
x4,y4
0 −50 −100 −150 −200
0
5
10
15
20
time(s)
25
30
35
Fig. 5. Synchronization of hyperchaotic Lü system x4 and y4 .
10
hat(e)1 hat(e)2 hat(e)3 hat(e)4
5 0 −5 −10 −15 −20
0
5
10
time(s)
15
20
Fig. 6. Synchronization errors of hyperchaotic Chen system.
25
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H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
8 ^x_ 1 > > > > < ^_ x2 > > ^x_ 3 > > :_ ^x4
¼ að^x2 ^x1 Þ þ ^x4 ; ¼ d^x1 ^x1 ^x3 þ c^x2 ;
ð31Þ
¼ ^x1 ^x2 b^x3 ; ¼ ^x2 ^x3 þ r^x4 ;
where ^ xi ði ¼ 1; . . . ; 4Þ are system state variables. a; b; c; d, and r are real constants. When a ¼ 35; b ¼ 3; c ¼ 12; d ¼ 7; 0 6 r 6 0:085, system (31) is chaotic; when a ¼ 35; b ¼ 3; c ¼ 12; d ¼ 7; 0:085 < r 6 0:798, system (31) is hyperchaotic; when a ¼ 35; b ¼ 3; c ¼ 12; d ¼ 7; 0:798 < r 6 0:9, system (31) is periodic.
30
hat(x)1 hat(y)1
25 20
x1,y1
15 10 5 0 −5 −10 −15 −20
0
5
10
15
20
time(s)
25
30
35
^1 . Fig. 7. Synchronization of hyperchaotic Chen system ^ x1 and y
30
hat(x)2 hat(y)2
25 20 15
x2,y2
10 5 0 −5 −10 −15 −20 0
5
10
15
20
time(s)
25
30
35
^2 . Fig. 8. Synchronization of hyperchaotic Chen system ^ x2 and y
45
hat(x)3 hat(y)3
40 35
x3,y3
30 25 20 15 10 5 0 −5
0
5
10
15
20
time(s)
25
30
^3 . Fig. 9. Synchronization of hyperchaotic Chen system ^ x3 and y
35
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H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
Let us consider system (31) as the master system, the slave system is
8 ^_ 1 y > > > >
^2 y ^1 Þ þ y ^4 þ v 1 ; ¼ aðy ^3 þ c y ^1 y ^1 y ^2 þ v 2 ; ¼ dy
ð32Þ
^1 y ^3 ; ^2 by ¼y ^3 þ r y ^2 y ^4 þ v 3 ; ¼y
v 1 ; v 2 and v 3 are the controllers to be determined to realize hyperchaotic synchronization. Denote ^1 ^x1 ; ^e1 ¼ y
^e2 ¼ y ^2 ^x2 ;
^e3 ¼ y ^3 ^x3 ;
^e4 ¼ y ^4 ^x4 :
From (31) and (32) we can get the following error system
8 ^e_ > > > 1 > < ^_ e2 > ^e_ 3 > > > :_ ^e4
¼ að^e2 ^e1 Þ þ ^e4 þ v 1 ; ¼ d^e1 ^e1 ^e3 ^e1 ^x3 ^x1 ^e3 þ c^e2 þ v 2 ;
ð33Þ
¼ b^e3 þ ^e1 ^e2 þ ^e1 ^x2 þ ^x1 ^e2 ; ¼ ^e2 ^e3 þ ^e2 ^x3 þ ^e3 ^x2 þ r^e4 þ v 3 :
Design the linear controller as
v 1 ¼ a^e2 ^e4 ; v 2 ¼ L1 ^e2 ðL1 > cÞ; v 3 ¼ L2 ^e4 ðL2 > rÞ: Then the error system (33) with such linear controller becomes
8 ^e_ 1 ¼ a^e1 ; > > > > < ^e_ ¼ d^e ^e ^e ^e ^x ^x ^e þ ðc L Þ^e ; 2 1 1 3 1 3 1 3 1 2 > > ^e_ 3 ¼ b^e3 þ ^e1 ^e2 þ ^e1 ^x2 þ ^x1 ^e2 ; > > :_ ^e4 ¼ ^e2 ^e3 þ ^e2 ^x3 þ ^e3 ^x2 þ ðr L2 Þ^e4 :
ð34Þ
We want to know whether system (34) is asymptotically stable. Now we analyze this stability problem of the error system step by step. Similarly as we have done for the hyperchaotic Lü system, this process is mainly divided into three steps. Step 1 The first equation of the system (34) is
^e_ 1 ¼ a^e1 : As a ¼ 35 > 0, it can be proved easily that this system is globally asymptotically stable. The detailed proof process is omitted here. Step 2 Consider the second and third equation in system (34)
(
^e_ 2 ¼ d^e1 ^e1 ^e3 ^e1 ^x3 ^x1 ^e3 þ ðc L1 Þ^e2 ; ^e_ 3 ¼ b^e3 þ ^e1 ^e2 þ ^e1 ^x2 þ ^x1 ^e2 :
ð35Þ
Take ^e1 as the virtual input of system (35). Then the next problem is whether the system is exponentially stable without the virtual input ^e1 . The unforced system is
100
hat(x)4 hat(y)4
x4,y4
50 0 −50 −100 −150
0
5
10
15
time(s)
20
25
30
^4 . Fig. 10. Synchronization of hyperchaotic Chen system ^ x4 and y
35
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H. Wang et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1910–1920
(
^e_ 2 ¼ ^x1 ^e3 þ ðc L1 Þ^e2 ; ^e_ 3 ¼ b^e3 þ ^x1 ^e2 :
ð36Þ
Chose the candidate Lyapunov function for system (36) as
V2 ¼
1 2 1 2 ^e þ ^e : 2 2 2 3
ð37Þ
Calculate the derivation of V 2 along the trajectories of system (36) and we can get
V_2 ¼ ^e2 ð^x1 ^e3 þ ðc L1 Þ^e2 Þ þ ^e3 ðb^e3 þ ^x1 ^e2 Þ ¼ ðL1 cÞ^e22 b^e23 6 k2 ^e22 þ ^e23
ð38Þ
where k2 ¼ minfL1 c; bg. From Lemma 1, system (36) is globally exponentially stable. Additional with Theorem 1, system (35) is globally asymptotically stable if we consider ^ e1 as virtual input. Step 3 Finally, let us analyze the stability problem of the last equation in system (34)
^e_ 4 ¼ ^e2 ^e3 þ ^e2 ^x3 þ ^e3 ^x2 þ ðr L2 Þ^e4 :
ð39Þ
Take ^ e2 ; ^ e3 as the virtual input, then the unforced system is
^e_ 4 ¼ ðL2 rÞ^e4 :
ð40Þ
Since L2 > r, this system is globally exponentially stable at e4 ¼ 0. As we can see from the above analysis process, e2 and e3 are globally asymptotically stable. From Theorem 1, the system (39) is globally asymptotically stable. So the error system (33) is globally asymptotically stable with the linear controller
v 1 ¼ a^e2 ^e4 ; v 2 ¼ L1 ^e2 ðL1 > cÞ; v 3 ¼ L2 ^e4 ðL2 > rÞ: Notation. In the above analysis, whether system (35) and (39) satisfy the global Lipschitz condition is not discussed. This is because we can analyze it in the same way as we do for the hyperchaotic Lü system in the appendix. So the discussion of this question is omitted here. 4. Simulation results In this section, simulation results for the hyperchaotic Lü and Chen systems are given, respectively. Fourth order Runge–Kutta integration method is used to solve the differential equations with time step length 0.0001. The parameters of hyperchaotic Lü system are selected as a1 ¼ 36; b1 ¼ 3; c1 ¼ 20; r1 ¼ 1:3. The Lü system exhibits hyperchaotic behavior. Controller gains are chosen as k1 ¼ c1 þ 1; k2 ¼ r 1 þ 1. The initial values of the master and slave systems are x1 ð0Þ ¼ 5; x2 ð0Þ ¼ 8; x3 ð0Þ ¼ 1; x4 ð0Þ ¼ 3 and y1 ð0Þ ¼ 3; y2 ð0Þ ¼ 4; y3 ð0Þ ¼ 5; y4 ð0Þ ¼ 5, respectively. Fig. 1 displays the time response of the controlled synchronization error system (23). Figs. 2–5 show the state response between the master system (21) and the slave system (22). From the figures we can see that the asymptotical synchronization of the hyperchotic Lü system is realized via the proposed linear feedback control. Chose the parameters of hyperchaotic Chen system as a ¼ 35; b ¼ 3; c ¼ 12; d ¼ 7; r ¼ 0:5 and the controller gain as L1 ¼ c þ 1; L2 ¼ r þ 1. The initial states of the master and slave systems are x1 ð0Þ ¼ 3; x2 ð0Þ ¼ 5; x3 ð0Þ ¼ 5; x4 ð0Þ ¼ 3 and y1 ð0Þ ¼ 1; y2 ð0Þ ¼ 2; y3 ð0Þ ¼ 2; y4 ð0Þ ¼ 3, respectively. Fig. 6 shows the time response of the synchronization error system (33). Figs. 7–10 provide the state trajectories of the master and slave system. From Figs. 6–10 we can see the slave system (32) asymptotically synchronize the master system (31) with the proposed simple linear controller. 5. Conclusion In this paper, linear controllers are proposed for hyperchaotic synchronization of the hyperchaotic Lü and Chen systems, respectively. Since the states of the hyperchaotic system are bounded, we can use the input-to-state theory in the cascade system to design the controller step by step. The linear feedback control presented is simple and easy to be implemented in real engineering process. Numerical simulations results illustrate the effectiveness of the proposed synchronization method. Appendix A Now we prove that the system (22) satisfies the first condition of Theorem 1, i.e. it meets the global Lipshitz condition. Let us introduce a lemma firstly. Lemma 4 [26]. Consider the nonlinear time-varying system x_ ¼ f ðx; tÞ, where x 2 U Rn ; t 2 J. If there exists a differentiable function VðÞ : U J ! R satisfying the following conditions: kðÞ : U ! R such that (1) There exist a positive constant k0 and a scalar function
k0 kxk2 6 Vðx; tÞ 6 kðxÞkxk2 ;
8ðx; tÞ 2 U J:
ð41Þ
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(2) There exist positive constants k1 > 0 and 2 _ Vðx; tÞjx¼f _ ðx;tÞ 6 k1 kðxÞkxk þ e;
e P 0 such that
8ðx; tÞ 2 U J:
ð42Þ
then the solution of the system x_ ¼ f ðx; tÞ is bounded by
kxk2 6
1 e Vðx0 ; t 0 ÞekV ðtt0 Þ þ ð1 ek1 ðtt0 Þ Þ 6 M 2 ; k0 k0 k1
ð43Þ
where M 2 ¼ k10 Vðx0 ; t 0 Þ þ k0ek1 . Then we can get that kxk 6 M.
Proof. Consider the following differential equation:
y_ ¼ k1 y þ ;
y0 ¼ Vðx0 ; t 0 Þ:
ð44Þ
then
dy ¼ k1 y þ ¼ k1 y : dt k1
ð45Þ
Then, we can get the following differential equation:
dðy k1 Þ y k1
¼ k1 dt:
ð46Þ
Integrate this differential equation with the initial condition y0 ¼ Vðx0 ; t0 Þ and we can get
y ¼ y0 ek1 ðtt0 Þ þ
e k1
1 ek1 ðtt0 Þ :
ð47Þ
From the comparison Lemma [20] we know
Vðx; tÞ 6 Vðx0 ; t 0 Þek1 ðtt0 Þ þ
e k1
1 ek1 ðtt0 Þ :
ð48Þ
Then
kxk2 6
1 e Vðx0 ; t 0 Þek1 ðtt0 Þ þ 1 ek1 ðtt0 Þ 6 M 2 ; k0 k0 k1
ð49Þ
where M2 ¼ k10 Vðx0 ; t 0 Þ þ k0ek1 . h Lemma 5 [27]. Let x; y be two n-dimensional vectors and q be a positive real number. Then the following inequality holds
2xT y 6 qxT x þ q1 yT y: Now we turn to prove that the system (22) satisfies the global Lipshitz condition. Derivations of computations concerning Step 2 of Section 3.1. Consider the system (22) as follows:
e_ 2 ¼ e1 e3 e1 x3 x1 e3 þ c1 e2 k1 e2 ; e_ 3 ¼ b1 e3 þ e1 e2 þ e1 x2 þ x1 e2 :
ð50Þ
Since system (21) is a hyperchaotic system, its state trajectories are bounded. There exist positive constants ni such that jxi j < ni ði ¼ 1; 2; 3; 4Þ. Consider the following positive function
V 3 ðe2 ; e3 Þ ¼ e22 þ e23 :
ð51Þ
Its derivation along the trajectories of system (50) is
V_3 ðe2 ; e3 Þ ¼ 2e2 ðe1 e3 e1 x3 x1 e3 þ c1 e2 k1 e2 Þ þ 2e3 ðb1 e3 þ e1 e2 þ e1 x2 þ x1 e2 Þ ¼ 2ðk1 c1 Þe22 2b1 e23 2e1 e2 x3 þ 2e1 e3 x2 6 ð2k1 2c1 e1 Þe22 ð2b1 e2 Þe23 þ 6 ð2k1 2c1 e1 Þe22 ð2b1 e2 Þe23 þ M;
1
e1
e21 x23 þ
1
e2
e21 x22 ð52Þ
where e1 and e2 are two positive constants and selected such that k1 c1 e1 > 0; b1 e2 > 0. The positive constant M is M ¼ e11 e21 ð0Þn23 þ e12 e21 ð0Þn22 . From Lemma 4, states of system (50) are bounded. So there exist positive constants rj ðj ¼ 2; 3Þ such that jej j < rj ðj ¼ 2; 3Þ.
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Denote f ¼ ðe2 ; e3 ÞT and system (50) can be rewritten as
f_ ¼ gðf; tÞ ¼
g 1 ðf; tÞ g 2 ðf; tÞ
¼
e1 e3 e1 x3 x1 e3 þ c1 e2 k1 e2 b1 e3 þ e1 e2 þ e1 x2 þ x1 e2
:
ð53Þ
Then
jg 1 ðf f0 Þ; tj ¼ j e1 e3 e1 x3 x1 e3 þ ðc1 k1 Þe2 þ e01 e03 þ e01 x3 þ x1 e03 ðc1 k1 Þe02 j 6 je03 þ x3 ke1 e01 j þ jc1 k1 ke2 e02 j þ je1 þ x1 ke3 e03 j 6 ðr3 þ n3 Þje1 e01 j þ jc1 k1 ke2 e02 j þ ðje1 ð0Þj þ n1 Þje3 e03 j:
ð54Þ
For g 2 ðf; tÞ, we have
jg 2 ðf f0 Þ; tj ¼ j b1 e3 þ e1 e2 þ e1 x2 þ x1 e2 ðb1 e03 þ e01 e02 þ e01 x2 þ x1 e02 Þj 6 je02 þ x2 ke1 e01 j þ jx1 þ e1 ke2 e02 j þ b1 je3 e03 j 6 ðr2 þ n2 Þje1 e01 j þ ðn1 þ je1 ð0ÞjÞje2 e02 j þ b1 je3 e03 j:
ð55Þ
Then
g ðf f0 Þ; t 2 6 ðjg 1 ðf f0 Þ; tj þ jg 2 ðf f0 Þ; tjÞ2 6 ðg je1 e0 j þ g je2 e0 j þ g je3 e0 jÞ2 ; kgðf f0 Þ; tk2 ¼ 1 1 2 3 1 2 3 0 g 2 ðf f Þ; t
ð56Þ
g1 ¼ r2 þ r3 þ n2 þ n3 ; g2 ¼ jc1 k1 j þ n1 þ je1 ð0Þj; g3 ¼ je1 ð0Þj þ n1 þ b1 :
ð57Þ
where
Inequality (56) implies that gðf; tÞ is global Lipschitz in ðe2 ; e3 ÞT . Derivations of computations concerning Step 3 of Section 3.1. Consider the following system
e_ 4 ¼ e1 e3 þ e1 x3 þ e3 x1 þ r 1 e4 k2 e4 :
ð58Þ
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