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Applied Mathematics and Computation xxx (2008) xxx–xxx
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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
Adaptive control and synchronization of a novel hyperchaotic system
with uncertain parameters
Xiaobing Zhou a,*, Yue Wu a, Yi. Li a, Hongquan Xue b
a
b
School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China
School of Business Management, Xi’an University of Technology, Xi’an, Shanxi 710048, PR China
a r t i c l e
i n f o
Article history:
Available online xxxx
Keywords:
Hyperchaotic system
Adaptive control
Adaptive synchronization
Lyapunov stability theory
a b s t r a c t
In this paper, the problems of control and synchronization of a novel hyperchaotic system
with uncertain parameters are studied. Based on the Lyapunov stability theory and the
adaptive control theory, this uncertain hyperchaotic system is suppressed to its unstable
equilibrium. Furthermore, synchronization between two identical uncertain hyperchaotic
systems is achieved by proposing an adaptive control law and a parameter estimation
update law. Numerical simulations are presented to demonstrate the analytical results.
Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction
Since Rössler [1] proposed a four-dimensional ordinary differential equation to exhibit hyperchaos in 1979, a large
amount of researches have been done on hyperchaos [2–17], and many more hyperchaotic systems have been proposed,
such as hyperchaotic Lorenz system [8], hyperchaotic Chen system [9], hyperchaotic Lü system [10], just to name a few.
Hyperchaotic systems possess more complex dynamical behaviors than chaotic systems such as having more than one positive Lyapunov exponent, therefore, they have broader potential applications, particularly in secure communications.
Over the years, a wide variety of methods for control and synchronization of hyperchaotic systems have been developed,
such as linear and nonlinear feedback [5,11], adaptive control [12,13], backstepping design [14,15], and sliding mode control
[16]. Some of the aforementioned methods and many other existing methods are based on the exactly knowing of the system’s structure and parameters. But in practical situations, some or all of the system’s parameters cannot be exactly known
in priori. Therefore, it is necessary to consider control and synchronization of hyperchaotic systems in the presence of uncertain parameters.
Very recently, Chen et al. [17] proposed a novel hyperchaotic system by introducing state feedback control and constant
multipliers to the two quadratic terms in the system reported in [18]. This novel hyperchaotic system takes the following
form:
8
x_ ¼ aðy xÞ þ eyz;
>
>
>
< y_ ¼ cx dxz þ y þ u;
ð1Þ
>
z_ ¼ xy bz;
>
>
:
u_ ¼ ky;
where x; y; z and u are state variables, and a; b; c; d; e and k are constant parameters.
* Corresponding author.
E-mail address: zhouxb@uestc.edu.cn (X. Zhou).
0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2008.04.004
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System (1) has only one unstable equilibrium Oð0; 0; 0; 0Þ and has bigger positive Lyapunov exponents than those already
known hyperchaotic systems. It can generate complex dynamics within wide parameter ranges, including periodic orbit,
quasi-periodic orbit, chaos and hyperchaos [17]. In particular, when a ¼ 35; b ¼ 4:9; c ¼ 25; d ¼ 5; e ¼ 35 and varying k
from 10 to 126, or a ¼ 35; c ¼ 25; d ¼ 5; e ¼ 35; k ¼ 100 and ranging b between 3.8 and 11, system (1) exhibits
hyperchaos.
In the remainder of this paper, hyperchaos in the novel hyperchaotic system (1) with uncertain parameters is controlled
to its unstable equilibrium and synchronization between two identical uncertain novel hyperchaotic systems is achieved by
applying the Lyapunov stability theory and the adaptive control theory. Numeric simulations are given for the purpose of
illustration and verification.
2. Adaptive control of the novel hyperchaotic system
In this section, the novel hyperchaotic system (1) with uncertain parameters will be suppressed to its unstable equilibrium Oð0; 0; 0; 0Þ by applying the Lyapunov stability theory and the adaptive control theory.
Consider the following controlled system:
8
x_ ¼ aðy xÞ þ eyz þ v1 ðtÞ;
>
>
>
<
y_ ¼ cx dxz þ y þ u þ v2 ðtÞ;
> z_ ¼ xy bz þ v3 ðtÞ;
>
>
:
u_ ¼ ky þ v4 ðtÞ;
ð2Þ
where a; b; c; d; e and k are uncertain parameters, and vi ðtÞ ði ¼ 1; 2; 3; 4Þ are controllers to be determined.
Construct the following positive definite Lyapunov function
~2 þ ~c2 þ d
~2 þ ~
~2 Þ;
~ ~c; d;
~ ~
~ ¼ 1 ðx2 þ y2 þ z2 þ u2 þ a
~2 þ b
~; b;
e2 þ k
Vðx; y; z; u; a
e; kÞ
2
ð3Þ
~ ¼ b b1 ; ~c ¼ c c1 ; d
~ ¼ d d1 ; ~
~ ¼ k k1 . a1 ; b1 ; c1 ; d1 ; e1 ; k1 are estimate values of
~ ¼ a a1 ; b
where a
e ¼ e e1 and k
a; b; c; d; e; k, respectively.
Calculating the time derivative of the Lyapunov function (3) along the trajectory of system (2) yields
~_
~_ þ ~
~k
~_ þ ~c~c_ þ d
~d
~b
~ ~c; d;
~ ~
~ ¼ xx_ þ yy_ þ zz_ þ uu_ þ a
_
~_ þ b
~; b;
~a
e~e_ þ k
Vðx;
y; z; u; a
e; kÞ
¼ xðaðy xÞ þ eyz þ v1 ðtÞÞ þ yðcx dxz þ y þ u þ v2 ðtÞÞ
þ zðxy bz þ v3 ðtÞÞ þ uðky þ v4 ðtÞÞ
~b_ 1 ~cc_ 1 d
~d_ 1 ~ee_ 1 k
~k_ 1 :
~a_ 1 b
a
ð4Þ
Choose the following adaptive control law:
8
v1 ðtÞ ¼ a1 ðy xÞ e1 yz x;
>
>
>
<
v2 ðtÞ ¼ d1 xz c1 x 2y u;
>
v3 ðtÞ ¼ ðb1 1Þz xy;
>
>
:
v4 ðtÞ ¼ k1 y u;
and a parameter estimation update law as follows:
8
a_ 1 ¼ xðy xÞ;
>
>
>
>
>
>
b_ 1 ¼ z2 ;
>
>
>
< c_ ¼ xy;
1
>
> d_ 1 ¼ xyz;
>
>
>
>
e_ 1 ¼ xyz;
>
>
>
:_
k1 ¼ uy:
ð5Þ
ð6Þ
With the above choices, the time derivative of the Lyapunov function becomes
~ ~c; d;
~ ~
~ ¼ x2 y2 z2 u2 < 0:
_
~; b;
Vðx;
y; z; u; a
e; kÞ
ð7Þ
Since the Lyapunov function (3) is positive definite and its derivative is negative definite in the neighborhood of the zero
solution for system (2). According to the Lyapunov stability theory, the equilibrium solution Oð0; 0; 0; 0Þ of the controlled system (2) is asymptotically stable, namely, the controlled system (2) can asymptotically converge to the equilibrium
Oð0; 0; 0; 0Þ with the adaptive control law (5) and the parameter estimation update law (6).
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3. Adaptive synchronization of the novel hyperchaotic system
In this section, synchronization between two identical novel hyperchaotic systems with uncertain parameters is achieved
based on the Lyapunov stability theory and the adaptive control theory.
Suppose the drive and response systems are given as the following forms:
8
x_ 1 ¼ aðy1 x1 Þ þ ey1 z1 ;
>
>
>
<
y_ 1 ¼ cx1 dx1 z1 þ y1 þ u1 ;
ð8Þ
>
_
>
> z1 ¼ x1 y1 bz1 ;
:
u_ 1 ¼ ky1 ;
and
8
x_ 2 ¼ aðy2 x2 Þ þ ey2 z2 þ v1 ðtÞ;
>
>
>
< y_ ¼ cx dx z þ y þ u þ v ðtÞ;
2
2
2 2
2
2
2
>
_ 2 ¼ x2 y2 bz2 þ v3 ðtÞ;
z
>
>
:
u_ 2 ¼ ky2 þ v4 ðtÞ;
ð9Þ
where a; b; c; d; e and k are uncertain parameters, and vi ðtÞ ði ¼ 1; 2; 3; 4Þ are controllers to be determined for achieving
synchronization between the drive system (8) and the response system (9).
Subtracting the drive system (8) from the response system (9) yields the following error dynamical system:
8
e_ 1 ¼ aðe2 e1 Þ þ eðy2 z2 y1 z1 Þ þ v1 ðtÞ;
>
>
>
< e_ ¼ ce dðx z x z Þ þ e þ e þ v ðtÞ;
2
1
2 2
1 1
2
4
2
ð10Þ
>
e_ 3 ¼ be3 þ x2 y2 x1 y1 þ v3 ðtÞ;
>
>
:
e_ 4 ¼ ke2 þ v4 ðtÞ;
where e1 ¼ x2 x1 ; e2 ¼ y2 y1 ; e3 ¼ z2 z1 and e4 ¼ u2 u1 .
Construct a positive definite Lyapunov function as follows:
~ ~c; d;
~ ~e; kÞ
~ ¼ 1 ðe2 þ e2 þ e2 þ e2 þ a
~2 þ ~c2 þ d
~2 þ ~e2 þ k
~2 Þ;
~; b;
~2 þ b
Vðe1 ; e2 ; e3 ; e4 ; a
ð11Þ
2
3
4
2 1
~ ¼ b b;
~c ¼ c c; d
~ ¼ d d;
~
~ ¼ k k:
a
c; d;
e; k
are estimate values of
~ ¼aa
; b
; b;
where a
e ¼ e e and k
a; b; c; d; e; k, respectively.
Calculating the time derivative of the Lyapunov function (11) along the trajectory of system (10) arrives at
~_
~_ þ ~e~e_ þ k
~k
~_ þ ~c~c_ þ d
~d
~b
~ ~c; d;
~ ~e; kÞ
~ ¼ e1 e_ 1 þ e2 e_ 2 þ e3 e_ 3 þ e4 e_ 4 þ a
_ 1 ; e2 ; e3 ; e4 ; a
~_ þ b
~; b;
~a
Vðe
¼ e1 ðaðe2 e1 Þ þ eðy2 z2 y1 z1 Þ þ v1 ðtÞÞ þ e2 ðce1 dðx2 z2 x1 z1 Þ þ e2 þ e4 þ v2 ðtÞÞ
~b
_ ~cc_ d
~d
_ ~ee_ k
~k:
_
~a
_ b
þ e3 ðbe3 þ x2 y2 x1 y1 þ v3 ðtÞÞ þ e4 ðke2 þ v4 ðtÞÞ a
Choose the following adaptive control law:
8
ðe2 e1 Þ eðy2 z2 y1 z1 Þ;
v1 ðtÞ ¼ e1 a
>
>
>
<
2 z2 x1 z1 Þ 2e2 e4 ;
v2 ðtÞ ¼ ce1 þ dðx
1Þe3 ;
> v3 ðtÞ ¼ x2 y2 þ x1 y1 þ ðb
>
>
:
v4 ðtÞ ¼ ke2 e4 ;
and a parameter estimation update law as follows:
8
_ ¼ e1 ðe2 e1 Þ;
a
>
>
>
>
>
_ ¼ e2 ;
>
b
>
3
>
>
>
< c_ ¼ e e ;
1 2
>
_ ¼ e2 ðx2 z2 x1 z1 Þ;
>
d
>
>
>_
>
>
e ¼ e1 ðy2 z2 y1 z1 Þ;
>
>
>
:
_ ¼ e2 e4 :
k
ð12Þ
ð13Þ
ð14Þ
With the above choices, the time derivative of the Lyapunov function becomes
~ ~c; d;
~ ~e; kÞ
~ ¼ e2 e2 e2 e2 < 0:
_ 1 ; e2 ; e3 ; e4 ; a
~; b;
Vðe
1
2
3
4
ð15Þ
According to the Lyapunov stability theory, the error dynamical system (10) can converge to the origin asymptotically, i.e.
limt!1 keðtÞk ¼ 0, where eðtÞ ¼ ½e1 ; e2 ; e3 ; e4 T . Therefore, the synchronization between two identical novel hyperchaotic systems is achieved with the adaptive control law (13) and the parameter estimation update law (14).
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15
x
y
z
u
10
5
0
-5
-10
-15
-20
-25
0
1
2
3
t
4
5
6
Fig. 1. Time responses of states x; y; z; u for the controlled system (2).
40
10
x1
x2
y1
y2
8
30
6
20
4
10
2
0
0
-10
-2
-20
-4
-30
-6
-40
0
1
2
3
t
4
5
9
6
0
1
2
3
t
4
5
15
z1
z2
8
-8
6
u1
u2
10
5
7
0
6
-5
5
-10
4
-15
-20
3
-25
2
1
0
-30
-35
1
2
3
t
4
5
6
0
1
2
3
t
4
5
6
Fig. 2. Time responses of states xi ; yi ; zi ; ui ði ¼ 1; 2Þ for the drive system (8) and the response system (9).
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10
5
e1
e2
e
3
e4
5
0
-5
-10
-15
0
1
2
3
t
4
5
6
Fig. 3. Time responses of states e1 ; e2 ; e3 ; e4 for the error dynamical system (10).
4. Illustrative examples
In this section, two numerical examples are presented to illustrate the theoretical analysis. In the following numerical
simulations, the fourth-order Runge–Kutta method is employed with time step size 0.001. The system parameters are selected as a ¼ 35; b ¼ 4:9; c ¼ 25; d ¼ 5; e ¼ 35 and k ¼ 22.
Example 1. For the control of the novel hypechaotic system, we consider the controlled system (2). The initial values of
system (2) are given as xð0Þ ¼ 8; yð0Þ ¼ 2; zð0Þ ¼ 4 and uð0Þ ¼ 6, the initial values of the parameter estimation update law
are a1 ð0Þ ¼ b1 ð0Þ ¼ c1 ð0Þ ¼ d1 ð0Þ ¼ e1 ð0Þ ¼ k1 ð0Þ ¼ 0:1. Fig. 1 shows the time responses of states x; y; z; u for the controlled
systems (2). From Fig. 1, we can conclude that the novel hyperchaotic system is suppressed to its unstable equilibrium
Oð0; 0; 0; 0Þ with the adaptive control law(5) and the parameter estimation update law (6).
Example 2. For the synchronization of the novel hypechaotic system, we consider the drive system (8) and the response
system (9). The initial values for systems (8) and (9) are taken as x1 ð0Þ ¼ 8; y1 ð0Þ ¼ 2; z1 ð0Þ ¼ 4; u1 ð0Þ ¼ 6 and
x2 ð0Þ ¼ 17; y2 ð0Þ ¼ 4; z2 ð0Þ ¼ 5; u2 ð0Þ ¼ 2, respectively. Thus, the initial errors are e1 ð0Þ ¼ 9; e2 ð0Þ ¼ 2; e3 ð0Þ ¼
ð0Þ ¼ bð0Þ
1; e4 ð0Þ ¼ 4. And the initial values of the parameter estimation update law are a
¼
cð0Þ ¼ dð0Þ ¼ eð0Þ ¼ kð0Þ ¼ 0:1. Fig. 2 shows the time responses of states xi ; yi ; zi ; ui ði ¼ 1; 2Þ determined by the drive system (8) and the response system (9) with the adaptive control law (13) and the parameter estimation update law (14). The
time responses of states e1 ; e2 ; e3 ; e4 for the error dynamical system (10) are shown in Fig. 3. From Figs. 2 and 3, we can
conclude that the drive system (8) and the response system (9) are asymptotically synchronized with the adaptive control
law (13) and the parameter estimation update law (14).
5. Conclusions
This paper has addressed the problems of control and synchronization of a novel hyperchaotic system with uncertain
parameters. Based on the Lyapunov stability theory and the adaptive control theory, this novel hyperchaotic system is suppressed to its unstable equilibrium. In addition, an adaptive control law and a parameter estimation update law are proposed
to achieve synchronization between two identical novel hyperchaotic systems with uncertain parameters. Numerical examples are given to demonstrate and verify the analytical results.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
O.E. Rössler, An equation for hyperchaos, Phys. Lett. A 71 (1979) 155–157.
T. Kapitaniak, K.-E. Thylwe, I. Cohen, J. Wojewoda, Chaos–hyperchaos transition, Chaos Soliton Fract. 5 (1995) 2003–2011.
F. Wang, C. Liu, Hyperchaos evolved from the Liu chaotic system, Chin. Phys. 15 (2006) 963–968.
S. Nikolov, S. Clodong, Hyperchaos–chaos–hyperchaos transition in modified Rössler systems, Chaos Soliton Fract. 28 (2006) 252–263.
Z. Yan, Controlling hyperchaos in the new hyperchaotic Chen system, Appl. Math. Comput. 168 (2005) 1239–1250.
Q. Li, X.-S. Yang, F. Yang, Hyperchaos in Hopfield-type neural networks, Neurocomputing 67 (2005) 275–280.
C. Li, X. Liao, K.-W. Wong, Lag synchronization of hyperchaos with application to secure communications, Chaos Soliton Fract. 23 (2005) 183–193.
Y. Li, W.K.S. Tang, G. Chen, Hyperchaos evolved from the generalized Lorenz equation, Int. J. Circuit Theory Appl. 33 (2005) 235–251.
Y. Li, W.K.S. Tang, G. Chen, Generating hyperchaos via state feedback control, Int. J. Bifur. Chaos 15 (2005) 3367–3375.
Please cite this article in press as: X. Zhou et al., Adaptive control and synchronization of a novel hyperchaotic system ...,
Appl. Math. Comput (2008), doi:10.1016/j.amc.2008.04.004
ARTICLE IN PRESS
6
X. Zhou et al. / Applied Mathematics and Computation xxx (2008) xxx–xxx
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
A. Chen, J. Lu, J. Lü, S. Yu, Generating hyperchaotic Lü attractor via state feedback control, Physica A: Stat. Mech. Appl. 364 (2006) 103–110.
F. Wang, C. Liu, A new criterion for chaos and hyperchaos synchronization using linear feedback control, Phys. Lett. A 360 (2006) 274–278.
Q. Jia, Adaptive control and synchronization of a new hyperchaotic system with unknown parameters, Phys. Lett. A 362 (2007) 424–429.
M.T. Yassen, Adaptive control and synchronization of a modified Chua’s circuit system, Appl. Math. Comput. 135 (2003) 113–128.
H. Zhang, X. Ma, M. Li, J. Zou, Controlling and tracking hyperchaotic Rössler system via active backstepping design, Chaos Soliton Fract. 26 (2005) 353–
361.
G.H. Li, S.P. Zhou, K. Yang, Generalized projective synchronization between two different chaotic systems using active backstepping control, Phys. Lett.
A 355 (2006) 326–330.
M.-J. Jang, C.-L. Chen, C.-K. Chen, Sliding mode control of hyperchaos in Rössler systems, Chaos Soliton Fract. 14 (2002) 1465–1476.
Z. Chen, Y. Yang, G. Qi, Z. Yuan, A novel hyperchaos system only with one equilibrium, Phys. Lett. A 360 (2007) 696–701.
G. Qi, G. Chen, S. Du, Z. Chen, Z. Yuan, Analysis of a new chaotic system, Physica A: Stat. Mech. Appl. 352 (2005) 295–308.
Please cite this article in press as: X. Zhou et al., Adaptive control and synchronization of a novel hyperchaotic system ...,
Appl. Math. Comput (2008), doi:10.1016/j.amc.2008.04.004