- Email: [email protected]
Synchronization of linearly coupled unified chaotic systems based on linear balanced feedback scheme with constraints
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Chaos, Solitons and Fractals 40 (2009) 1041–1049 www.elsevier.com/locate/chaos
Synchronization of linearly coupled unified chaotic systems based on linear balanced feedback scheme with constraints Heng-Hui Chen a
a,*
, Chaio-Shiung Chen b, Ching-I Lee
c
Department of Mechanical Engineering, HsiuPing Institute of Technology, No. 11, Gungye Road, Dali City, Taichung 412, Taiwan, ROC b Department of Mechanical and Automation Engineering, Dayeh University, Taiwan, ROC c Department of Automation Engineering, Nan Kai Insititute of Technology, Taiwan, ROC Accepted 29 August 2007
Abstract This paper investigates the synchronization of unidirectional and bidirectional coupled unified chaotic systems. A balanced coupling coefficient control method is presented for global asymptotic synchronization using the Lyapunov stability theorem and a minimum scheme with no constraints/constraints. By using the result of the above analysis, the balanced coupling coefficients are then designed to achieve the chaos synchronization of linearly coupled unified chaotic systems. The feasibility and effectiveness of the proposed chaos synchronization scheme are verified via numerical simulations. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction In 1963, Lorenz [1] found the first classical chaotic attractor in a three-dimensional autonomous system derived form a simplified model of earth atmospheric convection system. As the first chaotic model, the Lorenz system has became a paradigm of chaos research. In 1999, Chen found another similar but topologically not equivalent chaotic attractor [2], as the dual of the Lorenz system. In 2002, Lu¨ et al. [3,4] found a new chaotic system, bearing the name of the Lu¨ system, which bridges the gap between the Lorenz and Chen attractors and then presented a unified chaotic system based on three typical chaotic systems: Lorenz system, Chen system, and Lu¨ system. Afterward, Lu¨ et al. [4] analyzed the stability of synchronization chaotic solution of linear coupled unified systems and attained some sufficient conditions for global synchronization by using a method to decompose the coupled system and estimate the maximum characteristic root. Since then, various researchers have investigated the linearly coupled synchronization problems of the aforementioned unified chaotic systems. This coupling configuration is a bidirectional coupling [5] when both systems are connected in such a way that they mutually influence each other’s behavior. This situation typically occurs in physiology, e.g. between cardiac and respiratory systems or between interacting neurons or in nonlinear optics, e.g. coupled laser systems with feedback. On the contrary, when the evolution of one of the coupled systems is unaltered *
Corresponding author. Tel./fax: +886 4 24961108. E-mail addresses: [email protected], [email protected] (H.-H. Chen).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.090
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H.-H. Chen et al. / Chaos, Solitons and Fractals 40 (2009) 1041–1049
by the coupling, the resulting configuration is called unidirectional coupling or drive – response coupling [5]. Typical examples are communication with chaos. Recently, there are some results reported about the linearly bidirectional coupled unified systems, such as the work of Li et al. [6] derived some sufficient conditions for asymptotic synchronization of the system by mathematical theory that is similar Lyapunov method and Park [7] also derived a stability criterion for asymptotic synchronization by using the Lyapunov stability and the linear matrix inequality (LMI) framework. Besides, Wang et al. [8] designed a passive controller for synchronization of unidirectional y-coupled unified chaotic systems on the basis of the property of the passive system in 2007. This paper considers the synchronization of linearly unidirectional and bidirectional coupled unified chaotic systems. Balanced coupling coefficients of linearly coupled unified systems are derived using the Lyapunov stability theorem and the minimum scheme [9]. The derived results give only sufficient conditions for synchronization of system. Thus, we can minimize balanced coupling coefficients further with respect to variables of the Lyapunov candidate function. Compared with the above results, the sum of coupling coefficients is smaller than that obtained by solving the LMI in Ref. [7]. Furthermore, balanced coupling coefficients with constrains are considered to deal with the problem of gain saturation. The proposed balance coupling coefficients with no constrains/constrains scheme has the advantage that the relative minimum coupled coefficients can be obtained in the analytic form without predetermining them to check the stability criterion. The effectiveness of the proposed scheme is demonstrated via its application to the synchronization of Chen’s chaotic system.
2. Controller design under gain constrains In this section, a systematic design process of chaos synchronization of linearly coupled unified systems is provided based on linear balanced feedback scheme with constraint conditions. The idea of linear balanced feedback control with constraints is to choose a suitable set of linear feedback gains d i minimizing a function f ðd 1 ; d 2 ; . . . ; d n ) subject to r independent constraints in the form of gj ðd 1 ; d 2 ; . . . ; d n Þ ¼ 0; j ¼ 1; . . . ; r. Consider the following nonlinear differential equations that describe the unified chaotic system, as unidirectional coupled system for j ¼ 1 and bidirectional coupled system for j ¼ 2, are x_ ¼ Ax þ FðxÞ þ ðj 1Þu; y_ ¼ Ay þ FðyÞ u;
ð1Þ
where x ¼ ½x1 x2 . . . xn T 2 Rn and y ¼ ½y 1 y 2 . . . y n T 2 Rn are the state vector, A 2 Rnn is a constant matrix, F is a continuous nonlinear function, and u ¼ De is a coupling coefficient function, in which D ¼ diagfd 1 ; d 2 ; . . . ; d r ; . . . ; d n g is a diagonal matrix and e ¼ y x ¼ ½e1 e2 . . . en T is the state error. D is composed of two parts: one part where coupling coefficients under no saturation constraints, the other part where coupling coefficients d 1 ; . . . ; d r under the saturation constraints d j 6 sj ; j ¼ 1; . . . ; r, i.e., gj ¼ sj d j mnþj ¼ 0:
ð2Þ
That means 0 6 mnþj ¼ sj d j 6 sj ;
j ¼ 1; . . . ; r:
ð3Þ
Assuming that the following condition is held: Fðx þ eÞ FðxÞ ¼ Je þ FðeÞ
ð4Þ
nn
is the Jacobian matrix evaluated at y ¼ x. In fact, most of chaotic systems can be where J ¼ oFðyÞ=oyjy¼x 2 R described by (1) and (4), which will be further illustrated by an example in Section 3. From Eq. (1) to Eq. (4) the error dynamics for the system have the form e_ ¼ Be þ FðeÞ
ð5Þ
where B ¼ A þ J jD. The aim of synchronization is to make limt!1 keðtÞk ¼ 0. In this paper, we introduce linear balanced feedback control under gain constraints to design controller to synchronize linearly coupled unified chaotic systems. Construct a Lyapunov function V ðeÞ ¼ eT Pe;
ð6Þ
H.-H. Chen et al. / Chaos, Solitons and Fractals 40 (2009) 1041–1049
1043
and the matrix P 2 Rnn is a positive definite matrix, then V ðeÞ is a positive definite function. The derivative of the Lyapunov function along the trajectory of system (5): V_ ¼ e_ T Pe þ eT P_e ¼ eT ðBT P þ PBÞe þ ðFT Pe þ eT PFÞ ¼ eT Qe 6 ET ME; nn
is a positive definite matrix of variables x, and M 2 R where Q 2 R If there exists a positive definite matrix P such as
nn
ð7Þ
is a positive definite matrix.
FT Pe ¼ eT PF ¼ 0;
ð8Þ
i.e., the higher order terms of variables e in Eq. (7) vanish, then it is a global synchronization problem. Assuming that the parameters of coupled unified chaotic system are known and the states of both systems are measurable. One may achieve the synchronization by selecting a set of linear feedback gains D to make the matrix M is a positive definite function. The steps involved in the balanced coupling coefficient design procedure are summarized in the following. Initially, the coupling coefficients d i are derived from the positive definite matrix M: An assumption is made that all the principal minor determinants corresponding to the symmetric matrix M conform to Di ¼ jM ii j ¼ mi > 0;
i ¼ 1; 2; . . . ; n;
ð9Þ
where mi are independent variables. Solving Eq. (9) gives d i ¼ S i ðm1 ; m2 ; . . . ; mi Þ;
i ¼ 1; 2; . . . ; n;
ð10Þ
and od i –0; omi
od j ¼ 0; omi
j ¼ 1; 2; . . . ; i 1;
i ¼ 1; 2; . . . ; n:
ð11Þ
In the second step of the coefficient design process, the sum of the coupling coefficients under saturation conditions is minimized, i.e., f ¼ Minðd 1 þ . . . þ d n Þ takes on a minimum value under the independent constraints gj ¼ sj d j mnþj ¼ 0. The third step of the design process involves identifying the minima of the functions describing specific variables mi . One would form the auxiliary function r X kj gj ðmnþj Þ; ð12Þ uðm1 ; . . . ; mnþr Þ ¼ f ðm1 ; . . . ; mn Þ þ j¼1
where k1 ; . . . ; kr are unknown constants called Lagrange multipliers, mnþj will be specified, and write down the necessary conditions for rendering u a relative minimum with no constraints as follows: umi ¼
ou ¼ 0; omi
i ¼ 1; . . . ; n at pðm1 ; m2 . . . ; mn Þ:
ð13Þ
Substitute solution in Eq. (13) into the following constraint conditions to determine kj . ukj ¼
ou ¼ 0; okj
j ¼ 1; . . . ; r:
ð14Þ
Thus, finally, the desired point p can be found such that f can have a minimum value with constraints. We know the desired point p subject to mnþj in Eq. (3) is restricted to a region R with a finite boundary. Hence, it may happen that an extreme value is taken on a boundary point where umnþj is differ from zero for j ¼ 1; . . . ; r. In order to locate an absolute minimum in a finite region, it is necessary to explore all these possibilities.
3. Synchronization of the unified chaotic systems Consider the following unified chaotic system described by z ¼ Az þ FðzÞ
ð15Þ
where z is a state vector of the system, A is the matrix of the system parameter, and FðxÞ is the nonlinear part of the system as follows: 3 2 3 2 3 2 0 ð25a þ 10Þ 25a þ 10 0 z1 7 6 7 6 7 6 29a 1 0 z ¼ 4 z2 5; A ¼ 4 28 35a 5; FðzÞ ¼ 4 z1 z3 5; a 2 ½0; 1: 0 0 ð8 þ aÞ=3 z3 z1 z2
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The system is chaotic for any a 2 ½0; 1. When a 2 ½0; 0:8Þ, system (1) is called the generalized Lorenz chaotic system; when a ¼ 0:8, it is a Lu¨ chaotic system; whena 2 ð0:8; 1, it is called the generalized Chen chaotic system. Consider the identical unified systems in the form as follows x_ ¼ Ax þ FðxÞ þ ðj 1Þu; y_ ¼ Ay þ FðyÞ u;
ð16Þ
T
T
where x ¼ ½x1 x2 x3 , y ¼ ½y 1 y 2 y 3 ; u ¼ De is a coupling coefficient function, and D ¼ diagfd 1 ; d 2 ; d 3 g is a diagonal matrix, and e ¼ y x ¼ ½e1 e2 e3 T is the state error. We have Fðx þ eÞ FðxÞ ¼ Je þ FðeÞ;
ð17Þ
where 2
0 6 J ¼ 4 x3 x2
3 0 7 x1 5; 0
0 0 x1
3 0 7 6 FðeÞ ¼ 4 e1 e3 5: e1 e2 2
The error dynamics for the system in Eq. (16) are given by e_ ¼ Be þ FðeÞ
ð18Þ
where B ¼ A þ J jD, 2 25a þ 10 ð25a þ 10 þ jd 1 Þ 6 B ¼ 4 28 35a x3 29a 1 jd 2 x2
3
0 x1
7 5:
ðð8 þ aÞ=3 þ jd 3 Þ
x1
The problem of chaos synchronization between linearly coupled chaotic systems can be translated into a problem of how to realize the asymptotical stabilization of the error dynamics (18). So the aim is to design the coupling coefficients d i for stabilizing the error dynamical system (18) at origin. Choose a Lyapunov function candidate as V ¼ eT Pe;
ð19Þ T
where P ¼ diagfI 1 ; I 2 ; I 3 g is a positive definite matrix such that F Pe ¼ eT PF ¼ 0. For simplification, without loss general, one may choose I 1 ¼ qI 0 and I 2 ¼ I 3 ¼ I 0 where I 0 ; q > 0. The derivative of the Lyapunov function along the trajectory of the system in (18) has the form V_ ¼ e_ T Pe þ eT P_e ¼ eT ðBT P þ PBÞe þ ðFT Pe þ eT PFÞ ¼ eT Qe 6 ET ME
ð20Þ
T
where E ¼ ½je1 j je2 j je3 j and 2 c1 q þ 2jd 1 q c23 þ x3 6 c4 þ 2jd 2 Q ¼ I 0 4 c23 þ x3 x2
0
x2 0
3 7 5;
2
c1 q þ 2jd 1 q 6 M ¼ I 0 4 c23 þ Z
c5 þ 2jd 3
Z
c23 þ Z c4 þ 2jd 2
Z 0
0
c5 þ 2jd 3
3 7 5;
where c23 ¼ c2 þ c3 q; c1 ¼ 2ð25a þ 10Þ; c2 ¼ 35a 28; c3 ¼ ð25a þ 10Þ; c4 ¼ 2ð29a 1Þ; c5 ¼ 2ð8 þ aÞ=3; and Z ¼ X sign c23 for sign c23 ¼ 1 if c23 P 0 or for sign c23 ¼ 1 if c23 6 0. X is the maximum upper bounds of the absolute values of variables x2 and x3 . According to Lyapunov stability theory, the error dynamics (18) is asymptotically stable if V_ is a negative definite function, i.e., the matrix M is a positive definitive matrix. By Sylvester’s theorem, all principal minors of M are strictly positive. We have jd 1 ¼ ðm1 c1 qÞ=ð2qÞ; jd 2 ¼ ðc4 m1 þ ðc23 X Þ2 þ m2 Þ=ð2m1 Þ; jd 3 ¼ ðm3 m1 c5 m1 m2 þ X 2 ðc23 X Þ2 þ X 2 m2 Þ=ð2m1 m2 Þ;
ð21Þ
where m1 ; m2 and m3 are independent positive constants and the case c23 6 0, i.e., q P ð35a 28Þ=ð25a þ 10Þ, is considered.
H.-H. Chen et al. / Chaos, Solitons and Fractals 40 (2009) 1041–1049
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Then the states of two coupled unified chaotic systems are globally asymptotically synchronized if positive constants m1 ; m2 and m3 exist. According to balanced coefficient design process proposed by Chen [9], the balanced coupling coefficients with no constraints can be obtained by solving the function f ¼ Minfd 1 þ d 2 þ d 3 g. From Eq. (21), it is clear that an extreme þ value of d 3 ðm3 Þ is taken on at a boundary point as m 3 approaches zero, i.e., m3 ¼ 2em2 ; e ! 0 . If fm1 ¼ 0; f m2 ¼ 0 and þ 2 fm1 m1 > 0; f m1 m1 fm2 m2 > fm1 m2 at a point p m1 ; m2 ; 0 , then f has a relative minimum at that point. Thus the values of the desired parameter mi are found to be pffiffiffi ð22Þ m1 ¼ qð2X c23 Þ; m2 ¼ X ðX c23 Þ; m3 ¼ 0þ : The corresponding balanced coupling coefficients are pffiffiffi pffiffiffi jd 1 ¼ ½ð2X c23 Þ= q c1 =2; jd 2 ¼ ½ðX c23 Þ= q c4 =2;
pffiffiffi jd 3 ¼ ðX = q c5 Þ=2 þ e:
ð23Þ
In practice, the gain realization often bumps into the problem of gain saturated. Therefore, we will discuss how under the gain saturated limiting condition to choose a suitable set of balanced feedback gains. In designing the balanced coupling coefficients d i with constrains, the method proposed in this study aims to minimize the sum of the coupling coefficients f ¼ d 1 þ d 2 þ d 3 under some saturation conditions in the form of 0 6 d 1 6 s1 and 0 6 d 3 6 s2 . Assuming the independent saturation constraints as: g1 ¼ s1 d 1 m4 ¼ 0 and g2 ¼ s2 d 3 m5 ¼ 0, one would form the auxiliary function u ¼ d 1 þ d 2 þ d 3 þ k1 g1 þ k2 g2 ;
a
ð24Þ
b
40
160 140
d1,d2,d 3
d1+d 2+d 3
ρ=1
20
0
X = 50.
120 100 80 60
-20
0
10
20
30
40
40
50
0
c
10
X = 50.
m2
1000
m1
5
x 10 2
1.5
X = 50.
10
4
d
1500
5
ρ
X
1
500 0.5
0 0
5
ρ
10
0
0
ρ
Fig. 1. Variations of balanced coupling coefficients and desired parameters m1 ¼ m1 and m2 ¼ m2 as computed from Eqs. (23) and (22), respectively: (a) balanced coupling coefficients d 1 ðÞ; d 2 ðÞ and d 3 ð:Þ versus X at fixed parameter q ¼ 1, (b) the sum of balanced coupling coefficients versus q, (c) the corresponding desired parameter m1 ¼ m1 versus q, (d) the corresponding desired parameter m2 ¼ m2 versus q.
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H.-H. Chen et al. / Chaos, Solitons and Fractals 40 (2009) 1041–1049
where k1 and k2 are unknown constants, and write down the necessary conditions: ou ¼ 0; i ¼ 1; 2; 3 at p m1 ; m2 ; m3 : u mi ¼ omi
ð25Þ
ou Substitute solution in Eq. (25) into the following constraint conditions: ukj ¼ ok ¼ 0, to determine k1 and k2 . j Thus, finally, the desired gains are found to be
d 2 ¼ ðc4 m1 þ ðX c23 Þ2 þ m2 Þ=ð2m1 Þ;
d 1 ¼ s1 m4 ¼ d 1c ;
where m1 ¼ qð2jd 1c þ c1 Þ, 2 q > X =ð4j2 d 1c d 3c þ 2jc1 d 3c þ
2
2
m2
d 3 ¼ s2 m5 ¼ d 3c ;
ð26Þ 2
2
m3
þ
¼ X ðX c23 Þ =ðqð4j d 1c d 3c þ 2jc1 d 3c þ 2jc5 d 1c þ c1 c5 Þ X Þ, ¼0 , and 2jc5 d 1c þ c1 c5 Þ. According to Eq. (26), f ¼ d 1 þ d 2 þ d 3 has a relative minimum further at a point q if the necessary condition of ¼ 0 is satisfied. The corresponding minimum values of f with constraints ðd 1 ; d 3 Þ ¼ ðd 1c ; d 3c Þ are shown in the next oq section.
4. Numerical results To verify the proposed method described above for linear balanced feedback scheme with no constraints/constrains, this section chooses Chen’s chaotic system for illustration purposes. The validity and effectiveness of the proposed method with no constraints can be demonstrated by comparing the coupling coefficients computed by the present scheme for the Chen’s chaotic systems with those (f ¼ d 1 þ d 2 þ d 3 ¼ 62:03 i.e., d 1 ¼ 7:18; d 2 ¼ 38:68; and d 3 ¼ 16:18Þ obtained by Park [11] using the LMI method. Note that the verification procedure assumes a value of X ¼ 50 for the unified system. When a ¼ 1 and j ¼ 2, Eq. (16) represents bidirectional coupled Chen’s chaotic systems. Fig. 1 reveals that coupling coefficients are changed when q or X is varied. From Eq. (23), Fig. 1a can show that d 1 ¼ 14:5; d 2 ¼ 33:5; and d 3 ¼ 11 i.e., f ¼ 59 where m1 ¼ 128 and m2 ¼ 3900, indicated by ‘‘*’’ for q ¼ 1 and X ¼ 50. When q ¼ 2:66, the sum of the coupling coefficients is minimized further and derived as d 1 ¼ 11:03; d 2 ¼ 34:86; and d 3 ¼ 6:17 i.e., f ¼ 52:06 where m1 ¼ 303 and m2 ¼ 6800 in Fig. 1b–d indicated by ‘‘o’’. For numerical simulation, we assume that initial conditions xð0Þ ¼ ½3 5 7T and yð0Þ ¼ ½4 6 8T are employed. The corresponding synchronization error dynamics of two unified Chen’s chaotic systems rapidly converges to zero as shown in Fig. 2. For j ¼ 1, the numerical results of unidirectional coupled Chen’s chaotic systems are similar (not shown). 1
e1
0.5 0
-0.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1 t(sec)
1.2
1.4
1.6
1.8
2
1
e2
0.5 0 -0.5
1
e3
0.5 0 -0.5
Fig. 2. Synchronization error dynamics of coupled Chen’s chaotic systems with coupling coefficients ðd 1 ; d 2 ; d 3 Þ ¼ ð11:03; 34:86; 6:17Þ.
H.-H. Chen et al. / Chaos, Solitons and Fractals 40 (2009) 1041–1049
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Figs. 3–5 present the simulation results for synchronization of Chen’s chaotic systems based on linear balanced feedback scheme with constraints ðd 1 ; d 3 Þ ¼ ðd 1c ; d 3c Þ. The corresponding minimum values of the sum of the coupling coefficients are shown in Table 1 and indicated by ‘‘o’’, ‘‘*’, ‘‘x’’, ‘‘+’’ in Fig. 3. The corresponding synchronization error dynamics of two coupled unified Chen’s chaotic systems with coupling coefficients ðd 1c ; d 3c Þ ¼ ð0; 0Þ i.e., y-coupled for example, eventually converges to zero as illustrated in Fig. 6. The above simulation results can be seen that the proposed control scheme successfully achieves synchronization of two unified chaotic systems.
3
10
d1c =0, d3c =0 d1c =5, d3c =5 d1c =10, d3c =10
d1c +d2+d3c
d1c =15, d3c =15
2
10
X = 50.
0
15
10
5
ρ Fig. 3. Variation of the sum of balanced coupling coefficients with constraints ðd 1 ; d 3 Þ ¼ ðd 1c ; d 3c Þ as computed from Eq. (26) versus q.
3
m1
10
2
10
d1c =0, d3c =0 d1c =5, d3c =5
X = 50.
d1c =10, d3c =10 d1c =15, d3c =15 0
5
10
15
ρ Fig. 4. Variations of desired parameter m1 ¼ m1 as computed from Eq. (26) versus q.
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H.-H. Chen et al. / Chaos, Solitons and Fractals 40 (2009) 1041–1049
5
m2
10
4
10
d1c =0, d3c =0 d1c =5, d3c =5
X = 50.
3
10
d1c =10, d3c =10 d1c =15, d3c =15 0
5
10
15
ρ Fig. 5. Variations of desired parameter m2 ¼ m2 as computed from Eq. (26) versus q. Table 1 The minima of the sum of the coupling coefficients subject to the constraints ðd 1 ; d 3 Þ ¼ ðd 1c ; d 3c Þ at (ðm1 ; m2 ; q Þ. ðd 1c ; d 3c Þ
d 1c þ d 2 þ d 3c
m1
m2
q
(0, 0) (5, 5) (10, 10) (15, 15)
139.6667 55.2640 53.1840 58.3226
919.3333 302.8791 243.8385 235.4719
209440 12025 4170 2170
13.1333 3.3653 2.2167 1.8113
1
e1
0.5 0
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
1.5
2
2.5
3
3.5
4
1
e2
0.5 0 -0.5
1.5
e3
1 0.5 0 -0.5
t (sec) Fig. 6. Synchronization error dynamics of coupled Chen’s chaotic systems with coupling coefficients ðd 1c ; d 2 ; d 3c Þ ¼ ð0; 139:67; 0Þ.
H.-H. Chen et al. / Chaos, Solitons and Fractals 40 (2009) 1041–1049
1049
5. Conclusion This study has investigated the problem of minimizing the sum of the coupling coefficients under saturation conditions for synchronization of two linearly coupled unified chaotic systems. By using the Lyapunov stability theorem and a minimum scheme, appropriate balanced coupling coefficients have been derived to ensure the synchronization of two coupled unified chaotic systems. Furthermore, a minimized controller is designed for chaos synchronization based on the variable of Lyapunov function candidates and balanced coupling coefficients with constraints. The feasibility and effectiveness of the synchronization scheme have been verified via numerical simulations.
Acknowledgement This research was supported by the National Science Council, Republic of China, under grant number NSC 932218-E-164-001.
References [1] Lorenz EN. Deterministic non-periodic flows. J Atmos Sci 1963;20:130–41. [2] Chen GR, Ueta T. Yet another chaotic attractor. Int J Bifurc Chaos 1999;9:1465–6. ˇ elikovsky˘ S. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurc Chaos [3] Lu¨ J, Chen G, Cheng D, C 2002;12:2917–26. [4] Lu¨ J, Zhou T, Zhang S. Chaos synchronization between linearly coupled chaotic systems. Chaos, Solitons & Fractals 2002;14:529–41. [5] Boccaletti S, Kurths J, Osipov G, Valladares DL, Zhouc CS. The synchronization of chaotic systems. Phys Rep 2002;366:1–101. [6] Li D, Lu JA, Wu X. Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems. Chaos, Solitons & Fractals 2005;23:79–85. [7] Park JH. Stability criterion for synchronization of linearly coupled unified chaotic systems. Chaos, Solitons & Fractals 2005;23:1319–25. [8] Wang F, Liu C. Synchronization of unified chaotic system based on passive control. Phys D 2007;225:55–60. [9] Chen HH. Global synchronization of chaotic systems via linear balanced feedback control. Appl Math Comp 2007;186:923–31.
Chaos, Solitons and Fractals 40 (2009) 1041–1049 www.elsevier.com/locate/chaos
Synchronization of linearly coupled unified chaotic systems based on linear balanced feedback scheme with constraints Heng-Hui Chen a
a,*
, Chaio-Shiung Chen b, Ching-I Lee
c
Department of Mechanical Engineering, HsiuPing Institute of Technology, No. 11, Gungye Road, Dali City, Taichung 412, Taiwan, ROC b Department of Mechanical and Automation Engineering, Dayeh University, Taiwan, ROC c Department of Automation Engineering, Nan Kai Insititute of Technology, Taiwan, ROC Accepted 29 August 2007
Abstract This paper investigates the synchronization of unidirectional and bidirectional coupled unified chaotic systems. A balanced coupling coefficient control method is presented for global asymptotic synchronization using the Lyapunov stability theorem and a minimum scheme with no constraints/constraints. By using the result of the above analysis, the balanced coupling coefficients are then designed to achieve the chaos synchronization of linearly coupled unified chaotic systems. The feasibility and effectiveness of the proposed chaos synchronization scheme are verified via numerical simulations. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction In 1963, Lorenz [1] found the first classical chaotic attractor in a three-dimensional autonomous system derived form a simplified model of earth atmospheric convection system. As the first chaotic model, the Lorenz system has became a paradigm of chaos research. In 1999, Chen found another similar but topologically not equivalent chaotic attractor [2], as the dual of the Lorenz system. In 2002, Lu¨ et al. [3,4] found a new chaotic system, bearing the name of the Lu¨ system, which bridges the gap between the Lorenz and Chen attractors and then presented a unified chaotic system based on three typical chaotic systems: Lorenz system, Chen system, and Lu¨ system. Afterward, Lu¨ et al. [4] analyzed the stability of synchronization chaotic solution of linear coupled unified systems and attained some sufficient conditions for global synchronization by using a method to decompose the coupled system and estimate the maximum characteristic root. Since then, various researchers have investigated the linearly coupled synchronization problems of the aforementioned unified chaotic systems. This coupling configuration is a bidirectional coupling [5] when both systems are connected in such a way that they mutually influence each other’s behavior. This situation typically occurs in physiology, e.g. between cardiac and respiratory systems or between interacting neurons or in nonlinear optics, e.g. coupled laser systems with feedback. On the contrary, when the evolution of one of the coupled systems is unaltered *
Corresponding author. Tel./fax: +886 4 24961108. E-mail addresses: [email protected], [email protected] (H.-H. Chen).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.090
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H.-H. Chen et al. / Chaos, Solitons and Fractals 40 (2009) 1041–1049
by the coupling, the resulting configuration is called unidirectional coupling or drive – response coupling [5]. Typical examples are communication with chaos. Recently, there are some results reported about the linearly bidirectional coupled unified systems, such as the work of Li et al. [6] derived some sufficient conditions for asymptotic synchronization of the system by mathematical theory that is similar Lyapunov method and Park [7] also derived a stability criterion for asymptotic synchronization by using the Lyapunov stability and the linear matrix inequality (LMI) framework. Besides, Wang et al. [8] designed a passive controller for synchronization of unidirectional y-coupled unified chaotic systems on the basis of the property of the passive system in 2007. This paper considers the synchronization of linearly unidirectional and bidirectional coupled unified chaotic systems. Balanced coupling coefficients of linearly coupled unified systems are derived using the Lyapunov stability theorem and the minimum scheme [9]. The derived results give only sufficient conditions for synchronization of system. Thus, we can minimize balanced coupling coefficients further with respect to variables of the Lyapunov candidate function. Compared with the above results, the sum of coupling coefficients is smaller than that obtained by solving the LMI in Ref. [7]. Furthermore, balanced coupling coefficients with constrains are considered to deal with the problem of gain saturation. The proposed balance coupling coefficients with no constrains/constrains scheme has the advantage that the relative minimum coupled coefficients can be obtained in the analytic form without predetermining them to check the stability criterion. The effectiveness of the proposed scheme is demonstrated via its application to the synchronization of Chen’s chaotic system.
2. Controller design under gain constrains In this section, a systematic design process of chaos synchronization of linearly coupled unified systems is provided based on linear balanced feedback scheme with constraint conditions. The idea of linear balanced feedback control with constraints is to choose a suitable set of linear feedback gains d i minimizing a function f ðd 1 ; d 2 ; . . . ; d n ) subject to r independent constraints in the form of gj ðd 1 ; d 2 ; . . . ; d n Þ ¼ 0; j ¼ 1; . . . ; r. Consider the following nonlinear differential equations that describe the unified chaotic system, as unidirectional coupled system for j ¼ 1 and bidirectional coupled system for j ¼ 2, are x_ ¼ Ax þ FðxÞ þ ðj 1Þu; y_ ¼ Ay þ FðyÞ u;
ð1Þ
where x ¼ ½x1 x2 . . . xn T 2 Rn and y ¼ ½y 1 y 2 . . . y n T 2 Rn are the state vector, A 2 Rnn is a constant matrix, F is a continuous nonlinear function, and u ¼ De is a coupling coefficient function, in which D ¼ diagfd 1 ; d 2 ; . . . ; d r ; . . . ; d n g is a diagonal matrix and e ¼ y x ¼ ½e1 e2 . . . en T is the state error. D is composed of two parts: one part where coupling coefficients under no saturation constraints, the other part where coupling coefficients d 1 ; . . . ; d r under the saturation constraints d j 6 sj ; j ¼ 1; . . . ; r, i.e., gj ¼ sj d j mnþj ¼ 0:
ð2Þ
That means 0 6 mnþj ¼ sj d j 6 sj ;
j ¼ 1; . . . ; r:
ð3Þ
Assuming that the following condition is held: Fðx þ eÞ FðxÞ ¼ Je þ FðeÞ
ð4Þ
nn
is the Jacobian matrix evaluated at y ¼ x. In fact, most of chaotic systems can be where J ¼ oFðyÞ=oyjy¼x 2 R described by (1) and (4), which will be further illustrated by an example in Section 3. From Eq. (1) to Eq. (4) the error dynamics for the system have the form e_ ¼ Be þ FðeÞ
ð5Þ
where B ¼ A þ J jD. The aim of synchronization is to make limt!1 keðtÞk ¼ 0. In this paper, we introduce linear balanced feedback control under gain constraints to design controller to synchronize linearly coupled unified chaotic systems. Construct a Lyapunov function V ðeÞ ¼ eT Pe;
ð6Þ
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and the matrix P 2 Rnn is a positive definite matrix, then V ðeÞ is a positive definite function. The derivative of the Lyapunov function along the trajectory of system (5): V_ ¼ e_ T Pe þ eT P_e ¼ eT ðBT P þ PBÞe þ ðFT Pe þ eT PFÞ ¼ eT Qe 6 ET ME; nn
is a positive definite matrix of variables x, and M 2 R where Q 2 R If there exists a positive definite matrix P such as
nn
ð7Þ
is a positive definite matrix.
FT Pe ¼ eT PF ¼ 0;
ð8Þ
i.e., the higher order terms of variables e in Eq. (7) vanish, then it is a global synchronization problem. Assuming that the parameters of coupled unified chaotic system are known and the states of both systems are measurable. One may achieve the synchronization by selecting a set of linear feedback gains D to make the matrix M is a positive definite function. The steps involved in the balanced coupling coefficient design procedure are summarized in the following. Initially, the coupling coefficients d i are derived from the positive definite matrix M: An assumption is made that all the principal minor determinants corresponding to the symmetric matrix M conform to Di ¼ jM ii j ¼ mi > 0;
i ¼ 1; 2; . . . ; n;
ð9Þ
where mi are independent variables. Solving Eq. (9) gives d i ¼ S i ðm1 ; m2 ; . . . ; mi Þ;
i ¼ 1; 2; . . . ; n;
ð10Þ
and od i –0; omi
od j ¼ 0; omi
j ¼ 1; 2; . . . ; i 1;
i ¼ 1; 2; . . . ; n:
ð11Þ
In the second step of the coefficient design process, the sum of the coupling coefficients under saturation conditions is minimized, i.e., f ¼ Minðd 1 þ . . . þ d n Þ takes on a minimum value under the independent constraints gj ¼ sj d j mnþj ¼ 0. The third step of the design process involves identifying the minima of the functions describing specific variables mi . One would form the auxiliary function r X kj gj ðmnþj Þ; ð12Þ uðm1 ; . . . ; mnþr Þ ¼ f ðm1 ; . . . ; mn Þ þ j¼1
where k1 ; . . . ; kr are unknown constants called Lagrange multipliers, mnþj will be specified, and write down the necessary conditions for rendering u a relative minimum with no constraints as follows: umi ¼
ou ¼ 0; omi
i ¼ 1; . . . ; n at pðm1 ; m2 . . . ; mn Þ:
ð13Þ
Substitute solution in Eq. (13) into the following constraint conditions to determine kj . ukj ¼
ou ¼ 0; okj
j ¼ 1; . . . ; r:
ð14Þ
Thus, finally, the desired point p can be found such that f can have a minimum value with constraints. We know the desired point p subject to mnþj in Eq. (3) is restricted to a region R with a finite boundary. Hence, it may happen that an extreme value is taken on a boundary point where umnþj is differ from zero for j ¼ 1; . . . ; r. In order to locate an absolute minimum in a finite region, it is necessary to explore all these possibilities.
3. Synchronization of the unified chaotic systems Consider the following unified chaotic system described by z ¼ Az þ FðzÞ
ð15Þ
where z is a state vector of the system, A is the matrix of the system parameter, and FðxÞ is the nonlinear part of the system as follows: 3 2 3 2 3 2 0 ð25a þ 10Þ 25a þ 10 0 z1 7 6 7 6 7 6 29a 1 0 z ¼ 4 z2 5; A ¼ 4 28 35a 5; FðzÞ ¼ 4 z1 z3 5; a 2 ½0; 1: 0 0 ð8 þ aÞ=3 z3 z1 z2
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The system is chaotic for any a 2 ½0; 1. When a 2 ½0; 0:8Þ, system (1) is called the generalized Lorenz chaotic system; when a ¼ 0:8, it is a Lu¨ chaotic system; whena 2 ð0:8; 1, it is called the generalized Chen chaotic system. Consider the identical unified systems in the form as follows x_ ¼ Ax þ FðxÞ þ ðj 1Þu; y_ ¼ Ay þ FðyÞ u;
ð16Þ
T
T
where x ¼ ½x1 x2 x3 , y ¼ ½y 1 y 2 y 3 ; u ¼ De is a coupling coefficient function, and D ¼ diagfd 1 ; d 2 ; d 3 g is a diagonal matrix, and e ¼ y x ¼ ½e1 e2 e3 T is the state error. We have Fðx þ eÞ FðxÞ ¼ Je þ FðeÞ;
ð17Þ
where 2
0 6 J ¼ 4 x3 x2
3 0 7 x1 5; 0
0 0 x1
3 0 7 6 FðeÞ ¼ 4 e1 e3 5: e1 e2 2
The error dynamics for the system in Eq. (16) are given by e_ ¼ Be þ FðeÞ
ð18Þ
where B ¼ A þ J jD, 2 25a þ 10 ð25a þ 10 þ jd 1 Þ 6 B ¼ 4 28 35a x3 29a 1 jd 2 x2
3
0 x1
7 5:
ðð8 þ aÞ=3 þ jd 3 Þ
x1
The problem of chaos synchronization between linearly coupled chaotic systems can be translated into a problem of how to realize the asymptotical stabilization of the error dynamics (18). So the aim is to design the coupling coefficients d i for stabilizing the error dynamical system (18) at origin. Choose a Lyapunov function candidate as V ¼ eT Pe;
ð19Þ T
where P ¼ diagfI 1 ; I 2 ; I 3 g is a positive definite matrix such that F Pe ¼ eT PF ¼ 0. For simplification, without loss general, one may choose I 1 ¼ qI 0 and I 2 ¼ I 3 ¼ I 0 where I 0 ; q > 0. The derivative of the Lyapunov function along the trajectory of the system in (18) has the form V_ ¼ e_ T Pe þ eT P_e ¼ eT ðBT P þ PBÞe þ ðFT Pe þ eT PFÞ ¼ eT Qe 6 ET ME
ð20Þ
T
where E ¼ ½je1 j je2 j je3 j and 2 c1 q þ 2jd 1 q c23 þ x3 6 c4 þ 2jd 2 Q ¼ I 0 4 c23 þ x3 x2
0
x2 0
3 7 5;
2
c1 q þ 2jd 1 q 6 M ¼ I 0 4 c23 þ Z
c5 þ 2jd 3
Z
c23 þ Z c4 þ 2jd 2
Z 0
0
c5 þ 2jd 3
3 7 5;
where c23 ¼ c2 þ c3 q; c1 ¼ 2ð25a þ 10Þ; c2 ¼ 35a 28; c3 ¼ ð25a þ 10Þ; c4 ¼ 2ð29a 1Þ; c5 ¼ 2ð8 þ aÞ=3; and Z ¼ X sign c23 for sign c23 ¼ 1 if c23 P 0 or for sign c23 ¼ 1 if c23 6 0. X is the maximum upper bounds of the absolute values of variables x2 and x3 . According to Lyapunov stability theory, the error dynamics (18) is asymptotically stable if V_ is a negative definite function, i.e., the matrix M is a positive definitive matrix. By Sylvester’s theorem, all principal minors of M are strictly positive. We have jd 1 ¼ ðm1 c1 qÞ=ð2qÞ; jd 2 ¼ ðc4 m1 þ ðc23 X Þ2 þ m2 Þ=ð2m1 Þ; jd 3 ¼ ðm3 m1 c5 m1 m2 þ X 2 ðc23 X Þ2 þ X 2 m2 Þ=ð2m1 m2 Þ;
ð21Þ
where m1 ; m2 and m3 are independent positive constants and the case c23 6 0, i.e., q P ð35a 28Þ=ð25a þ 10Þ, is considered.
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Then the states of two coupled unified chaotic systems are globally asymptotically synchronized if positive constants m1 ; m2 and m3 exist. According to balanced coefficient design process proposed by Chen [9], the balanced coupling coefficients with no constraints can be obtained by solving the function f ¼ Minfd 1 þ d 2 þ d 3 g. From Eq. (21), it is clear that an extreme þ value of d 3 ðm3 Þ is taken on at a boundary point as m 3 approaches zero, i.e., m3 ¼ 2em2 ; e ! 0 . If fm1 ¼ 0; f m2 ¼ 0 and þ 2 fm1 m1 > 0; f m1 m1 fm2 m2 > fm1 m2 at a point p m1 ; m2 ; 0 , then f has a relative minimum at that point. Thus the values of the desired parameter mi are found to be pffiffiffi ð22Þ m1 ¼ qð2X c23 Þ; m2 ¼ X ðX c23 Þ; m3 ¼ 0þ : The corresponding balanced coupling coefficients are pffiffiffi pffiffiffi jd 1 ¼ ½ð2X c23 Þ= q c1 =2; jd 2 ¼ ½ðX c23 Þ= q c4 =2;
pffiffiffi jd 3 ¼ ðX = q c5 Þ=2 þ e:
ð23Þ
In practice, the gain realization often bumps into the problem of gain saturated. Therefore, we will discuss how under the gain saturated limiting condition to choose a suitable set of balanced feedback gains. In designing the balanced coupling coefficients d i with constrains, the method proposed in this study aims to minimize the sum of the coupling coefficients f ¼ d 1 þ d 2 þ d 3 under some saturation conditions in the form of 0 6 d 1 6 s1 and 0 6 d 3 6 s2 . Assuming the independent saturation constraints as: g1 ¼ s1 d 1 m4 ¼ 0 and g2 ¼ s2 d 3 m5 ¼ 0, one would form the auxiliary function u ¼ d 1 þ d 2 þ d 3 þ k1 g1 þ k2 g2 ;
a
ð24Þ
b
40
160 140
d1,d2,d 3
d1+d 2+d 3
ρ=1
20
0
X = 50.
120 100 80 60
-20
0
10
20
30
40
40
50
0
c
10
X = 50.
m2
1000
m1
5
x 10 2
1.5
X = 50.
10
4
d
1500
5
ρ
X
1
500 0.5
0 0
5
ρ
10
0
0
ρ
Fig. 1. Variations of balanced coupling coefficients and desired parameters m1 ¼ m1 and m2 ¼ m2 as computed from Eqs. (23) and (22), respectively: (a) balanced coupling coefficients d 1 ðÞ; d 2 ðÞ and d 3 ð:Þ versus X at fixed parameter q ¼ 1, (b) the sum of balanced coupling coefficients versus q, (c) the corresponding desired parameter m1 ¼ m1 versus q, (d) the corresponding desired parameter m2 ¼ m2 versus q.
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H.-H. Chen et al. / Chaos, Solitons and Fractals 40 (2009) 1041–1049
where k1 and k2 are unknown constants, and write down the necessary conditions: ou ¼ 0; i ¼ 1; 2; 3 at p m1 ; m2 ; m3 : u mi ¼ omi
ð25Þ
ou Substitute solution in Eq. (25) into the following constraint conditions: ukj ¼ ok ¼ 0, to determine k1 and k2 . j Thus, finally, the desired gains are found to be
d 2 ¼ ðc4 m1 þ ðX c23 Þ2 þ m2 Þ=ð2m1 Þ;
d 1 ¼ s1 m4 ¼ d 1c ;
where m1 ¼ qð2jd 1c þ c1 Þ, 2 q > X =ð4j2 d 1c d 3c þ 2jc1 d 3c þ
2
2
m2
d 3 ¼ s2 m5 ¼ d 3c ;
ð26Þ 2
2
m3
þ
¼ X ðX c23 Þ =ðqð4j d 1c d 3c þ 2jc1 d 3c þ 2jc5 d 1c þ c1 c5 Þ X Þ, ¼0 , and 2jc5 d 1c þ c1 c5 Þ. According to Eq. (26), f ¼ d 1 þ d 2 þ d 3 has a relative minimum further at a point q if the necessary condition of ¼ 0 is satisfied. The corresponding minimum values of f with constraints ðd 1 ; d 3 Þ ¼ ðd 1c ; d 3c Þ are shown in the next oq section.
4. Numerical results To verify the proposed method described above for linear balanced feedback scheme with no constraints/constrains, this section chooses Chen’s chaotic system for illustration purposes. The validity and effectiveness of the proposed method with no constraints can be demonstrated by comparing the coupling coefficients computed by the present scheme for the Chen’s chaotic systems with those (f ¼ d 1 þ d 2 þ d 3 ¼ 62:03 i.e., d 1 ¼ 7:18; d 2 ¼ 38:68; and d 3 ¼ 16:18Þ obtained by Park [11] using the LMI method. Note that the verification procedure assumes a value of X ¼ 50 for the unified system. When a ¼ 1 and j ¼ 2, Eq. (16) represents bidirectional coupled Chen’s chaotic systems. Fig. 1 reveals that coupling coefficients are changed when q or X is varied. From Eq. (23), Fig. 1a can show that d 1 ¼ 14:5; d 2 ¼ 33:5; and d 3 ¼ 11 i.e., f ¼ 59 where m1 ¼ 128 and m2 ¼ 3900, indicated by ‘‘*’’ for q ¼ 1 and X ¼ 50. When q ¼ 2:66, the sum of the coupling coefficients is minimized further and derived as d 1 ¼ 11:03; d 2 ¼ 34:86; and d 3 ¼ 6:17 i.e., f ¼ 52:06 where m1 ¼ 303 and m2 ¼ 6800 in Fig. 1b–d indicated by ‘‘o’’. For numerical simulation, we assume that initial conditions xð0Þ ¼ ½3 5 7T and yð0Þ ¼ ½4 6 8T are employed. The corresponding synchronization error dynamics of two unified Chen’s chaotic systems rapidly converges to zero as shown in Fig. 2. For j ¼ 1, the numerical results of unidirectional coupled Chen’s chaotic systems are similar (not shown). 1
e1
0.5 0
-0.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1 t(sec)
1.2
1.4
1.6
1.8
2
1
e2
0.5 0 -0.5
1
e3
0.5 0 -0.5
Fig. 2. Synchronization error dynamics of coupled Chen’s chaotic systems with coupling coefficients ðd 1 ; d 2 ; d 3 Þ ¼ ð11:03; 34:86; 6:17Þ.
H.-H. Chen et al. / Chaos, Solitons and Fractals 40 (2009) 1041–1049
1047
Figs. 3–5 present the simulation results for synchronization of Chen’s chaotic systems based on linear balanced feedback scheme with constraints ðd 1 ; d 3 Þ ¼ ðd 1c ; d 3c Þ. The corresponding minimum values of the sum of the coupling coefficients are shown in Table 1 and indicated by ‘‘o’’, ‘‘*’, ‘‘x’’, ‘‘+’’ in Fig. 3. The corresponding synchronization error dynamics of two coupled unified Chen’s chaotic systems with coupling coefficients ðd 1c ; d 3c Þ ¼ ð0; 0Þ i.e., y-coupled for example, eventually converges to zero as illustrated in Fig. 6. The above simulation results can be seen that the proposed control scheme successfully achieves synchronization of two unified chaotic systems.
3
10
d1c =0, d3c =0 d1c =5, d3c =5 d1c =10, d3c =10
d1c +d2+d3c
d1c =15, d3c =15
2
10
X = 50.
0
15
10
5
ρ Fig. 3. Variation of the sum of balanced coupling coefficients with constraints ðd 1 ; d 3 Þ ¼ ðd 1c ; d 3c Þ as computed from Eq. (26) versus q.
3
m1
10
2
10
d1c =0, d3c =0 d1c =5, d3c =5
X = 50.
d1c =10, d3c =10 d1c =15, d3c =15 0
5
10
15
ρ Fig. 4. Variations of desired parameter m1 ¼ m1 as computed from Eq. (26) versus q.
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H.-H. Chen et al. / Chaos, Solitons and Fractals 40 (2009) 1041–1049
5
m2
10
4
10
d1c =0, d3c =0 d1c =5, d3c =5
X = 50.
3
10
d1c =10, d3c =10 d1c =15, d3c =15 0
5
10
15
ρ Fig. 5. Variations of desired parameter m2 ¼ m2 as computed from Eq. (26) versus q. Table 1 The minima of the sum of the coupling coefficients subject to the constraints ðd 1 ; d 3 Þ ¼ ðd 1c ; d 3c Þ at (ðm1 ; m2 ; q Þ. ðd 1c ; d 3c Þ
d 1c þ d 2 þ d 3c
m1
m2
q
(0, 0) (5, 5) (10, 10) (15, 15)
139.6667 55.2640 53.1840 58.3226
919.3333 302.8791 243.8385 235.4719
209440 12025 4170 2170
13.1333 3.3653 2.2167 1.8113
1
e1
0.5 0
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
1.5
2
2.5
3
3.5
4
1
e2
0.5 0 -0.5
1.5
e3
1 0.5 0 -0.5
t (sec) Fig. 6. Synchronization error dynamics of coupled Chen’s chaotic systems with coupling coefficients ðd 1c ; d 2 ; d 3c Þ ¼ ð0; 139:67; 0Þ.
H.-H. Chen et al. / Chaos, Solitons and Fractals 40 (2009) 1041–1049
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5. Conclusion This study has investigated the problem of minimizing the sum of the coupling coefficients under saturation conditions for synchronization of two linearly coupled unified chaotic systems. By using the Lyapunov stability theorem and a minimum scheme, appropriate balanced coupling coefficients have been derived to ensure the synchronization of two coupled unified chaotic systems. Furthermore, a minimized controller is designed for chaos synchronization based on the variable of Lyapunov function candidates and balanced coupling coefficients with constraints. The feasibility and effectiveness of the synchronization scheme have been verified via numerical simulations.
Acknowledgement This research was supported by the National Science Council, Republic of China, under grant number NSC 932218-E-164-001.
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