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Application of mode decomposition approach to the synchronization of the non-identical dynamical systems
Applied Mathematical Modelling 36 (2012) 779–791
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Application of mode decomposition approach to the synchronization of the non-identical dynamical systems q Lijun Pei ⇑, Benhua Qiu Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450001, PR China
a r t i c l e
i n f o
Article history: Received 7 October 2008 Received in revised form 26 May 2011 Accepted 1 July 2011 Available online 22 July 2011 Keywords: Synchronization Mode decomposition Non-identical dynamical systems
a b s t r a c t The synchronization problem of two different dynamical systems is considered by employing mode decomposition approach in this paper. Synchronization of non-identical coupled dynamical systems with non-chaotic attractors, i.e., equilibria, periodic and quasi-periodic solutions, is investigated analytically and numerically. Some results are obtained by this method. Some examples, supported by numerical simulation, are presented to illustrate the conciseness and effectiveness of the approach. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Synchronization in coupled dynamical systems is a subject of many theoretical papers over the last few years [1–4]. Our granted Project (NNSFC No. 10702065) is titled by Study on Synchronization Mechanism of the delayed coupling dynamical systems. It aims to find the mechanism and laws of the synchronization of the delayed coupling dynamical systems. It includes mainly three problems, i.e., the synchronization approach, stability and design of the synchronization schemes, the classification and switch of the generalized synchronization, the effect of time delay on the synchronization of the coupled dynamical systems. The considered approach in this paper, mode decomposition approach, is one of the main synchronization approaches including the generalized Hamiltonian systems and observer approach, parameters update law, exponential approach, inertial manifold approach, etc. Non-chaotic (periodic or quasi-periodic) synchronization of dynamic systems was explored by El Naschie (see Refs. [5–8]) as well as Ramaswamy et al. (see Refs. [9–11]). Communications security and engineering were put into effect in different ways [12,13], revealing that synchronization of systems with periodic dynamics is an important and well-known effect in the fields of physics, engineering as well as in other branches of sciences [14,15]. And the synchronization of systems with quasi-periodic or aperiodic dynamics, i.e., the synchronization of strange non-chaotic attractors, is also a possible scheme of communications security and can offer some advantages [9]. Thus synchronization of non-chaotic attractors should be also investigated but not only chaotic synchronization. In this paper, synchronization of non-chaotic attractors including equilibria, periodic and quasi-periodic attractors is considered. When identical systems are coupled, synchronization is the most obvious effect. However, experimental and even more real systems are often not fully identical, especially there are mismatches in parameters of the systems. It is thus important and also interesting to investigate synchronization behavior between non-identical systems. In general, it is difficult to achieve the synchronization for the master–salve chaotic systems with non-identical parameters, so the synchronization
q
This work is supported by National Natural Science Foundation of China (No. 10702065).
⇑ Corresponding author.
E-mail address: [email protected] (L. Pei). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.07.006
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L. Pei, B. Qiu / Applied Mathematical Modelling 36 (2012) 779–791
of coupled dynamical systems with periodic and quasi-periodic solutions is discussed in this paper. We have known the stability of synchronized motion usually means the stability of the complete synchronization manifold, i.e., x = y, or the stability of the zero solution of the synchronization error. When we deal with non-identical coupled systems, similar stability criteria can be formulated. In most case, most of rigorous theoretical results of synchronization are obtained by constructing Lyapunov function [16,17]. Unfortunately, this method can be applied only to some particular examples where the coupling matrix is generally required to be negative definite. However, a novel method, mode decomposition [18–20], of treating the stability of a synchronous state is proposed based on the Floquet theory. A rigorous criterion is then derived, which can also be applied to non-identical coupled systems. The paper is structured as follows. Firstly, we state the problem of the stability of systems’ synchronization, and then transform the stability of synchronization solution into a lower-dimensional problem by using mode decomposition method. In Section 2, we give the general analysis for non-identical coupled systems. In Sections 3–5, several typical examples are given to demonstrate the accuracy of theoretical analysis. At last, we give some numerical simulations to illustrate our findings and some useful conclusions. 2. The general analysis First we introduce the mode decomposition approach [19,20]. Consider two non-identical systems which are coupled in the following form:
(
dx dt dy dt
¼ f ðxÞ þ cðx; yÞ;
ð1Þ
¼ gðyÞ þ cðy; xÞ;
where x, y 2 Rn, and c(x, y) is smooth. The coupling term c(x, y) implies the coupling between both of the non-identical systems. Thus their couplings in system (1) are the bidirectional and same. In this paper these coupling terms c(x, y) and c(y, x) take the usual forms in [19,20]. Generally, synchronization occurs on an invariant manifold given by x = y = /(t), where /(t) is a nontrivial solution of the uncoupled system ( dx dt dy dt
¼ f ðxÞ;
ð2Þ
¼ gðyÞ:
Then, we have f(/) = g(/). The stability problem of the synchronization of the coupled systems can be formulated in a very general way by addressing the question of the stability of the CS synchronization manifold x = y, or equivalently by studying the temporal evolution of the synchronization error e = y x, then Eq. (1) lead to a certain linearized equation in e. This implies that the synchronization problem for a coupled system is equivalent to the stability of a zero solution of the linearized equation. The synchronization manifold is linearly stable if
lim keðtÞk ¼ lim kxðtÞ yðtÞk ¼ 0:
t!þ1
ð3Þ
t!þ1
To determine the behavior of e(t) in this limit, we consider a variational equation of Eq. (1) evaluated along the synchronous manifold (/(t), /(t))
e_ 1 e_ 2
¼
Df ð/Þ þ Dx cð/; /Þ
Dy cð/; /Þ
Dy cð/; /Þ
Dgð/Þ þ Dx cð/; /Þ
e1
e2
;
ð4Þ
where Df(/), Dg(/) is the Jacobian matrix of f(x), g(x) evaluated at /(t) whereas Dxc(/, /) and Dyc(/, /) are Jacobian matrices of c(x, y) evaluated at (/(t), /(t))T with respect to x and y, respectively. Furthermore, we introduce a mode decomposition to simplify Eq. (4). For clarity, let
P¼
I
I
I
I
;
ð5Þ
where I is an n n identical matrix. The inverse matrix of P is
P1 ¼
1 2
I
I
I
I
:
ð6Þ
:
ð7Þ
Let
e1
e2
¼P
w1 w2
It follows from Eqs. (5)–(7) that
_1 w _2 w
¼ P1
e_ 1 e_ 2
¼
Gð/Þ
0
0
Hð/Þ
w1 w2
;
ð8Þ
L. Pei, B. Qiu / Applied Mathematical Modelling 36 (2012) 779–791
781
where
f ð/Þ ¼ gð/Þ;
Gð/Þ ¼ Df ð/Þ þ Dx cð/; /Þ þ Dy cð/; /Þ;
Hð/Þ ¼ Df ð/Þ þ Dx cð/;
/Þ Dy cð/; /Þ:
Thus, the variational equation (4) is decomposed into two independent systems:
_ 1 ¼ ½Df ð/Þ þ Dx cð/; /Þ þ Dy cð/; /Þw1 ; w
ð9Þ
_ 2 ¼ ½Df ð/Þ þ Dx cð/; /Þ Dy cð/; /Þw2 : w
ð10Þ
and There are two special cases: Case 1: If Eq. (1) satisfy the conservative condition c(x, x) = 0 or c(x, x) c (constant), then Dxc(/, /) + Dyc(/, /) = 0, namely, Eqs. (9) and (10) can be written as
_ 1 ¼ Df ð/Þx1 ; w
ð11Þ
and
_ 2 ¼ ½Df ð/Þ þ 2Dx cð/; /Þx2 : w
ð12Þ
Case 2: If c(x, y) of Eq. (1) is symmetric with respect to x and y, then Dxc(/, /) = Dyc(/, /), Eqs. (9) and (10) can also be written as the similar form of (11) and (12). Such a decomposition has geometrical and theoretical explanations. Geometrically, the stability of a synchronous manifold does not depend upon the tangent motion whereas the transverse motion determines its stability. Theoretically, a clear explanation may be given by virtue of similar Floquet multiplier theory. According to the above decomposition and based on an extended Floquet theory, to make the considered synchronous state be stable, we impose a sufficient condition [18–20], that is, we force all the real parts of eigenvalues of matrix Df(/) + 2Dxc(/, /) to be uniformly negative in t, i.e., we require all eigenvalues of matrix
AðtÞ ¼
1 ½ðDf ð/Þ þ 2Dx cð/; /ÞÞ þ ðDf ð/Þ þ 2Dx cð/; /ÞÞT 2
ð13Þ
to be uniformly negative, this easily leads to the following theorem: Theorem 2.1. Suppose f(x) 2 C1(Rn), g(x) 2 C1(Rn) and c(x, y) 2 C1(Rn,Rn). Denote by (/1(t), /2(t))T a solution of Eq. (1). If the eigenvalues of matrix (13) are all uniformly negative in t, then /1(t) and /2(t) are synchronized in the sense of Eq. (3). Note that the conditions in Theorem 2.1 only suppose that c(x, y) is smooth. Especially in the linear coupling case, the corresponding condition on synchronization will become simpler. If f(x) = g(y), then systems (1) can be transformed into the form of identical systems.
3. Application to the equilibria’s synchronization First the simple case that the synchronization of the non-identical dynamical systems is considered. At the same time, the synchronized systems are asymptotically stable, i.e., they have the same asymptotically stable equilibria. This case of synchronization is called as the equilibria’s synchronization. Consider the following systems:
x_ 1 ¼ x1 ; x_ 2 ¼ x2 ;
ð14Þ
y_ 1 ¼ y2 ; y_ 2 ¼ 2y1 3y2 ;
ð15Þ
and
we set the coupling term c(x, y) as
cðx1 ; x2 ; y1 ; y2 Þ ¼
ða1 y2 þ b1 Þðy1 x1 Þ ða2 x1 þ b2 Þðy2 x2 Þ
;
ð16Þ
and the overall coupled system is then of the form:
8 x_ 1 ¼ x1 þ ða1 y2 þ b1 Þðy1 x1 Þ; > > > < x_ ¼ x þ ða x þ b Þðy x Þ; 2 2 2 1 2 2 2 > _ 1 ¼ y2 þ ða1 x2 þ b1 Þðx1 y1 Þ; y > > : y_ 2 ¼ 2y1 3y2 þ ða2 y1 þ b2 Þðx2 y2 Þ:
ð17Þ
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L. Pei, B. Qiu / Applied Mathematical Modelling 36 (2012) 779–791
To make systems (17) synchronous, it is required to appropriately choose value of a1, a2, b1 and b2. Suppose / = (/1, /2)T is a equilibrium of Eqs. (14) and (15), and introduce a matrix
AðtÞ :¼
1 ½ðDf ð/Þ þ 2Dx cð/; /Þ þ ðDf ð/Þ þ 2Dx cð/; /ÞÞÞT ; 2
where f(x) represents the right hand side of Eq. (14), c(x, y) is defined by Eq. (16), Df is the Jacobian matrix of f(x), and Dxc(x, y) is the Jacobian matrix of c(x, y) with respect to x. It follows form Eq. (17) that
Df ð/Þ þ 2Dx cð/; /Þ ¼
1 2ða1 /2 þ b1 Þ 0 ; 0 1 2ða2 /1 þ b2 Þ
by simple calculation we have
AðtÞ ¼
1 2ða1 /2 þ b1 Þ
0
0
1 2ða2 /1 þ b2 Þ
:
From Theorem 2.1, we see that a sufficient condition on the synchrony stability is that eigenvalues l1(t) and l2(t) of matrix A(t) are uniformly negative in t. This implies
l1 ðtÞ ¼ 1 2ð/2 a1 þ b1 Þ < 0; and
l2 ðtÞ ¼ 1 2ða2 /1 þ b2 Þ < 0: From the above two inequalities on a1, a2, b1 and b2, we can obtain the following conditions:
1 b1 > /2 a1 ; 2
ð18Þ
1 b2 > /1 a2 : 2
ð19Þ
and
Here the equilibrium point is (0, 0), thus we have b1 > 12 ; b2 > 12. To satisfy the above conditions, the parameter can be selected as b1 = 5, b2 = 1. Theorem 3.1. Suppose (/1, /2)T is a solution of systems (14) and (15) . If conditions (18) and (19) hold, then the nonlinearly nonidentical coupled systems (17) are synchronized in the sense of expression (3). Remark 1. The equilibria’s synchronization is the simplest synchronization. The other synchronization such as the periodic, quasi-periodic and chaotic synchronization are the problems that are the most important and interesting. Some synchronization will be considered in the following sections. Next, we present numerical simulations according to the theoretical results. In the following, Fig. 1 illustrate the time evolution of components x1(t) and x2(t). Fig. 2 shows time series of synchronization errors, i.e., e1 = y1 x1 and e2 = y2 x2 via time. From the Fig. 2 we see that these errors tend to zeros as t goes infinity. Fig. 3 The phase portrait of the coupled system (17) is displayed.
4. Application to the periodic synchronization 4.1. Coupled periodic oscillators In this section we take the following systems as an example of the periodic synchronization, and show the above theoretical analysis. The systems can be mathematically expressed as:
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < x_ 1 ¼ x2 þ x1 1 x21 þ x22 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : x_ 2 ¼ x1 þ x2 1 x2 þ x2 ; 1 2
ð20Þ
and
y_ 1 ¼ y2 ; y_ 2 ¼ y1 :
ð21Þ
They have the same periodic solution y12 + y22 = x12 + x22 = 1. For convenience, we let the coupling term c(x, y) be
cðx1 ; x2 ; y1 ; y2 Þ ¼
ða1 y2 þ b1 Þðy1 x1 Þ ða2 x1 þ b2 Þðy2 x2 Þ
:
ð22Þ
783
L. Pei, B. Qiu / Applied Mathematical Modelling 36 (2012) 779–791 0.1
0.1
0.05
0.05
0
0
-0.05
-0.05
-0.1
-0.1
0
5
10
15
20
25
30
0
5
10
15
20
25
30
Fig. 1. Time histories of the component x1 and x2 in the coupled system (17) are displayed respectively in Fig. 1(a) and (b), where the parameter values are taken as a1 = 0.2, b1 = 5, a2 = 0.5, b2 = 1, and initial conditions are x1(0) = 0.1, y1(0) = 1, x2(0) = 0.2, y2(0) = 0.3.
0.1
0.1
0.05
0.05
0
0
-0.05
-0.05
-0.1
-0.1
0
5
10
15
20
25
30
0
5
10
15
20
25
30
Fig. 2. Time histories of the synchronization errors e1, e2 in the coupled system (17) are displayed respectively in Fig. 2(a) and (b), where the parameter values are same to those in Fig. 1.
Then the coupled periodic systems are expressed as:
8 > > > x_ 1 > > > < x_ 2 > > > y_ 1 > > > :_ y2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x2 þ x1 1 x21 þ x22 þ ða1 y2 þ b1 Þðy1 x1 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x1 þ x2 1 x21 þ x22 þ ða2 x1 þ b2 Þðy2 x2 Þ;
ð23Þ
¼ y2 þ ða1 x2 þ b1 Þðx1 y1 Þ; ¼ y1 þ ða2 y1 þ b2 Þðx2 y2 Þ:
To make systems (23) synchronous, it is required to appropriately choose value of a1, a2, b1 and b2. Suppose /(t) = (/1(t), /2(t))T is a nontrivial solution of Eqs. (20) and (21), and introduce a matrix
AðtÞ :¼
1 ½ðDgð/Þ þ 2Dx cð/; /Þ þ ðDgð/Þ þ 2Dx cð/; /ÞÞÞT ; 2
where g(y) represents the right hand side of Eq. (21), c(x, y) is defined by Eq. (22), Dg is the Jacobian matrix of g(y), and Dxc(x, y) is the Jacobian matrix of c(x, y) with respect to x. It follows form Eq. (23) that
AðtÞ ¼
2ða1 /2 þ b1 Þ
0
0
2ða2 /1 þ b2 Þ
:
From Theorem 2.1, we see that a sufficient condition of the synchronization stability is that eigenvalues l1(t) and l2(t) of matrix A(t) are uniformly negative in t. This implies
l1 ðtÞ ¼ 2ð/2 a1 þ b1 Þ < 0; and
l2 ðtÞ ¼ 2ða2 /1 þ b2 Þ < 0:
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L. Pei, B. Qiu / Applied Mathematical Modelling 36 (2012) 779–791
0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.35
-0.3 -0.25
-0.2 -0.15
-0.1 -0.05
0
0.05
0.1
Fig. 3. The phase portrait of the coupled system (17) is displayed in Fig. 3, where parameter values are same to those in Fig. 1.
Note that the same periodic solution (/1(t), /2(t))T of the Eqs. (20) and (21) must be bounded. We assume m 6 /1(t), /2(t) 6 m. From the above two inequalities on a1, a2, b1 and b2. We can obtain the following conditions:
b1 > ja1 jm;
ð24Þ
b2 > ja2 jm:
ð25Þ
and
Finally, we conclude the above analysis in the following theorem Theorem 4.1. Suppose (/1(t), /2(t))T is a solution of Eqs. (20) and (21) , satisfying m 6 /1(t), /2(t) 6 m. If conditions (24) and (25) hold, then the nonlinearly non-identical coupled systems Eq. (23) are synchronized in the sense of Eq. (3). Next, we present numerical simulations according to the theoretical results. In the following, Fig. 4 illustrate the time evolution of components x1(t) and y1(t). Obviously, they oscillate around null. Moreover, they have the same frequency and amplitude of the oscillation when time t is sufficiently large. Fig. 5 shows time series of synchronization errors, i.e., e1 = y1 x1 and e2 = y2 x2 via time. From the Fig. 5 we see that these errors tend to zeros as t goes infinity. Fig. 6 is a projected attractor of the coupled systems on the phase space (y1, y2). 4.2. Another coupled periodic oscillators Consider the following systems:
(
x_ 1 ¼ x2 þ x1 1 x21 x22 ; x_ 2 ¼ x1 þ x2 1 x21 x22 ;
ð26Þ
y_ 1 ¼ y2 y21 þ y22 ; 2 y_ 2 ¼ y1 y1 þ y22 :
ð27Þ
and
(
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-1 0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
Fig. 4. Time series of the component x1 and y1 in coupled system (23) are displayed respectively in Fig. 4(a) and (b), where the parameter values are taken as a1 = 0.1, b1 = 3, a2 = 0.2, b2 = 5, and initial conditions are x1(0) = 1, y1(0) = 1, x2(0) = 0.02, y2(0) = 0.2.
785
L. Pei, B. Qiu / Applied Mathematical Modelling 36 (2012) 779–791
0
-0.05
-0.1
-0.15
-0.2 0
5
10
15
20
Fig. 5. The synchronization errors, i.e., e1 (solid line) and e2 (dashed line) in the coupled system (23) are displayed respectively in Fig. 5, where the parameter values are same to those in Fig. 4.
1
0.5
0
-0.5
-1 -1
-0.5
0
0.5
1
Fig. 6. A projected attractor of the coupled systems (23) on phase space (y1, y2) is displayed in Fig. 6, where parameter values are same to those in Fig. 4.
For convenience, we let the coupling term c(x, y) be
cðx1 ; x2 ; y1 ; y2 Þ ¼
ða1 y2 þ b1 Þðy1 x1 Þ 0
;
ð28Þ
and the overall coupled system is the following form:
8 x_ > > > 1 > < x_ 2 > _1 y > > > :_ y2
¼ x2 þ x1 1 x21 x22 þ ða1 y2 þ b1 Þðy1 x1 Þ; ¼ x1 þ x2 1 x21 x22 ; ¼ y2 y21 þ y22 þ ða1 x2 þ b1 Þðx1 y1 Þ; ¼ y1 y21 þ y22 :
ð29Þ
To make systems (29) synchronous, it is required to appropriately choose value of a1, a2, b1 and b2. Suppose /(t) = (/1(t), /2(t))T is a nontrivial solution of Eqs. (26) and (27). It follows from Eq. (29) that
AðtÞ ¼
1 /22 3/21 2ða1 /2 þ b1 Þ
2/1 /2
2/1 /2
1 /21 3/22
!
:
Next, investigate eigenvalues of A(t). For this, consider
l ð1 /2 3/2 2ða / þ b ÞÞ 2/1 /2 1 2
1 2 1 detðlI AÞ ¼
2/1 /2 l ð1 /21 3/22 Þ
¼ l2 þ l /22 þ 3/21 1 þ 2ða1 /2 þ b1 Þ þ /21 þ 3/22 1 þ /22 þ 3/21 1 þ 2ða1 /2 þ b1 Þ /21 þ 3/22 1 4/21 /22 : The eigenvalues of A(t) are all real since A(t) is symmetric. From Theorem 2.1, we know that if all these eigenvalues are uniformly negative in t, then coupled systems (29) have a stable synchronous solution in the sense of Eq. (3). For requirement, we obtain the following conditions:
/22 þ 3/21 1 þ 2ða1 /2 þ b1 Þ þ /21 þ 3/22 1 > 0; 2 /2 þ 3/21 1 þ 2ða1 /2 þ b1 Þ /21 þ 3/22 1 4/21 /22 > 0;
786
L. Pei, B. Qiu / Applied Mathematical Modelling 36 (2012) 779–791
since the solution (/1(t), /2(t))T of the Eqs. (26) and (27) is bounded, and the periodic solution satisfy /21 ðtÞ þ /22 ðtÞ ¼ 1. We assume m 6 /1(t), /2(t) 6 m. From the above two inequalities on a1, a2, b1, and b2. We can obtain the following conditions:
b1 > 1 þ ja1 jm;
ð30Þ
b1 > ja1 jm:
ð31Þ
From the fact that the periodic system is bounded, we see that the above inequalities hold if b1 is large enough. For this, we choose
b1 > ja1 jm:
ð32Þ
Next, we present numerical simulations to verify the theoretical results. Fig. 7 illustrate the time evolution of components x2(t) and y2(t). Obviously, they have the same frequency and amplitude of the oscillation when time t is sufficiently large. Fig. 8 shows time series of errors between corresponding variables in the coupled systems, i.e., e1 = y1 x1 and e2 = y2 x2 via time. From Fig. 8 we see that these errors tend to zeros as t goes infinity. Fig. 9 is a projected attractor of the coupled systems on the phase space (x1, x2). 5. Coupled quasi-periodic systems In this section, as an application of Theorem 2.1 in the case of quasi-periodic synchronization, we investigate synchronization in nonlinearly coupled quasi-periodic systems. Strangely enough, there are the synchronization of periodic solutions, pffiffiffi such as x1 ¼ 1; x2 ¼ 2, and x3 = 1, x4 = 3p. It can be written in the following form:
8 x_ 1 ¼ x2 ; > > > > < x_ ¼ x x x ðx þ x Þx ; 2 1 2 1 1 2 4 > _ ¼ x ; x > 3 4 > > :_ x4 ¼ ðx1 þ x2 Þx2 þ x1 x2 x3 ;
ð33Þ
8 y_ 1 ¼ y2 ; > > > > < y_ ¼ x x y ðx þ x Þy ; 2 3 4 1 3 4 4 > _ ¼ y ; y > 3 4 > > :_ y4 ¼ ðx3 þ x4 Þy2 þ x3 x4 y3 ;
ð34Þ
and
where the general solutions of quasi-periodic systems (33) and (34):
8 x1 > > > > < x2 > > x3 > > : x4
¼ r 1 cosx1 t þ r2 cosx2 t; ¼ r1 x1 sinx1 t r 2 x2 sinx2 t; ¼ r 1 sinx1 t þ r 2 sinx2 t; ¼ r 1 x1 cosx1 t þ r 2 x2 cosx2 t;
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1 0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
Fig. 7. Time histories of the component x2 and y2 in coupled system (29) are displayed respectively in Fig. 7(a) and (b), where the parameter values are taken as a1 = 0.1, b1 = 3, a2 = 0.2, b2 = 5, and initial conditions are x1(0) = 0.01, y1(0) = 1, x2(0) = 0.3, y2(0) = 0.2.
787
L. Pei, B. Qiu / Applied Mathematical Modelling 36 (2012) 779–791 1 0.8 0.6 0.4 0.2 0 0
5
10
15
20
25
30
35
40
45
50
Fig. 8. The synchronization errors, i. e., e1 (solid line) and e2 (dashed line) in the coupled system (29), ei is the opposite of ei, where the parameter values are same to those in Fig. 7.
1
0.5
0
-0.5
-1 -0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Fig. 9. A projected attractor of the coupled systems (29) on phase space (x1, x2) is illustrated in Fig. 9, where parameter values are same to those in Fig. 7.
and
8 y1 > > > y > 3 > : y4
¼ r 01 cosx3 t þ r 02 cosx4 t; ¼ r 01 x3 sinx3 t r02 x4 sinx4 t; ¼ r 01 sinx3 t þ r 02 sinx4 t; ¼ r 01 x3 cosx3 t þ r02 x4 cosx4 t;
where the coupled term c(x, y) is taken as
1 ða1 y4 þ b1 Þðy1 x1 Þ B ða x þ b Þðy x Þ C 2 C B 2 1 2 2 cðx1 ; x2 ; x3 ; x4 ; y1 ; y2 ; y3 ; y4 Þ ¼ B C: @ ða3 y2 þ b3 Þðy3 x3 Þ A 0
ð35Þ
ða4 x3 þ b4 Þðy4 x4 Þ Then the coupled quasi-periodic systems are expressed as
8 x_ 1 > > > > > x_ > > > 2 > > x_ > > > 3 > < x_ 4 > y_ 1 > > > > > y_ 2 > > > > > y_ 3 > > > : y_ 4
¼ x2 þ ða1 y4 þ b1 Þðy1 x1 Þ; ¼ x1 x2 x1 ðx1 þ x2 Þx4 þ ða2 x1 þ b2 Þðy2 x2 Þ; ¼ x4 þ ða3 y2 þ b3 Þðy3 x3 Þ; ¼ ðx1 þ x2 Þx2 þ x1 x2 x3 þ ða4 x3 þ b4 Þðy4 x4 Þ; ¼ y2 þ ða1 x4 þ b1 Þðx1 y1 Þ;
ð36Þ
¼ x3 x4 y1 ðx3 þ x4 Þy4 þ ða2 y1 þ b2 Þðx2 y2 Þ; ¼ y4 þ ða3 x2 þ b3 Þðx3 y3 Þ; ¼ ðx3 þ x4 Þy2 þ x3 x4 y3 þ ða4 y3 þ b4 Þðx4 y4 Þ:
We shall show that the coupled systems with different quasi-periodic solutions are synchronized at the periodic solutions for some a and b. Suppose /(t) = (/1(t), /2(t))T is a nontrivial solution of Eqs. (33) and (34), and we have
AðtÞ :¼
1 ½ðDf ð/Þ þ 2Dx cð/; /ÞÞ þ ðDf ð/Þ þ 2Dx cð/; /ÞÞT : 2
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L. Pei, B. Qiu / Applied Mathematical Modelling 36 (2012) 779–791
By simple calculation we have
0 B B AðtÞ ¼ B B @
2ða1 /4 þ b1 Þ 1 ð1 2
þ x1 x2 Þ
1 ð1 2
þ x1 x2 Þ
2ða2 /1 þ b2 Þ
0
0
0
0
0
0
1
C C C: þ x1 x2 Þ C 2ða3 /2 þ b3 Þ A 1 ð1 þ x1 x2 Þ 2ða1 /4 þ b4 Þ 2 0
0
1 ð1 2
The eigenvalues of A(t) are all real since A(t) is symmetric. From Theorem 2.1, we know that if all these eigenvalues are uniformly negative in t, then coupled systems (36) have a stable synchronous solution in the sense of Eq. (3). By applying the Sylvester’s Criterion-which provides a test for negative definite of a matrix-thus, we have that the mentioned matrix, if we choose ai and bi (i = 1, 2, 3, 4) such that
2ða1 /4 þ b1 Þ < 0; 2ða3 /2 þ b3 Þ < 0; 1 4ða1 /4 þ b1 Þða2 /1 þ b2 Þ ð1 þ x1 x2 Þ2 > 0; 4 1 4ða3 /2 þ b3 Þða4 /3 þ b4 Þ ð1 þ x1 x2 Þ2 > 0; 4 since the solution (/1, /2, /3, /4)T of quasi-periodic systems is bounded, we assume m 6 /i 6 m (i = 1, 2, 3, 4). From the above four inequalities on ai and bi, we can obtain the following conditions:
b1 > ja1 jm;
ð37Þ
b3 > ja3 jm;
ð38Þ
b2 > ja2 jm þ
1 ð1 þ x1 x2 Þ2 ; 16 b1 ja1 jm
ð39Þ
b4 > ja4 jm þ
1 ð1 þ x1 x2 Þ2 ; 16 b3 ja3 jm
ð40Þ
we see that the above conditions hold if ai and bi are large enough. Thus, for the coupled systems (36) we obtain the following theorem: Theorem 5.1. Suppose (/1, /2, /3, /4)T is a solution of quasi-periodic systems (33) and conditions (37)–(40) holds. Then the nonlinearly coupled systems (36) are synchronized in the sense of Eq. (3). Remarks 1. Note that periodic solutions of the quasi-periodic systems (33) and (34) exist, and there is only the synchronization of periodic solutions, instead of quasi-periodic solutions. Actually, suppose the coupled systems (36) are synchronized at the quasi-periodic solutions, without loss of generality, we assume that the quasi-periodic systems (33) and (34) have the same quasi-periodic solution, and set the solution as:
8 x11 > > > > < x22 > x33 > > > : x44
¼ r 11 cosx11 t þ r 22 cosx22 t; ¼ r 11 x11 sinx11 t r 22 x22 sinx22 t; ¼ r 11 sinx11 t þ r22 sinx22 t; ¼ r 11 x11 cosx11 t þ r 22 x22 cosx22 t:
If we substitute the solution in systems (33) and (34), then we have the following equalities hold:
x_ 22 ¼ r 11 x211 cosx11 t r 22 x222 cosx22 t ¼ x1 x2 ðr 11 cosx11 t þ r 22 cosx22 tÞ ðx1 þ x2 Þðr11 x11 cosx11 t þ r 22 x22 cosx22 tÞ ¼ x3 x4 ðr 11 cosx11 t þ r 22 cosx22 tÞ ðx3 þ x4 Þðr11 x11 cosx11 t þ r 22 x22 cosx22 tÞ; we can obtain the following forms:
x1 x2 r11 ðx1 þ x2 Þr11 x11 ¼ x3 x4 r11 ðx3 þ x4 Þr11 x11 ; x1 x2 r22 ðx1 þ x2 Þr22 x22 ¼ x3 x4 r22 ðx3 þ x4 Þr22 x22 ;
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1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
Fig. 10. Time histories of the components x1 and x3 in the coupled systems (36) are displayed respectively in Fig. 10(a) and (b), where the parameter values are taken as a1 = 0.3, b1 = 5, a2 = 0.5, b2 = 12, a3 = 0.2, b3 = 6, a4 = 0.1, b4 = 8, and initial conditions are x1(0) = 0.01, y1(0) = 1, x2(0) = 0.2, y2(0) = 0.3, x3(0) = 0.03, y3(0) = 2, x4(0) = 0.2, y4(0) = 0.1.
i.e., we have
x11 ¼ x22 ¼
x1 x2 x3 x4 : x1 þ x2 x3 x4
Hence,we have proved the above results. 2. When x1, x2, x3 and x4 is different pffiffiffi from each other, there pffiffiffi exist the transient synchronization of quasi-periodic solutions, such as x1 ¼ 1; x2 ¼ 2 and x3 ¼ 0:5; x4 ¼ 3. Whereas the synchronization of quasi-periodic solutions disappears, when t is large enough. Thereafter, the synchronization does not exists. Finally, we present numerical simulations to verify the theoretical results. Fig. 10(a) and (b) illustrate the time series of components x1(t) and x3(t). Obviously, they have the same frequency and amplitude of the oscillation when time t is sufficiently large. Fig. 11 shows time series of errors between corresponding variables in the coupled systems, i.e., (a) e1 = y1 x1, 0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3 0
5
10
15
20
0
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
0
5
10
15
20
0
5
10
15
5
10
15
20
20
Fig. 11. Synchronization errors e1, e2, e3 and e4 in the coupled system (36) are displayed respectively in Fig. 11(a)–(d), where the parameter values are same to those in Fig. 10.
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L. Pei, B. Qiu / Applied Mathematical Modelling 36 (2012) 779–791 1.5
2
1
1.5 1
0.5
0.5 0 0 -0.5 -0.5 -1
-1 -1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Fig. 12. The projected attractors of the coupled systems (36) on phase space (x1, x3) and (y1, y3) are displayed respectively in Fig. 12(a) and (b), where parameter values are same to those in Fig. 10.
(b) e2 = y2 x2, (c) e3 = y3 x3 and (d) e4 = y4 x4 via time. From Fig. 11 we see that these errors tend to zeros as t goes infinity. Fig. 12(a) and (b) are two projected attractor of the coupled systems on the phase space (x1, x3) and (y1, y3). 6. Conclusion In this paper, we introduce a theoretical approach, i.e., mode decomposition, to treat the synchronization problem of nonidentical coupled dynamical systems. For such a coupled system, we first propose a mode decomposition which decomposes the linearized equations in a synchronous state into motion in two directions, i.e., the transverse direction and the parallel direction. Then, we derive by such a decomposition a rigorous sufficient condition which guarantees, if satisfied, that the overall coupled systems reach the desired synchronization. More importantly, such a decomposition has geometrical and theoretical explanations. Geometrically, the stability of a synchronous manifold does not depend upon the tangent motion whereas the transverse motion determines its stability. Theoretically, a clear explanation may be given by virtue of similar Floquet multiplier theory. Finally, we present several numerical examples to verify the theoretical results. The results are obtained as the following: First, the synchronization of the non-identical dynamical systems on the non-chaotic oscillators is reached by employing the mode decomposition approach in this paper. At the best knowledge of the authors, this approach rarely even never is applied to the synchronization of the non-identical dynamical systems. And our work is the generalization of Xiong [18], Zhou [19,20] to the case of the non-identical dynamical systems’ synchronization. The other result is the non-identical dynamical systems on quasi-periodic oscillators will synchronize at the periodic state but not at the quasi-periodic state, although the transient quasi-periodic synchronization will appear. We especially emphasize that the synchronization problem of identical and non-identical coupled systems can be treated in a united framework according to the above proposed theory. The synchronization of the non-identical dynamical systems on chaotic oscillators by the mode decomposition approach will be our direct next work. Since the dynamical systems on the non-chaotic oscillators are the specially constructed cases but not the general, the synchronization problem of the non-identical dynamical systems on the all general kinds of oscillators will be the part of our further work. It is very interesting to study the generalized synchronization of the identical and non-identical dynamical systems by the mode decomposition approach. Because of the ubiquity of the time delay, the synchronization of the delayed dynamical systems employing the mode decomposition approach is also very interesting and important. The above two problems are also the other part of our further work. Acknowledgments The authors acknowledge the financial support from the National Natural Science Foundation of China under Grant No. 10702065. They also are grateful of the reviewers for their valuable and helpful suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
D. McMillen et al, Synchronizing genetic relaxation oscillators by intercell signaling, PNAS 99 (2002) 679–684. L. Chen et al, Noise-induced cooperative behavior in a multi-cell system, Bioinformatics 21 (2005) 2722–2729. A.T. Winfree, The Geometry of Biological Time, Springer-Verlag, New York, 1980. J.J. Collins, I.N. Steward, Coupled nonlinear oscillation and the symmetries of animal gaits, J. Nonlinear Sci. 3 (1993) 349–392. M.S. El Naschie, On turbulence and complex dynamic in a four-dimensional Peano–Hilbert space, J. Franklin Inst. 330 (1993) 183–198. M.S. El Naschie, Multi-dimensional Cantor sets in classical and quantum mechanics, Chaos Soliton. Fract. 2 (1992) 211–220. M.S. El Naschie, Complex dynamic in a 4D Peano–Hilbert space, Nuovo. Cimento. B 107 (1992) 583–594. M.S. El Naschie, Average symmetry stability and ergodicity of multidimensional Cantor sets, Nuovo. Cimento. B 109 (1994) 149–157. R. Ramaswamy, Synchronization of strange nonchaotic attractors, Phys. Rev. E 56 (1997) 7294–7296. D.S.V. Maria et al, Synchronization of regular and chaotic systems, Phys. Rev. E 46 (1992) 7359–7362.
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[11] H. Rong, P.G. Vaidya, Analysis and synthesis of synchronous periodic and chaotic system, Phys. Rev. E 46 (1992) 7387–7392. [12] P. Woafo, R.A. Kraenkel, Synchronization: stability and duration time, Phys. Rev. E 65 (2002) 036225–036230. [13] R. Yamapi, O.J.B. Chabi, Synchronization of the regular and chaotic states of electromechanical devices with and without delay, Int. J. Bifur. Chaos 14 (2004) 171–181. [14] A.A. Andronov et al, Theory of Oscillators, Pergamon, Oxford, 1996. [15] S.H. Stogatz, Nonlinear Dynamics and Chaos, Reading, Addison-Wesley, MA, 1994. [16] N.F. Rulkov et al, Mutual synchronization of chaotic self-oscillators with dissipative coupling, Int. J. Bifur. Chaos 2 (1992) 669–676. [17] C.W. Chua, L.O. Chua, A unified framework for synchronization and control of dynamical systems, Int. J. Bifur. Chaos 4 (1994) 979–998. [18] X.H. Xiong et al, Mode decomposition for a synchronous state and its applications, Chaos Soliton. Fract. 31 (2007) 718–725. [19] T.S. Zhou et al, Synchronization stability of three chaotic systems with linear coupling, Phys. Lett. A 301 (2002) 231–240. [20] T.S. Zhou, Stability of echo waves in linearly couple oregonators, Phys. Lett. A 320 (2003) 116–126.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Application of mode decomposition approach to the synchronization of the non-identical dynamical systems q Lijun Pei ⇑, Benhua Qiu Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450001, PR China
a r t i c l e
i n f o
Article history: Received 7 October 2008 Received in revised form 26 May 2011 Accepted 1 July 2011 Available online 22 July 2011 Keywords: Synchronization Mode decomposition Non-identical dynamical systems
a b s t r a c t The synchronization problem of two different dynamical systems is considered by employing mode decomposition approach in this paper. Synchronization of non-identical coupled dynamical systems with non-chaotic attractors, i.e., equilibria, periodic and quasi-periodic solutions, is investigated analytically and numerically. Some results are obtained by this method. Some examples, supported by numerical simulation, are presented to illustrate the conciseness and effectiveness of the approach. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Synchronization in coupled dynamical systems is a subject of many theoretical papers over the last few years [1–4]. Our granted Project (NNSFC No. 10702065) is titled by Study on Synchronization Mechanism of the delayed coupling dynamical systems. It aims to find the mechanism and laws of the synchronization of the delayed coupling dynamical systems. It includes mainly three problems, i.e., the synchronization approach, stability and design of the synchronization schemes, the classification and switch of the generalized synchronization, the effect of time delay on the synchronization of the coupled dynamical systems. The considered approach in this paper, mode decomposition approach, is one of the main synchronization approaches including the generalized Hamiltonian systems and observer approach, parameters update law, exponential approach, inertial manifold approach, etc. Non-chaotic (periodic or quasi-periodic) synchronization of dynamic systems was explored by El Naschie (see Refs. [5–8]) as well as Ramaswamy et al. (see Refs. [9–11]). Communications security and engineering were put into effect in different ways [12,13], revealing that synchronization of systems with periodic dynamics is an important and well-known effect in the fields of physics, engineering as well as in other branches of sciences [14,15]. And the synchronization of systems with quasi-periodic or aperiodic dynamics, i.e., the synchronization of strange non-chaotic attractors, is also a possible scheme of communications security and can offer some advantages [9]. Thus synchronization of non-chaotic attractors should be also investigated but not only chaotic synchronization. In this paper, synchronization of non-chaotic attractors including equilibria, periodic and quasi-periodic attractors is considered. When identical systems are coupled, synchronization is the most obvious effect. However, experimental and even more real systems are often not fully identical, especially there are mismatches in parameters of the systems. It is thus important and also interesting to investigate synchronization behavior between non-identical systems. In general, it is difficult to achieve the synchronization for the master–salve chaotic systems with non-identical parameters, so the synchronization
q
This work is supported by National Natural Science Foundation of China (No. 10702065).
⇑ Corresponding author.
E-mail address: [email protected] (L. Pei). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.07.006
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of coupled dynamical systems with periodic and quasi-periodic solutions is discussed in this paper. We have known the stability of synchronized motion usually means the stability of the complete synchronization manifold, i.e., x = y, or the stability of the zero solution of the synchronization error. When we deal with non-identical coupled systems, similar stability criteria can be formulated. In most case, most of rigorous theoretical results of synchronization are obtained by constructing Lyapunov function [16,17]. Unfortunately, this method can be applied only to some particular examples where the coupling matrix is generally required to be negative definite. However, a novel method, mode decomposition [18–20], of treating the stability of a synchronous state is proposed based on the Floquet theory. A rigorous criterion is then derived, which can also be applied to non-identical coupled systems. The paper is structured as follows. Firstly, we state the problem of the stability of systems’ synchronization, and then transform the stability of synchronization solution into a lower-dimensional problem by using mode decomposition method. In Section 2, we give the general analysis for non-identical coupled systems. In Sections 3–5, several typical examples are given to demonstrate the accuracy of theoretical analysis. At last, we give some numerical simulations to illustrate our findings and some useful conclusions. 2. The general analysis First we introduce the mode decomposition approach [19,20]. Consider two non-identical systems which are coupled in the following form:
(
dx dt dy dt
¼ f ðxÞ þ cðx; yÞ;
ð1Þ
¼ gðyÞ þ cðy; xÞ;
where x, y 2 Rn, and c(x, y) is smooth. The coupling term c(x, y) implies the coupling between both of the non-identical systems. Thus their couplings in system (1) are the bidirectional and same. In this paper these coupling terms c(x, y) and c(y, x) take the usual forms in [19,20]. Generally, synchronization occurs on an invariant manifold given by x = y = /(t), where /(t) is a nontrivial solution of the uncoupled system ( dx dt dy dt
¼ f ðxÞ;
ð2Þ
¼ gðyÞ:
Then, we have f(/) = g(/). The stability problem of the synchronization of the coupled systems can be formulated in a very general way by addressing the question of the stability of the CS synchronization manifold x = y, or equivalently by studying the temporal evolution of the synchronization error e = y x, then Eq. (1) lead to a certain linearized equation in e. This implies that the synchronization problem for a coupled system is equivalent to the stability of a zero solution of the linearized equation. The synchronization manifold is linearly stable if
lim keðtÞk ¼ lim kxðtÞ yðtÞk ¼ 0:
t!þ1
ð3Þ
t!þ1
To determine the behavior of e(t) in this limit, we consider a variational equation of Eq. (1) evaluated along the synchronous manifold (/(t), /(t))
e_ 1 e_ 2
¼
Df ð/Þ þ Dx cð/; /Þ
Dy cð/; /Þ
Dy cð/; /Þ
Dgð/Þ þ Dx cð/; /Þ
e1
e2
;
ð4Þ
where Df(/), Dg(/) is the Jacobian matrix of f(x), g(x) evaluated at /(t) whereas Dxc(/, /) and Dyc(/, /) are Jacobian matrices of c(x, y) evaluated at (/(t), /(t))T with respect to x and y, respectively. Furthermore, we introduce a mode decomposition to simplify Eq. (4). For clarity, let
P¼
I
I
I
I
;
ð5Þ
where I is an n n identical matrix. The inverse matrix of P is
P1 ¼
1 2
I
I
I
I
:
ð6Þ
:
ð7Þ
Let
e1
e2
¼P
w1 w2
It follows from Eqs. (5)–(7) that
_1 w _2 w
¼ P1
e_ 1 e_ 2
¼
Gð/Þ
0
0
Hð/Þ
w1 w2
;
ð8Þ
L. Pei, B. Qiu / Applied Mathematical Modelling 36 (2012) 779–791
781
where
f ð/Þ ¼ gð/Þ;
Gð/Þ ¼ Df ð/Þ þ Dx cð/; /Þ þ Dy cð/; /Þ;
Hð/Þ ¼ Df ð/Þ þ Dx cð/;
/Þ Dy cð/; /Þ:
Thus, the variational equation (4) is decomposed into two independent systems:
_ 1 ¼ ½Df ð/Þ þ Dx cð/; /Þ þ Dy cð/; /Þw1 ; w
ð9Þ
_ 2 ¼ ½Df ð/Þ þ Dx cð/; /Þ Dy cð/; /Þw2 : w
ð10Þ
and There are two special cases: Case 1: If Eq. (1) satisfy the conservative condition c(x, x) = 0 or c(x, x) c (constant), then Dxc(/, /) + Dyc(/, /) = 0, namely, Eqs. (9) and (10) can be written as
_ 1 ¼ Df ð/Þx1 ; w
ð11Þ
and
_ 2 ¼ ½Df ð/Þ þ 2Dx cð/; /Þx2 : w
ð12Þ
Case 2: If c(x, y) of Eq. (1) is symmetric with respect to x and y, then Dxc(/, /) = Dyc(/, /), Eqs. (9) and (10) can also be written as the similar form of (11) and (12). Such a decomposition has geometrical and theoretical explanations. Geometrically, the stability of a synchronous manifold does not depend upon the tangent motion whereas the transverse motion determines its stability. Theoretically, a clear explanation may be given by virtue of similar Floquet multiplier theory. According to the above decomposition and based on an extended Floquet theory, to make the considered synchronous state be stable, we impose a sufficient condition [18–20], that is, we force all the real parts of eigenvalues of matrix Df(/) + 2Dxc(/, /) to be uniformly negative in t, i.e., we require all eigenvalues of matrix
AðtÞ ¼
1 ½ðDf ð/Þ þ 2Dx cð/; /ÞÞ þ ðDf ð/Þ þ 2Dx cð/; /ÞÞT 2
ð13Þ
to be uniformly negative, this easily leads to the following theorem: Theorem 2.1. Suppose f(x) 2 C1(Rn), g(x) 2 C1(Rn) and c(x, y) 2 C1(Rn,Rn). Denote by (/1(t), /2(t))T a solution of Eq. (1). If the eigenvalues of matrix (13) are all uniformly negative in t, then /1(t) and /2(t) are synchronized in the sense of Eq. (3). Note that the conditions in Theorem 2.1 only suppose that c(x, y) is smooth. Especially in the linear coupling case, the corresponding condition on synchronization will become simpler. If f(x) = g(y), then systems (1) can be transformed into the form of identical systems.
3. Application to the equilibria’s synchronization First the simple case that the synchronization of the non-identical dynamical systems is considered. At the same time, the synchronized systems are asymptotically stable, i.e., they have the same asymptotically stable equilibria. This case of synchronization is called as the equilibria’s synchronization. Consider the following systems:
x_ 1 ¼ x1 ; x_ 2 ¼ x2 ;
ð14Þ
y_ 1 ¼ y2 ; y_ 2 ¼ 2y1 3y2 ;
ð15Þ
and
we set the coupling term c(x, y) as
cðx1 ; x2 ; y1 ; y2 Þ ¼
ða1 y2 þ b1 Þðy1 x1 Þ ða2 x1 þ b2 Þðy2 x2 Þ
;
ð16Þ
and the overall coupled system is then of the form:
8 x_ 1 ¼ x1 þ ða1 y2 þ b1 Þðy1 x1 Þ; > > > < x_ ¼ x þ ða x þ b Þðy x Þ; 2 2 2 1 2 2 2 > _ 1 ¼ y2 þ ða1 x2 þ b1 Þðx1 y1 Þ; y > > : y_ 2 ¼ 2y1 3y2 þ ða2 y1 þ b2 Þðx2 y2 Þ:
ð17Þ
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To make systems (17) synchronous, it is required to appropriately choose value of a1, a2, b1 and b2. Suppose / = (/1, /2)T is a equilibrium of Eqs. (14) and (15), and introduce a matrix
AðtÞ :¼
1 ½ðDf ð/Þ þ 2Dx cð/; /Þ þ ðDf ð/Þ þ 2Dx cð/; /ÞÞÞT ; 2
where f(x) represents the right hand side of Eq. (14), c(x, y) is defined by Eq. (16), Df is the Jacobian matrix of f(x), and Dxc(x, y) is the Jacobian matrix of c(x, y) with respect to x. It follows form Eq. (17) that
Df ð/Þ þ 2Dx cð/; /Þ ¼
1 2ða1 /2 þ b1 Þ 0 ; 0 1 2ða2 /1 þ b2 Þ
by simple calculation we have
AðtÞ ¼
1 2ða1 /2 þ b1 Þ
0
0
1 2ða2 /1 þ b2 Þ
:
From Theorem 2.1, we see that a sufficient condition on the synchrony stability is that eigenvalues l1(t) and l2(t) of matrix A(t) are uniformly negative in t. This implies
l1 ðtÞ ¼ 1 2ð/2 a1 þ b1 Þ < 0; and
l2 ðtÞ ¼ 1 2ða2 /1 þ b2 Þ < 0: From the above two inequalities on a1, a2, b1 and b2, we can obtain the following conditions:
1 b1 > /2 a1 ; 2
ð18Þ
1 b2 > /1 a2 : 2
ð19Þ
and
Here the equilibrium point is (0, 0), thus we have b1 > 12 ; b2 > 12. To satisfy the above conditions, the parameter can be selected as b1 = 5, b2 = 1. Theorem 3.1. Suppose (/1, /2)T is a solution of systems (14) and (15) . If conditions (18) and (19) hold, then the nonlinearly nonidentical coupled systems (17) are synchronized in the sense of expression (3). Remark 1. The equilibria’s synchronization is the simplest synchronization. The other synchronization such as the periodic, quasi-periodic and chaotic synchronization are the problems that are the most important and interesting. Some synchronization will be considered in the following sections. Next, we present numerical simulations according to the theoretical results. In the following, Fig. 1 illustrate the time evolution of components x1(t) and x2(t). Fig. 2 shows time series of synchronization errors, i.e., e1 = y1 x1 and e2 = y2 x2 via time. From the Fig. 2 we see that these errors tend to zeros as t goes infinity. Fig. 3 The phase portrait of the coupled system (17) is displayed.
4. Application to the periodic synchronization 4.1. Coupled periodic oscillators In this section we take the following systems as an example of the periodic synchronization, and show the above theoretical analysis. The systems can be mathematically expressed as:
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < x_ 1 ¼ x2 þ x1 1 x21 þ x22 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : x_ 2 ¼ x1 þ x2 1 x2 þ x2 ; 1 2
ð20Þ
and
y_ 1 ¼ y2 ; y_ 2 ¼ y1 :
ð21Þ
They have the same periodic solution y12 + y22 = x12 + x22 = 1. For convenience, we let the coupling term c(x, y) be
cðx1 ; x2 ; y1 ; y2 Þ ¼
ða1 y2 þ b1 Þðy1 x1 Þ ða2 x1 þ b2 Þðy2 x2 Þ
:
ð22Þ
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25
30
Fig. 1. Time histories of the component x1 and x2 in the coupled system (17) are displayed respectively in Fig. 1(a) and (b), where the parameter values are taken as a1 = 0.2, b1 = 5, a2 = 0.5, b2 = 1, and initial conditions are x1(0) = 0.1, y1(0) = 1, x2(0) = 0.2, y2(0) = 0.3.
0.1
0.1
0.05
0.05
0
0
-0.05
-0.05
-0.1
-0.1
0
5
10
15
20
25
30
0
5
10
15
20
25
30
Fig. 2. Time histories of the synchronization errors e1, e2 in the coupled system (17) are displayed respectively in Fig. 2(a) and (b), where the parameter values are same to those in Fig. 1.
Then the coupled periodic systems are expressed as:
8 > > > x_ 1 > > > < x_ 2 > > > y_ 1 > > > :_ y2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x2 þ x1 1 x21 þ x22 þ ða1 y2 þ b1 Þðy1 x1 Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ x1 þ x2 1 x21 þ x22 þ ða2 x1 þ b2 Þðy2 x2 Þ;
ð23Þ
¼ y2 þ ða1 x2 þ b1 Þðx1 y1 Þ; ¼ y1 þ ða2 y1 þ b2 Þðx2 y2 Þ:
To make systems (23) synchronous, it is required to appropriately choose value of a1, a2, b1 and b2. Suppose /(t) = (/1(t), /2(t))T is a nontrivial solution of Eqs. (20) and (21), and introduce a matrix
AðtÞ :¼
1 ½ðDgð/Þ þ 2Dx cð/; /Þ þ ðDgð/Þ þ 2Dx cð/; /ÞÞÞT ; 2
where g(y) represents the right hand side of Eq. (21), c(x, y) is defined by Eq. (22), Dg is the Jacobian matrix of g(y), and Dxc(x, y) is the Jacobian matrix of c(x, y) with respect to x. It follows form Eq. (23) that
AðtÞ ¼
2ða1 /2 þ b1 Þ
0
0
2ða2 /1 þ b2 Þ
:
From Theorem 2.1, we see that a sufficient condition of the synchronization stability is that eigenvalues l1(t) and l2(t) of matrix A(t) are uniformly negative in t. This implies
l1 ðtÞ ¼ 2ð/2 a1 þ b1 Þ < 0; and
l2 ðtÞ ¼ 2ða2 /1 þ b2 Þ < 0:
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0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.35
-0.3 -0.25
-0.2 -0.15
-0.1 -0.05
0
0.05
0.1
Fig. 3. The phase portrait of the coupled system (17) is displayed in Fig. 3, where parameter values are same to those in Fig. 1.
Note that the same periodic solution (/1(t), /2(t))T of the Eqs. (20) and (21) must be bounded. We assume m 6 /1(t), /2(t) 6 m. From the above two inequalities on a1, a2, b1 and b2. We can obtain the following conditions:
b1 > ja1 jm;
ð24Þ
b2 > ja2 jm:
ð25Þ
and
Finally, we conclude the above analysis in the following theorem Theorem 4.1. Suppose (/1(t), /2(t))T is a solution of Eqs. (20) and (21) , satisfying m 6 /1(t), /2(t) 6 m. If conditions (24) and (25) hold, then the nonlinearly non-identical coupled systems Eq. (23) are synchronized in the sense of Eq. (3). Next, we present numerical simulations according to the theoretical results. In the following, Fig. 4 illustrate the time evolution of components x1(t) and y1(t). Obviously, they oscillate around null. Moreover, they have the same frequency and amplitude of the oscillation when time t is sufficiently large. Fig. 5 shows time series of synchronization errors, i.e., e1 = y1 x1 and e2 = y2 x2 via time. From the Fig. 5 we see that these errors tend to zeros as t goes infinity. Fig. 6 is a projected attractor of the coupled systems on the phase space (y1, y2). 4.2. Another coupled periodic oscillators Consider the following systems:
(
x_ 1 ¼ x2 þ x1 1 x21 x22 ; x_ 2 ¼ x1 þ x2 1 x21 x22 ;
ð26Þ
y_ 1 ¼ y2 y21 þ y22 ; 2 y_ 2 ¼ y1 y1 þ y22 :
ð27Þ
and
(
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-1 0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
Fig. 4. Time series of the component x1 and y1 in coupled system (23) are displayed respectively in Fig. 4(a) and (b), where the parameter values are taken as a1 = 0.1, b1 = 3, a2 = 0.2, b2 = 5, and initial conditions are x1(0) = 1, y1(0) = 1, x2(0) = 0.02, y2(0) = 0.2.
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0
-0.05
-0.1
-0.15
-0.2 0
5
10
15
20
Fig. 5. The synchronization errors, i.e., e1 (solid line) and e2 (dashed line) in the coupled system (23) are displayed respectively in Fig. 5, where the parameter values are same to those in Fig. 4.
1
0.5
0
-0.5
-1 -1
-0.5
0
0.5
1
Fig. 6. A projected attractor of the coupled systems (23) on phase space (y1, y2) is displayed in Fig. 6, where parameter values are same to those in Fig. 4.
For convenience, we let the coupling term c(x, y) be
cðx1 ; x2 ; y1 ; y2 Þ ¼
ða1 y2 þ b1 Þðy1 x1 Þ 0
;
ð28Þ
and the overall coupled system is the following form:
8 x_ > > > 1 > < x_ 2 > _1 y > > > :_ y2
¼ x2 þ x1 1 x21 x22 þ ða1 y2 þ b1 Þðy1 x1 Þ; ¼ x1 þ x2 1 x21 x22 ; ¼ y2 y21 þ y22 þ ða1 x2 þ b1 Þðx1 y1 Þ; ¼ y1 y21 þ y22 :
ð29Þ
To make systems (29) synchronous, it is required to appropriately choose value of a1, a2, b1 and b2. Suppose /(t) = (/1(t), /2(t))T is a nontrivial solution of Eqs. (26) and (27). It follows from Eq. (29) that
AðtÞ ¼
1 /22 3/21 2ða1 /2 þ b1 Þ
2/1 /2
2/1 /2
1 /21 3/22
!
:
Next, investigate eigenvalues of A(t). For this, consider
l ð1 /2 3/2 2ða / þ b ÞÞ 2/1 /2 1 2
1 2 1 detðlI AÞ ¼
2/1 /2 l ð1 /21 3/22 Þ
¼ l2 þ l /22 þ 3/21 1 þ 2ða1 /2 þ b1 Þ þ /21 þ 3/22 1 þ /22 þ 3/21 1 þ 2ða1 /2 þ b1 Þ /21 þ 3/22 1 4/21 /22 : The eigenvalues of A(t) are all real since A(t) is symmetric. From Theorem 2.1, we know that if all these eigenvalues are uniformly negative in t, then coupled systems (29) have a stable synchronous solution in the sense of Eq. (3). For requirement, we obtain the following conditions:
/22 þ 3/21 1 þ 2ða1 /2 þ b1 Þ þ /21 þ 3/22 1 > 0; 2 /2 þ 3/21 1 þ 2ða1 /2 þ b1 Þ /21 þ 3/22 1 4/21 /22 > 0;
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since the solution (/1(t), /2(t))T of the Eqs. (26) and (27) is bounded, and the periodic solution satisfy /21 ðtÞ þ /22 ðtÞ ¼ 1. We assume m 6 /1(t), /2(t) 6 m. From the above two inequalities on a1, a2, b1, and b2. We can obtain the following conditions:
b1 > 1 þ ja1 jm;
ð30Þ
b1 > ja1 jm:
ð31Þ
From the fact that the periodic system is bounded, we see that the above inequalities hold if b1 is large enough. For this, we choose
b1 > ja1 jm:
ð32Þ
Next, we present numerical simulations to verify the theoretical results. Fig. 7 illustrate the time evolution of components x2(t) and y2(t). Obviously, they have the same frequency and amplitude of the oscillation when time t is sufficiently large. Fig. 8 shows time series of errors between corresponding variables in the coupled systems, i.e., e1 = y1 x1 and e2 = y2 x2 via time. From Fig. 8 we see that these errors tend to zeros as t goes infinity. Fig. 9 is a projected attractor of the coupled systems on the phase space (x1, x2). 5. Coupled quasi-periodic systems In this section, as an application of Theorem 2.1 in the case of quasi-periodic synchronization, we investigate synchronization in nonlinearly coupled quasi-periodic systems. Strangely enough, there are the synchronization of periodic solutions, pffiffiffi such as x1 ¼ 1; x2 ¼ 2, and x3 = 1, x4 = 3p. It can be written in the following form:
8 x_ 1 ¼ x2 ; > > > > < x_ ¼ x x x ðx þ x Þx ; 2 1 2 1 1 2 4 > _ ¼ x ; x > 3 4 > > :_ x4 ¼ ðx1 þ x2 Þx2 þ x1 x2 x3 ;
ð33Þ
8 y_ 1 ¼ y2 ; > > > > < y_ ¼ x x y ðx þ x Þy ; 2 3 4 1 3 4 4 > _ ¼ y ; y > 3 4 > > :_ y4 ¼ ðx3 þ x4 Þy2 þ x3 x4 y3 ;
ð34Þ
and
where the general solutions of quasi-periodic systems (33) and (34):
8 x1 > > > > < x2 > > x3 > > : x4
¼ r 1 cosx1 t þ r2 cosx2 t; ¼ r1 x1 sinx1 t r 2 x2 sinx2 t; ¼ r 1 sinx1 t þ r 2 sinx2 t; ¼ r 1 x1 cosx1 t þ r 2 x2 cosx2 t;
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1 0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
Fig. 7. Time histories of the component x2 and y2 in coupled system (29) are displayed respectively in Fig. 7(a) and (b), where the parameter values are taken as a1 = 0.1, b1 = 3, a2 = 0.2, b2 = 5, and initial conditions are x1(0) = 0.01, y1(0) = 1, x2(0) = 0.3, y2(0) = 0.2.
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5
10
15
20
25
30
35
40
45
50
Fig. 8. The synchronization errors, i. e., e1 (solid line) and e2 (dashed line) in the coupled system (29), ei is the opposite of ei, where the parameter values are same to those in Fig. 7.
1
0.5
0
-0.5
-1 -0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Fig. 9. A projected attractor of the coupled systems (29) on phase space (x1, x2) is illustrated in Fig. 9, where parameter values are same to those in Fig. 7.
and
8 y1 > > >
¼ r 01 cosx3 t þ r 02 cosx4 t; ¼ r 01 x3 sinx3 t r02 x4 sinx4 t; ¼ r 01 sinx3 t þ r 02 sinx4 t; ¼ r 01 x3 cosx3 t þ r02 x4 cosx4 t;
where the coupled term c(x, y) is taken as
1 ða1 y4 þ b1 Þðy1 x1 Þ B ða x þ b Þðy x Þ C 2 C B 2 1 2 2 cðx1 ; x2 ; x3 ; x4 ; y1 ; y2 ; y3 ; y4 Þ ¼ B C: @ ða3 y2 þ b3 Þðy3 x3 Þ A 0
ð35Þ
ða4 x3 þ b4 Þðy4 x4 Þ Then the coupled quasi-periodic systems are expressed as
8 x_ 1 > > > > > x_ > > > 2 > > x_ > > > 3 > < x_ 4 > y_ 1 > > > > > y_ 2 > > > > > y_ 3 > > > : y_ 4
¼ x2 þ ða1 y4 þ b1 Þðy1 x1 Þ; ¼ x1 x2 x1 ðx1 þ x2 Þx4 þ ða2 x1 þ b2 Þðy2 x2 Þ; ¼ x4 þ ða3 y2 þ b3 Þðy3 x3 Þ; ¼ ðx1 þ x2 Þx2 þ x1 x2 x3 þ ða4 x3 þ b4 Þðy4 x4 Þ; ¼ y2 þ ða1 x4 þ b1 Þðx1 y1 Þ;
ð36Þ
¼ x3 x4 y1 ðx3 þ x4 Þy4 þ ða2 y1 þ b2 Þðx2 y2 Þ; ¼ y4 þ ða3 x2 þ b3 Þðx3 y3 Þ; ¼ ðx3 þ x4 Þy2 þ x3 x4 y3 þ ða4 y3 þ b4 Þðx4 y4 Þ:
We shall show that the coupled systems with different quasi-periodic solutions are synchronized at the periodic solutions for some a and b. Suppose /(t) = (/1(t), /2(t))T is a nontrivial solution of Eqs. (33) and (34), and we have
AðtÞ :¼
1 ½ðDf ð/Þ þ 2Dx cð/; /ÞÞ þ ðDf ð/Þ þ 2Dx cð/; /ÞÞT : 2
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By simple calculation we have
0 B B AðtÞ ¼ B B @
2ða1 /4 þ b1 Þ 1 ð1 2
þ x1 x2 Þ
1 ð1 2
þ x1 x2 Þ
2ða2 /1 þ b2 Þ
0
0
0
0
0
0
1
C C C: þ x1 x2 Þ C 2ða3 /2 þ b3 Þ A 1 ð1 þ x1 x2 Þ 2ða1 /4 þ b4 Þ 2 0
0
1 ð1 2
The eigenvalues of A(t) are all real since A(t) is symmetric. From Theorem 2.1, we know that if all these eigenvalues are uniformly negative in t, then coupled systems (36) have a stable synchronous solution in the sense of Eq. (3). By applying the Sylvester’s Criterion-which provides a test for negative definite of a matrix-thus, we have that the mentioned matrix, if we choose ai and bi (i = 1, 2, 3, 4) such that
2ða1 /4 þ b1 Þ < 0; 2ða3 /2 þ b3 Þ < 0; 1 4ða1 /4 þ b1 Þða2 /1 þ b2 Þ ð1 þ x1 x2 Þ2 > 0; 4 1 4ða3 /2 þ b3 Þða4 /3 þ b4 Þ ð1 þ x1 x2 Þ2 > 0; 4 since the solution (/1, /2, /3, /4)T of quasi-periodic systems is bounded, we assume m 6 /i 6 m (i = 1, 2, 3, 4). From the above four inequalities on ai and bi, we can obtain the following conditions:
b1 > ja1 jm;
ð37Þ
b3 > ja3 jm;
ð38Þ
b2 > ja2 jm þ
1 ð1 þ x1 x2 Þ2 ; 16 b1 ja1 jm
ð39Þ
b4 > ja4 jm þ
1 ð1 þ x1 x2 Þ2 ; 16 b3 ja3 jm
ð40Þ
we see that the above conditions hold if ai and bi are large enough. Thus, for the coupled systems (36) we obtain the following theorem: Theorem 5.1. Suppose (/1, /2, /3, /4)T is a solution of quasi-periodic systems (33) and conditions (37)–(40) holds. Then the nonlinearly coupled systems (36) are synchronized in the sense of Eq. (3). Remarks 1. Note that periodic solutions of the quasi-periodic systems (33) and (34) exist, and there is only the synchronization of periodic solutions, instead of quasi-periodic solutions. Actually, suppose the coupled systems (36) are synchronized at the quasi-periodic solutions, without loss of generality, we assume that the quasi-periodic systems (33) and (34) have the same quasi-periodic solution, and set the solution as:
8 x11 > > > > < x22 > x33 > > > : x44
¼ r 11 cosx11 t þ r 22 cosx22 t; ¼ r 11 x11 sinx11 t r 22 x22 sinx22 t; ¼ r 11 sinx11 t þ r22 sinx22 t; ¼ r 11 x11 cosx11 t þ r 22 x22 cosx22 t:
If we substitute the solution in systems (33) and (34), then we have the following equalities hold:
x_ 22 ¼ r 11 x211 cosx11 t r 22 x222 cosx22 t ¼ x1 x2 ðr 11 cosx11 t þ r 22 cosx22 tÞ ðx1 þ x2 Þðr11 x11 cosx11 t þ r 22 x22 cosx22 tÞ ¼ x3 x4 ðr 11 cosx11 t þ r 22 cosx22 tÞ ðx3 þ x4 Þðr11 x11 cosx11 t þ r 22 x22 cosx22 tÞ; we can obtain the following forms:
x1 x2 r11 ðx1 þ x2 Þr11 x11 ¼ x3 x4 r11 ðx3 þ x4 Þr11 x11 ; x1 x2 r22 ðx1 þ x2 Þr22 x22 ¼ x3 x4 r22 ðx3 þ x4 Þr22 x22 ;
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1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
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50
0
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Fig. 10. Time histories of the components x1 and x3 in the coupled systems (36) are displayed respectively in Fig. 10(a) and (b), where the parameter values are taken as a1 = 0.3, b1 = 5, a2 = 0.5, b2 = 12, a3 = 0.2, b3 = 6, a4 = 0.1, b4 = 8, and initial conditions are x1(0) = 0.01, y1(0) = 1, x2(0) = 0.2, y2(0) = 0.3, x3(0) = 0.03, y3(0) = 2, x4(0) = 0.2, y4(0) = 0.1.
i.e., we have
x11 ¼ x22 ¼
x1 x2 x3 x4 : x1 þ x2 x3 x4
Hence,we have proved the above results. 2. When x1, x2, x3 and x4 is different pffiffiffi from each other, there pffiffiffi exist the transient synchronization of quasi-periodic solutions, such as x1 ¼ 1; x2 ¼ 2 and x3 ¼ 0:5; x4 ¼ 3. Whereas the synchronization of quasi-periodic solutions disappears, when t is large enough. Thereafter, the synchronization does not exists. Finally, we present numerical simulations to verify the theoretical results. Fig. 10(a) and (b) illustrate the time series of components x1(t) and x3(t). Obviously, they have the same frequency and amplitude of the oscillation when time t is sufficiently large. Fig. 11 shows time series of errors between corresponding variables in the coupled systems, i.e., (a) e1 = y1 x1, 0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
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0
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5
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20
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-1
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20
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15
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20
Fig. 11. Synchronization errors e1, e2, e3 and e4 in the coupled system (36) are displayed respectively in Fig. 11(a)–(d), where the parameter values are same to those in Fig. 10.
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L. Pei, B. Qiu / Applied Mathematical Modelling 36 (2012) 779–791 1.5
2
1
1.5 1
0.5
0.5 0 0 -0.5 -0.5 -1
-1 -1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Fig. 12. The projected attractors of the coupled systems (36) on phase space (x1, x3) and (y1, y3) are displayed respectively in Fig. 12(a) and (b), where parameter values are same to those in Fig. 10.
(b) e2 = y2 x2, (c) e3 = y3 x3 and (d) e4 = y4 x4 via time. From Fig. 11 we see that these errors tend to zeros as t goes infinity. Fig. 12(a) and (b) are two projected attractor of the coupled systems on the phase space (x1, x3) and (y1, y3). 6. Conclusion In this paper, we introduce a theoretical approach, i.e., mode decomposition, to treat the synchronization problem of nonidentical coupled dynamical systems. For such a coupled system, we first propose a mode decomposition which decomposes the linearized equations in a synchronous state into motion in two directions, i.e., the transverse direction and the parallel direction. Then, we derive by such a decomposition a rigorous sufficient condition which guarantees, if satisfied, that the overall coupled systems reach the desired synchronization. More importantly, such a decomposition has geometrical and theoretical explanations. Geometrically, the stability of a synchronous manifold does not depend upon the tangent motion whereas the transverse motion determines its stability. Theoretically, a clear explanation may be given by virtue of similar Floquet multiplier theory. Finally, we present several numerical examples to verify the theoretical results. The results are obtained as the following: First, the synchronization of the non-identical dynamical systems on the non-chaotic oscillators is reached by employing the mode decomposition approach in this paper. At the best knowledge of the authors, this approach rarely even never is applied to the synchronization of the non-identical dynamical systems. And our work is the generalization of Xiong [18], Zhou [19,20] to the case of the non-identical dynamical systems’ synchronization. The other result is the non-identical dynamical systems on quasi-periodic oscillators will synchronize at the periodic state but not at the quasi-periodic state, although the transient quasi-periodic synchronization will appear. We especially emphasize that the synchronization problem of identical and non-identical coupled systems can be treated in a united framework according to the above proposed theory. The synchronization of the non-identical dynamical systems on chaotic oscillators by the mode decomposition approach will be our direct next work. Since the dynamical systems on the non-chaotic oscillators are the specially constructed cases but not the general, the synchronization problem of the non-identical dynamical systems on the all general kinds of oscillators will be the part of our further work. It is very interesting to study the generalized synchronization of the identical and non-identical dynamical systems by the mode decomposition approach. Because of the ubiquity of the time delay, the synchronization of the delayed dynamical systems employing the mode decomposition approach is also very interesting and important. The above two problems are also the other part of our further work. Acknowledgments The authors acknowledge the financial support from the National Natural Science Foundation of China under Grant No. 10702065. They also are grateful of the reviewers for their valuable and helpful suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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