Chapter 5 Generalized Synchronization of Coupled Systems
5.2 Theoretical Analysis
Consider the following unidirectional coupled nonautonomous systems ( , ), containing the origin. Assume that the solutions of Eq. (5.1) have a priori bounds then they must exist for infinite time. That is, for given ( , ,t0 x z0 ˆ0)∈Ω ∩ Ω1 2 the solution
0 0 ˆ0 ˆ 0 0 ˆ0
x ( ; , x , z ) z ( ; , x , z )T t t T t t T
⎡ ⎤
⎣ ⎦ of Eq. (5.1) exists for t≥ . At the first, we recall t0 the definition of generalized synchronization.
Definition 5.1 The system (5.1) is generalized synchronized if there is a continuous
79
In Eq. (5.1) u is the coupling function or the controlling term. In order to investigate the transversal stability of synchronization manifold, define z=H( )x and e= −zˆ z to be the state error. Herein, the function H∈ is differentiable and C1 can be arbitrary assigned to increase the complication of synchronization. Then the error equations can be written as
ˆ ( , )t ˆ ( , , )t ˆ H( ) error e but also a function of x . As a result, the traditional Lyapunov direct method can not be used. On the other hand, the variational equation or Lyapunov exponents may be used to clarify transversal stability. As mentioned before, there is a drawback that we can only calculate finite evolution time in computer simulation but infinite evolution time is needed by definition of Lyapunov exponent.
Herein, we add the upper half (lower half also works) of Eq. (5.1) with ˆz synchronization manifold is stable in transversal directions. This can be done via stability with respect to partial variables.
In the following, we choose u( , , )t z zˆ = Γ −(z zˆ) and ( , ) H ( , )
80
where 0 is still an equilibrium point of the second equation of Eqs. (5.4) as synchronization occurs.
Theorem 5.1 The partial state e asymptotically approaches to 0 in Eq. (5.4) if LIn − Γ is negative definite, i.e. the system in Eq. (5.1) is in generalized synchronization if LIn − Γ is negative definite.
Proof Choose a function 1 ( , )
2
V x e = e e that is positive definite with respect to T e and with infinitesimal upper bound. Then its time derivative along the solution of Eq.
(5.3) is uniformly asymptotically approaches to 0 in Eq. (5.4) by partial stability theory.
Hence the system in Eq. (5.1) is in generalized synchronization if LIn− Γ is is global if f is globally Lipschitz.
5.3 Numerical Illustrated Examples
Example 5.1 Autonomous case: Lorenz systems
1 1 2 1 chapter, one needs to estimate the Lipschitz constant at the beginning. By
81 81
Cauchy-Schwarz inequality, it can be derived for any
21 22 23 1 11 12 13
z x Ax b to be an affine mapping, then the response system becomes ˆ= (Φ−1( ))ˆ − Γ − Φ(ˆ ( )),
z A f z z x
where { , , }Γ =diag γ " γ and zˆ=[z zˆ ˆ ˆ1 2 z3]T . LIn− Γ is negative definite if γ =88. First, select Φ be a reflection, that is A= −I and b=0. With the initial value [x0T zˆ0T T] =[10 10 10 0.5 0.5 0.5]T , the simulated results are shown in Fig.5.1-5.4. As expectation, the projections of synchronized manifold in Fig.5.2 are diagonal-like and reflected to vertical axis. Compare Fig.5.4 with Fig.5.3, the phase portrait of response system in Fig.5.4 is reflected to the phase portrait of driving system in Fig.5.3. This case is also called anti-synchronization of chaos. With the same initial condition, let
1 1 0
the simulated results are shown in Fig. 5.5 and Fig. 5.7, respectively. The projections of synchronized manifold are no longer diagonal-like but more complicated.
Example 5.2 Nonautonomous case: An extended equation of the coupled Duffing systems is written as
1 2 1
82 82
= Φ( )= +
z x Ax b . By Cauchy-Schwarz inequality, it can be derived for any
21 22 1 11 12 can be obtained as
2 2 The simulated results are shown in Fig. 5.8-5.11. Fig. 5.9 shows that the projections of synchronized manifold are complicated. Fig. 5.10 and Fig. 5.11 show that the phase portraits of the driving and response systems are different.
83 83
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 5 10
e1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 5 10
e2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 5 10
t (sec) e3
Fig. 5.1 e e and 1, 2 e versus time.3
84 84
-10 -5 0 5 10 15
-10 0 10
x1 x4
-15 -10 -5 0 5 10 15 20
-20 -10 0 10 20
x2 x5
15 20 25 30 35 40
-40 -30 -20 -10
x3 x6
Fig. 5.2 Projections of synchronized manifold.
85 85
-20 -15 -10 -5 0 5 10 15 20
-40 -20 0 20 40
5 10 15 20 25 30 35 40 45
x1 y1
z1
Fig. 5.3 Phase portrait of the driving system.
86 86
-20 -15 -10 -5 0 5 10 15 20
-40 -20 0 20 40 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5
x2 y2
z2
Fig. 5.4 Phase portrait of the response system.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.5 1 1.5
e1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-30 -20 -10 0
e2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 5 10 15 20
t (sec) e3
Fig. 5.5 e e and 1, 2 e versus time.3
88 88
-15 -10 -5 0 5 10 15
-10 -5 0 5 10
x1 x4
-20 -15 -10 -5 0 5 10 15 20
-20 0 20 40 60 80
x2 x5
10 15 20 25 30 35 40
-50 0 50
x3 x6
Fig. 5.6 Projections of synchronized manifold.
89 89
-15 -10
-5 0
5
10 15
-40 -20 0 20 40 60 80 -60 -40 -20 0 20 40 60
x2 y2
z2
Fig. 5.7 Phase portrait of the response system.
90 90
0 5 10 15
-0.15 -0.1 -0.05 0 0.05
e1
0 5 10 15
-0.3 -0.2 -0.1 0 0.1
t (sec) e2
Fig. 5.8 e and 1 e versus time.2
91 91
-1.53 -1 -0.5 0 0.5 1 1.5
4 5 6 7
x1 x3
-0.82 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
3 4 5 6 7 8
x2 x4
Fig. 5.9 Projections of synchronized manifold.
92 92
-1.5 -1 -0.5 0 0.5 1 1.5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
x1 y1
Fig. 5.10 Phase portrait of the driving system.
93 93
3 3.5 4 4.5 5 5.5 6 6.5 7
2 3 4 5 6 7 8
x2 y2
Fig. 5.11 Phase portrait of the response system.
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Chapter 6
Conclusions
Chaos synchronization is an important research topic in these years. There are several methods to guarantee the emergence of chaos synchronization but there is no easy unified criterion in general. Most of them are suitable for a specific kind of system or even for a special system. Herein, a general scheme for both unidirectional and mutual coupled systems is proposed to achieve chaos synchronization via partial stability theory. It can overcome two drawbacks. First, it is difficulty to use the traditional Lyapunov method since the state error equation is not a pure function of state error in general. Second, zero crossing of Lyapunov exponent whose definition needs infinite evolution time is used as a criterion of chaos synchronization widely but we can only calculate finite evolution time in computer simulation. The benefit of this scheme is that the usage of the partial stability theory is similar to the traditional Lyapunov method. Superficially, the order of the error dynamic equation is enlarged since it is replaced by an extended equation in this scheme. But only partial variables are manipulated in actual. Furthermore, many control techniques can be applied to synchronize coupled systems in this scheme.
Follows the procedure of the proposed scheme, the unidirectional coupled systems are discussed first and three sufficient criteria are derived. One of them is suitable for systems without perturbation and the other two are suitable for systems under two kinds of perturbations, vanishing and nonvanishing, respectively. Second, the effort is concentrated on synchronization of mutual coupled systems. Similar to the unidirectional case, three theorems are proven to ensure the occurrence of synchronization. One of them is suitable for systems without perturbation and the other two are suitable for systems under two kinds of perturbations, vanishing and nonvanishing, respectively.
In previous six criteria, to guarantee the emergence of synchronization a matrix equation should be satisfied and the estimation of Lipschitz constant is needed.
Moreover, the estimate of Lipschitz constant is often conservative. To overcome these two shortcomings, this matrix equation and the estimation of Lipschitz constant are replaced by adopting an adaptive coupling gain and an adaptive estimator,
95 95
respectively. As a result, a simple and convenient adaptive synchronization of chaotic systems is realized for both unidirectional and mutual coupled systems. It is easier and more convenient to use this method for synchronization of both unidirectional and mutual coupled systems than the six theorems in chapter 2 and 3. Furthermore, to increase the convergent rate of state error dynamics we only need to set a larger initial condition of the adaptive equation.
The synchronization discussed indicates the identical synchronization (or complete synchronization) in the foregoing results. Another kind of synchronization called generalized synchronization which means that there is a functional relation between the states of driving and response systems as time goes to infinity are studied in the chapter 5. This function can increase the complication of synchronization.
Similar to the chapter 2, a scheme to achieve chaos generalized synchronization via partial stability is proposed. One theorem is proven to ensure generalized synchronization for a general kind of unidirectional coupled nonautonomous systems by linear feedback.
Several examples are simulated numerically to illustrate the theoretical analyses.
All the criteria derived in this dissertation work for regular and chaotic, linear and nonlinear systems, autonomous and nonautonomous systems. Hence, the proposed scheme to achieve chaos synchronization is successful.
96 96
Appendix
The content of this appendix follows [78-80]. Consider a differential system ( , )t
=
x f x , (A1)
where f:[ , )t0 ∞ × Ω →R , n f( , )t 0 =0 ∀ ∈t I [ , )t0 ∞ and Ω ⊂ R is a region n containing the origin. Assume that f is smooth enough to ensure that the solution of (A1) exists uniquely. To shorten the notation, write x=( ,y1 "ym, , ,z1 " zn m− )T ,
We assume that the solution of (A1) is z-extendable, i.e. any solution of (A1) exists for all t≥t0 and y( )t ≤H , H is a constant. Write independent of t in the definition of y-stable. 0
The solution of (A1) is asymptotically stable with respect to y (asymptotically y-stable) if it is (1) y-stable and (2) y-attractive, i.e. ∀ ∈t0 [0, )∞ ,
y x . The solution of (A1) is uniformly asymptotically y-stable if it is (1) uniformly y-stable and (2) uniformly y-attractive, i.e. δ′( , )t0 ε′ is independent of t and 0 T t( ,0 x0, )ε′ is independent of t x in the definition of 0, 0 y-attractive.
The solution of (A1) is globally y- attractive if Bδ = R in the definition of n y-attractive. Furthermore, if Bδ = R and n ∃δ′( ) 0t0 > can be replaced by ∀δ the solution of (A1) is globally uniformly y- attractive. The solution of (A1) is globally asymptotically y- stable if it is (1) y-stable and (2) globally y-attractive. The solution
97 97
of (A1) is globally uniformly asymptotically y- stable if it is (1) uniformly y-stable and (2) globally uniformly y-attractive.
The next definition extended the notation of definite functions to partial variables.
Let V t( , )x ∈C t
(
[ , )0 ∞ ×R R with ( , )n,)
V t 0 =0 and V is in the domain Q . Definition A2 A t implicit positive (negative) semi-definite function V x is called ( ) positive (negative) definite with respect to y if V x can vanish only when ( ) y=0 .A positive (negative) semi-definite function V t x is called positive (negative) ( , ) definite with respect to y if there is a positive (negative) definite function W y such ( ) that V t( , )x ≥W( )y ( ( , )V t x ≤W( )y ). function V t x( , ) possesses an infinitesimal upper bound with respect to
1, , k ( )
Theorem A1 Suppose there exists a positive definite function V t x with respect ( , ) to x1, ," xk (k≤n) such that ( , )V t x is positive semi-definite or vanishes, then the undisturbed motion is stable with respect to x1, ," xk (k ≤n).
Theorem A2 Suppose there exists a positive definite function V t x with respect ( , ) to x1, ," xk (k≤n) such that ( , )V t x possesses an infinitesimal upper bound and
98
then the origin is uniformly asymptotically y-stable.
Theorem A4 Suppose there exist a function V :[0, )∞ ×Ω×Rm→R such that for
then the origin is globally asymptotically y-stable.
If there is perturbation in the system, the stability of motion is different. Consider differential equation of a system under constantly acting perturbation
( , )t ( , )t motion x=0 of system (A1) is said to be y-stable under constantly acting perturbation small on the average, if ∀ >ε 0, t0 > , 0 ∀ >T 0, ∃δ ε1( , , ) 0t T0 > ,
2( , , ) 0t T0
δ ε > such that whenever x0 <δ ε1( , , )t T0 and
99 small at each instant, are uniform. This is also called total stability.
Theorem A5 Suppose there exist a function V :[0, )∞ ×Ω×Rm→R such that for
then the solution of system (A2) is y-stable under constantly acting perturbation small at each instant.
Theorem A6 Suppose there exist a function V :[0, )∞ ×Ω×Rm→R such that for
then the solution of system (A2) is y-stable under constantly acting perturbation small on the average.
Corollary A1 The functions f and Df x are continuous and bounded in Q . If ( ) the invariant set
{
x y=0 is uniformly asymptotically stable, then it is uniformly}
stable under constantly acting perturbation small on the average.
100 100
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Paper List
1. Zheng-Ming Ge and Yen-Sheng Chen, 2005, “Synchronization of Mutual Coupled Chaotic Systems via Partial Stability Theory”, submitted to Physics Lett. A.
2. Zheng-Ming Ge and Yen-Sheng Chen, 2004, “Synchronization of Unidirectional Coupled Chaotic Systems via Partial Stability”, Chaos, Solitons & Fractals 21, pp.101-111. (SCI, Impact Factor: 1.526)
3. Zheng-Ming Ge and Yen-Sheng Chen, 2005, “Adaptive Synchronization of Unidirectional and Mutual Coupled Chaotic Systems”, Chaos, Solitons &
Fractals 26, pp.881-888. (SCI, Impact Factor: 1.526)
4. Zheng-Ming Ge and Yen-Sheng Chen, 2005, “Diffeomorphic Synchronization of Unidirectional and Mutual Coupled Chaotic Systems”, submitted to Chaos, Solitons & Fractals.
5. Zheng-Ming Ge, Tsung-Chih Yu and Yen-Sheng Chen, 2003, “Chaos Synchronization of a Horizontal Platform System”, Journal of Sound and Vibration 268, pp.731-749. (SCI, Impact Factor: 0.828)
6. Zheng-Ming Ge, Chia-Yang Yu and Yen-Sheng Chen, 2004, “Chaos Synchronization and Anticontrol of a Rotationally Supported Simple Pendulum”, JSME Int. J. Series C, Vol. 47, No.1, pp.233-241. (SCI)
7. Zheng-Ming Ge, Chun-Chi Lin and Yen-Sheng Chen, 2004, “Chaos, Chaos Control and Synchronization of Vibrometer System”, Proc. Instn Mech. Engrs Vol. 218, Part C: J. Mechanical Engineer Science, pp.1001-1020. (SCI)
8. Zheng-Ming Ge, Jui-Wen Cheng and Yen-Sheng Chen, 2004, “Chaos Anticontrol and Synchronization of Three Time Scales Brushless DC Motor
107 107
System”, Chaos, Solitons & Fractals 22, pp.1165-1182. (SCI, Impact Factor:
1.526)
9. Zheng-Ming Ge, Ching-Ming Chang and Yen-Sheng Chen, 2004, “Anti-Control of Chaos of Single Time Scale Brushless DC Motors and Chaos Synchronization of Different Order Systems”, accepted for publication by Chaos, Solitons & Fractals. (SCI, Impact Factor: 1.526)
10. Zheng-Ming Ge, Chun-Lai Hsiao and Yen-Sheng Chen, 2004, “Nonlinear dynamics and Chaos Control for a Time Delay Duffing System”, Int. J. of Nonlinear Sciences and Numerical Simulation 6(2), pp.187-199. (SCI, Impact Factor: 0.483).
11. Zheng-Ming Ge, Chun-Lai Hsiao and Yen-Sheng Chen, 2004, “Chaos and Chaos Control for a Two-Degree-of-Freedom Heavy Symmetric Gyroscope”, submitted to Chaos, Solitons & Fractals.