Different Translation Pragmatical Synchronization of New Ge-Ku-Mathieu

Chapter 4 Different Translation Pragmatical Generalized Synchronization by

4.3 Different Translation Pragmatical Synchronization of New Ge-Ku-Mathieu

Using different translation pragmatical synchronization by stability theory of partial region, we can choose a Lyapunov function in the form of a positive definite function in first quadrant:

2 p, in the first quadrant. The Lyapunov asymptotical stability theorem is not satisfied.

We can not obtain that common origin of error dynamics (4.14) and parameter dynamics (4.17) is asymptotically stable. By pragamatical asymptotically stability theorem, D is a 11-manifold, n=11 and the number of error state variables p=3. When

1 2 3 0

e e  e and a, b, c, d , g, h, l , p, take arbitrary values, V 0, so X is of 3 dimensions i.e. p=3, m=n-p=11-3=8, m+1<n is satisfied. According to the pragmatical asymptotically stability theorem, error vector e approaches zero and the estimated parameters also approach the uncertain parameters. The equilibrium point is pragmatically asymptotically stable. Under the assumption of equal probability, it is actually asymptotically stable. The simulation results are shown in Figs. 4.4-4.7.

31 dynamics always happens in the first quadrant e coordinate system.

1 2 state vector of Lorenz system:

1 2 1 dynamics without controller always exists in first quadrant as shown in Fig. 4.8.

Our aim is lim 0

Using different translation pragmatical synchronization by stability theory of partial region, we can choose a Lyapunov function in the form of a positive definite function in first quadrant:

33 p, in the first quadrant. The Lyapunov asymptotical stability theorem is not satisfied.

We can not obtain that common origin of error dynamics (4.24) and parameter dynamics (4.27) is asymptotically stable. By pragamatical asymptotically stability theorem, D is a 11-manifold, n=11 and the number of error state variables p=3. When

1 2 3 0

e e  e and a, b, c, d , g, h, l , p, take arbitrary values, V 0, so X is of 3 dimensions i.e. p=3, m=n-p=11-3=8, m+1<n is satisfied. According to the pragamatical asymptotically stability theorem, error vector e approaches zero and the estimated parameters also approach the uncertain parameters. The equilibrium point is pragmatically asymptotically stable. Under the assumption of equal probability, it is actually asymptotically stable. The simulation results are shown in Figs. 4.9-4.12.

Case 3.

The following chaotic systems are two translated master and slave Ge-Ku-Mathieu (GKM) systems of which the old origin is translated to

1 2 3

(x ,x x, )(350,350,350) , ( ,y y y₁ ₂, ₃)(50,50,50) to guarantee the error dynamics always happens in the first quadrant of e coordinate system.

1 2 the Ge-Ku-van der Pol system is a chaotic system.

We find that the error dynamic without controller always exists in first quadrant as shown in Fig. 4.13.

Using different translation pragmatical synchronization by stability theory of partial region, we can choosen a Lyapunov function in the form of a positive definite function in first quadrant:

2 p, in the first quadrant. The Lyapunov asymptotical stability theorem is not satisfied.

We can not obtain that common origin of error dynamics (4.34) and parameter dynamics (4.37) is asymptotically stable. By pragamatical asymptotically stability theorem, D is a 11-manifold, n=11 and the number of error state variables p=3. When

1 2 3 0

4.4 Summary

In this chapter, a new strategy to achieve chaos synchronization by the different translation pragmatical synchronization using stability theory of partial region is proposed. The pragmatical asymptotical stability theorem fills the vacancy between the actual asymptotical stability and mathematical asymptotical stability, the conditions of the Lyapunov function for pragmatical asymptotical stability are lower than that for traditional asymptotical stability. By using the different translation pragmatical synchronization by stability theory of partial region, with the same conditions for Lyapunov function, V 0, V 0, as that in current scheme of adaptive synchronization, we not only obtain the generalized synchronization of chaotic systems but also prove strictly that the estimated parameters approach the uncertain values and the Lyapunov function is simple linear homogeneous function for error states, the controllers are more simple and have less simulation error because they are in lower degree than that of traditional controllers.

It is important to note that K , ₁ K are not arbitrary, two proper values must ₂ chosen to make that the error dynamics always in first quadrant, so give two more insurances for secret communication than other synchronization methods.

Fig. 4.1 Coordinate translation.

Fig. 4.2 Coordinate translation.

Fig. 4.3 Phase portrait of the error dynamic for Case 1.

Fig. 4.4 Time histories of x , _i y for Case 1. _i

Fig. 4.5 Time histories of errors for Case 1.

Fig. 4.6 Time histories of parameter errors for Case 1.

Fig. 4.7 Time histories of parameter errors for Case 1.

Fig. 4.8 Phase portrait of the error dynamic for Case 2.

Fig. 4.9 Time histories of x , _i y for Case 2. _i

Fig. 4.10 Time histories of errors for Case 2.

Fig. 4.11 Time histories of parameter errors for Case 2.

Fig. 4.12 Time histories of parameter errors for Case 2.

Fig. 4.13 Phase portrait of the error dynamic for Case 3.

Fig. 4.14 Time histories of x , _i y for Case 3. _i

Fig. 4.15 Time histories of errors for Case 3.

Fig. 4.16 Time histories of parameter errors for Case 3.

Fig. 4.17 Time histories of parameter errors for Case 3.

Chapter 5

Multiple Symplectic Synchronization for Ge-Ku-Mathieu System

5.1 Preliminary

In this Chapter, a new type of synchronization, multiple symplectic synchronization is studied. Symplectic synchronization and double symplectic synchronization are special cases of the multiple symplectic synchronization. When the double symplectic functions is extended to a more general form,

( , , ,z , , )w t  ( , , ,z , , )w t

G x y F x y , it is called “multiple symplectic synchronization”. The multiple symplectic synchronization may be applied to increase the security of secret communication due to the complexity of its synchronization form.

5.2 Multiple Symplectic Synchronization Scheme

Generalized synchronization refers to a functional relation between the state vectors of master and of slave, i.e. yF x( , )t , where x and y are the state vectors of master and slave. Recently, generalized synchronization is extended to a more general form, yF x y( , , )t . This means that the final desired state y of the “slave”

system not only depends upon the “master” system state x but also depends upon the state y itself. Therefore the “slave” system is not traditional pure slave obeying the master system completely but plays a role to determine the final desired state of the

“slave” system. This kind of synchronization, is called “symplectic synchronization”, and the “master” system is called Partner A, the “slave” system is called Partner B.

Since the symplectic functions are presented at both the right hand side and the left hand side of the equality, it is called double symplectic synchronization,

( , , )t  ( , , )t

5.3 Synchronization of Three Different Chaotic Systems Case 1.

Consider the Chen system is described by

1 2 1

The Lorenz system is described by

1 2 1

The controlled Ge-Ku-Mathieu(GKM) system is described by

 

Thus we design the controller as

1 2 1 1 1 1 1 2 1 1 1 2 1 2 1 1 1 3 2 synchronization is achieved. The phase portrait of the controlled Ge-Ku-Mathieu

system and the time histories of G( , , , )x y z t and F x y( , , , )z t and the time histories of the state errors are shown in Fig. 5.3 and Fig. 5.4 and Fig. 5.5, respectively.

Case 2.

The Lorenz system is described by

1 2 1

The controlled Ge-Ku-Mathieu(GKM) system is described by

 

where a 0.6,b5,c11,d 0.3,g8,h10,l0.5,p0.2, u



u u u1, 2, 3



^T is the controller, and the initial condition is y₁(0)0.01, y₂(0)0.01,y₃(0)0.01. Thus we design the controller as

1 2 3 1 1 1 1 2 1 1 1 2 1 2 1 1 2 synchronization is achieved. The phase portrait of the controlled Ge-Ku-Mathieu system and the time histories of G( , , , )x y z t and F x y( , , , )z t and the time histories of the state errors are shown in Fig. 5.7 and Fig. 5.8 and Fig. 5.9, respectively.

Case 3.

and our goal is to achieve the multiple symplectic synchronization

( , , , ) ( , , , ) G x y z t F x y z t .

Consider the sprott system is described by

1 2 3 chaotic attractor of the sprott system is shown in Fig. 5.10.

The Lorenz system is described by

1 2 1

The controlled Ge-Ku-Mathieu(GKM) system is described by

 

Thus we design the controller as

53 synchronization is achieved. The phase portrait of the controlled Ge-Ku-Mathieu system and the time histories of G( , , , )x y z t and F x y( , , , )z t and the time histories of the state errors are shown in Fig. 5.11 and Fig. 5.12 and Fig. 5.13, respectively.

5.4 Summary

A new type of synchronization, multiple symplectic synchronization, is studied in this Chapter. It is an extension of double symplectic synchronization. By applying active control, the multiple symplectic synchronization is achieved. The simulation results show that the proposed scheme is effective and feasible. Furthermore, the multiple symplectic synchronization of chaotic systems can be used to increase the security of secret communication.

Fig. 5.1 The chaotic attractor of the Chen system.

Fig. 5.2 The chaotic attractor of the Lorenz system.

Fig. 5.3 Phase portrait of a controlled new Ge-Ku-Mathieu system for Case 1.

Fig. 5.4 Time histories of G( , , , )x y z t and F x y( , , , )z t for Case 1.

Fig. 5.5 Time histories of the state errors for Case 1.

Fig. 5.6 The chaotic attractor of the Rossler system.

Fig. 5.7 Phase portrait of the controlled Ge-Ku-Mathieu system for Case 2.

Fig. 5.8 Time histories of G( , , , )x y z t and F x y( , , , )z t for Case 2.

Fig. 5.9 Time histories of the state errors for Case 2.

Fig. 5.10 The chaotic attractor of the sprott system.

Fig. 5.11 Phase portrait of the controlled Ge-Ku-Mathieu system for Case 3.

Fig. 5.12 Time histories of G( , , , )x y z t and F x y( , , , )z t for Case 3.

Fig. 5.13 Time histories of the state errors for Case 3.

Chapter 6

Robust Projective Anti-Synchronization of Nonautonomous Chaotic Systems with Stochastic Disturbance by Fuzzy

Logic Constant Controller 6.1 Preliminary

In this paper, a simplest fuzzy logic constant controller (FLCC) ,which is derived via fuzzy logic design and Lyapunov direct method, is presented for projective anti-synchronization of nonautonomous chaotic systems with uncertain and stochastic signals. Controllers in traditional Lyapunov direct method are always nonlinear and complicated. However, FLCC proposed are such simple controllers which are constant numbers, decided via the values of the upper and lower bounds of the error derivatives. This new method is used in projective anti-synchronization of nonautonomous chaotic systems with stochastic disturbance to show the robustness and effectiveness of FLCC.

6.2 Projective Chaos Anti-Synchronization by FLCC Scheme

Consider the following master chaotic system (A ) f( ) 

The slave system which can be either identical or different from the master, is ( )

B g

  

y y y u (6.2)

where y[ ,y y₁ ₂, y_n]^T Rⁿ denotes a state vector, B is an n n constant

coefficient matrix, g is a nonlinear vector function, and u[ ,u u₁ ₂, u_n]^TRⁿis the fuzzy logic controller needed to be designed.

For projective anti-synchronization, in order to make the chaos state y approaching the goal statex , define ex  ( y) x y as the state error,  hereis a constant. The chaos projective anti-synchronization is accomplished in the sense that [59]: From Eq. (6-4) we have the following error dynamics:

e x y [ (A  )x f ( )x  ] B[ y g ( )y (6.5) u ] According to Lyapunov direct method, we have the following Lyapunov function

to derive the fuzzy logic controller for projective anti-synchronization:

2 2 2

1 1

( , , , , ) 1( ) 0

m n 2 m n

V  f e  e  e  e e e  (6.6) The derivative of the Lyapunov function in Eq. (6.6) is:

V e e_{1 1} e e_{m m} e e_{n n} (6.7) If the vector controller in Eq. (6.5) can be suitably designed to achieve V0, then the zero solution e0 of Eq.(6.5) are asymptotically stable i.e the projective anti-synchronization is accomplished. Next, the design process of FLCC is introduced.

We use the error derivativee(t)



e e1, 2, ,e_m, ,e_n



^T, as the antecedent part of the proposed FLCC to design the control input u which is used in the consequent part of the proposed FLCC:

u



u u1, 2, ,u_m, ,u_n



^T (6.8)

where u is a constant column vector and accomplishes the objective to stabilize the error dynamics in Eq. (6.5).

The strategy of the FLCC designed is proposed as follow and the configuration of the strategy is shown in Fig. 6.1.

Assume the upper bound and lower bound of e_m are Z_m and –Z_m, then the FLCC can be design step by step:

(1) If e_m is detected as positive (e_m0), we design a controller fore_m0for

Therefore we have the following ith if–then fuzzy rule as:

Rule 1 : Ife_mis M1 Then um1 =e_m 0 (6.15)

M1,M₂and

M refer to the membership functions of positive (P), negative (N) and 3

zero (Z) separately which are presented in Fig. 6.2. For each case,u_mi, i= 1~3 is the i-th output of e_m, which is a constant controller. The centriod defuzzifier evaluates the output of all rules as follows:

Rule Antecedent Consequent Part

em u_mi

1 Negative (N) um1

2 Positive (P) ^u_m₂

3 Zero (Z) um3

With appropriate fuzzy logic constant controllers in Eq. (6.7), a negative definite derivatives of Lyapunov function V can be obtained and the asymptotical stability of Lyapunov theorem can be achieved.

Consequently, the processes of FLCC designed to control a system following the trajectory of a master system are getting the upper bound and lower bound of the error derivatives of the goal and control systems without any controller, i.e.

m m

m e Z

Z  

  . Through the fuzzy logic system which follows the rules of Eq. (6.9)

~ Eq. (6.17), a negative definite derivatives of Lyapunov function V can be

obtained and the asymptotically stability of Lyapunov theorem can be achieved.

6.3 Simulation Results

There are two examples in this Section. Each example is divided into two parts, projective anti-synchronization by FLCC and that by traditional method. In the end of each example, we give the simulation results of two controllers and list the tables and figures to show the effectiveness and robustness of our method.

Case 1

6.1 Projective Anti-Synchronization of Sprott Systems by New FLCC The Sprott 19 system [60] is:

1 2 chaos of the Sprott 19 system appears. The chaotic behavior of Eq. (6.19) is shown in Fig. 6.3.

6.1.1 Projective Anti-Synchronization of Nonautonomous Sprott 19 System by New FLCC

The nonautonomous Sprott 19 system is:

1 nonautonomous Sprott 19 system appears in Fig. 6.5.

The Sprott 22 system [60] is:

1 2 the Sprott 22 system is shown in Fig. 6.6.

The slave Sprott 22 system with controllers is:

The error vector for projective anti-synchronization is

1 1 1

67 derivatives without any controllers shown in Fig. 6.7.

FLCC are proposed in Part 1, 2 and 3 to make V₁ e₁e₁ 0 , 0

e e

V₂  ₂₂  andV₃ e₃e₃ 0. Hence we haveV V₁V₂ V₃ 0. It is clear that all of the rules in FLCC can lead that the Lyapunov function satisfies the asymptotical stability theorem. The simulation results are shown in Fig. 6.8 and Fig. 6.9. The projection of phase portraits of system (6.22) with chaotic behaviors is shown in Fig.

6.6.

6.1.2 Robust Projective Anti-Synchronization of Nonautonomous Sprott System with Stochastic Disturbance by FLCC

The master nonautonomous Sprott 19 system with stochastic disturbance is:

1 Sprott 19 system with stochastic disturbance appears in Fig. 6.11.

The slave system is the same as Eq. (6.22) and Lyapunov function derived

And time derivative of Lyapunov function is:

1 1 2 2 3 3

The maximum value and minimum valuewithout any controller can be observed by time histories of error derivatives shown in Fig 6.12. The robust projective anti-synchronization scheme to make V e e_{1 1}e e_{2 2}e e_{3 3}0. It is clear that all of the rules in FLCC can lead that the Lyapunov function satisfies the asymptotical

stability theorem. The simulation results are shown in Fig. 6.13 and Fig. 6.14.

6.1.3 Robust Projective Anti-Synchronization of Nonautonomous Sprott System with Stochastic Disturbance by Traditional Method

In order to lead the derivative of Lyapunov function in Eq. (6.30) to negative definite, we choose robust traditional nonlinear controllers as:

1 2 2 2 1 The derivative of Lyapunov function is negative definite and the error dynamics in Eq. (6.29) are going to achieve asymptotically stable. The simulation results are shown in Fig. 6.15 and Fig. 6.16.

6.1.4 FLCC Compared to Traditional Method

In this subsection, the controllers and numerical simulation results in subsection 6.1.2 and subsection 6.1.3 are listed in Tables 2 and 3 for comparison. Comparing two kinds of controller in Table 2 and two kinds of errors in Table 3, it is clear to find out that (1) The controllers in FLCC designing are much simpler than traditional ones; (2) The performance of the error convergence of states by FLCC is much better than that by traditional method.

Consequently, even the system contains noise and parameter uncertainty, the FLCC can still remain the high performance to synchronize the two chaotic systems with uncertainty and stochastic signals exactly and efficiently.

Table 2 The controllers of FLCC and of traditional method.

Controller u1

FLCC Traditional method

1 10

Zm  [ ( x₂  ₂) y₂e₁]

u2 Z_m₂ 20 [ ( x₃  ₂) y₃e₂]

u3 Z_m₃ 50

1 3 2 2 1 2

3 2 1 3

{ [ (1 ) ]

sin }

a x bx x x

cy y y e



       

   

Table 3 Errors data after the action of controllers.

Time

Case 2

6.2 Projective Anti-Synchronization of Sprott System and Ge-Ku-Mathieu Systems

6.2.1 Chaos Projective Anti-Synchronization of Nonautonomous Sprott 19 System and Ge-Ku-Mathieu (GKM) System by New FLCC

The nonautonomous Sprott 19 system is:

1 chaos of the nonautonomous Sprott 19 system with stochastic disturbance appears in Fig. 6.17. well.u₁,u₂andu₃are FLCC to anti-synchronize projectively the slave GKM system to the master one.

Part 3: derivatives without any controller shown in Fig. 6.18.

FLCC are proposed in Part 1, 2 and 3 and make satisfies the asymptotical stability theorem. The simulation results are shown in Fig.

6.19 and Fig. 6.20.

6.2.2 Robust Projective Anti-Synchronization of Nonautonomous Sprott System with Stochastic Disturbance and GKM System by FLCC

The master noautonomous Sprott 19 system with stochastic disturbance is:

1 a=-0.6, b=2.75, chaos of the nonautonomous Sprott 19 system with stochastic signal appears. The chaotic behavior of Eq. (6.40) is shown in Fig. 6.21.

The slave system was same as Eq. (6.34) andLyapunov functionderived through Eqs. (6.35) ~(6.39).

Time derivative of Lyapunov function is:

1 1 2 2 3 3

The maximum value and minimum valuewithout any controller can be observed in time histories of error derivatives shown in Fig 6.22. The projective anti-synchronization scheme to make V e e_{1 1}e e_{2 2}e e_{3 3}0. It is clear that all of the rules in FLCC can lead that the Lyapunov function satisfies the asymptotical stability theorem. The simulation results are shown in Fig. 6.23 and Fig. 6.24.

6.2.3 Robust Projective Anti-Synchronization of Nonautonomous Sprott 19 System with Stochastic Disturbance and GKM System by Traditional Method

According to Eq. (6.42), we design complicated controllers to anti-synchronize projectively chaotic systems in subsection 6.2.2 by traditional method.

We choose controllers are

And we can obtain

V  e e₁ ₁  e e₂ ₂ e e₃ ₃0 (6.44) The derivative of Lyapunov function is negative definite and the error dynamics in Eq. (6.41) achieves asymptotical stability. The simulation results are shown in Fig.

6.25 and Fig. 6.26.

6.2.4 FLCC Compared to Traditional Method

In this case, the controllers and numerical simulation results of subsection 6.2.2 and subsection 6.2.3 are listed in Table 4 and Table 5 for comparison. The mater and slave systems are more complex than Case 1, but the good-robustness and high performance can be still achieved through FLCC. The two main superiorities are still existed: (1) The controllers in FLCC designing are much simpler than traditional ones;

(2) The performance of the convergence of error states by FLCC is much better than by traditional method.

Table 4 The controller of FLCC and of traditional method.

Controller

Table 5 Errors data after the action of controllers.

Time after the

6.4 Summary

A simplest fuzzy controller (FLCC) is introduced to projective anti-synchronization of non-autonomous chaotic systems with stochastic disturbance.

Three main contributions can be concluded: (1) High performance of the convergence of error states in synchronization; (2) Good robustness in projective anti-synchronization of the chaotic systems with stochastic disturbance; (3) Simple constant controllers are used, which can be easily obtained.

Fig. 6.1. The configuration of fuzzy logic controller.

Fig. 6.2. Membership function.

Fig. 6.3. Projections of phase portrait of chaotic Sprott No.19 system with a=-0.6,

b=2.75.

Fig. 6.4. ₁ is pulse generator.

Fig. 6.5. Projections of phase portrait of nonautonomous chaotic Sprott 19 system and

a=-0.6, b=2.75.

Fig. 6.6. Projections of phase portrait of chaotic Sprott 22 system with controllers.

Fig. 6.7. Time histories of error derivatives for master and slave Sprott nonautonomous chaotic systems without controllers.

Fig. 6.8. Time histories of errors for Case1 (nonautonomous system) the FLCC is

added after 30s.

Fig. 6.9. Time histories of states for Case1 (nonautonomous system) the FLCC is

added after 30s.

Fig. 6.10. The stochastic signal of ₂ is band-limited white noise(PSD=0.1).

Fig. 6.11. Projections of phase portrait of nonautonomous chaotic Sprott 19 system with stochastic disturbance ₂, a=-0.6 and b=2.75.

Fig. 6.12. Time histories of error derivatives for master and slave Sprott chaotic systems without controllers.

Fig. 6.13. Time histories of errors for subsection 3.1.2, the FLCC is applied after 30s.

Fig. 6.14. Time histories of states for subsection 3.1.2, the FLCC is applied after 30s.

Fig. 6.15. Time histories of errors for subsection 3.1.3 the traditional nonlinear

controller is applied after 30s.

Fig. 6.16. Time histories of states for subsection 3.1.3 the traditional nonlinear

controller is applied after 30s.

Fig. 6.17. Projections of phase portrait of nonautonomous chaotic Sprott 19 system

with stochastic disturbance where a=-0.6, b=2.75.

Fig. 6.18. Time histories of error derivatives for subsection 3.2.1.

Fig. 6.19. Time histories of errors for section 3.2 where FLCC is added after 30s.

Fig. 6.20. Time histories of states for subsection 2-3.2 the FLCC is coming into after

30s.

Fig. 6.21. Projections of phase portrait of nonautonomous chaotic Sprott 19 system

with stochastic disturbance where a=-0.6, b=2.75.

在文檔中 Ge-Ku-Mathieu系統及Sprott19, 22系統的渾沌及渾沌同步 (頁 40-0)