Synchronization scheme

Among many kinds of synchronizations [2-6], the generalized synchronization is investigated [7-8]. This means that we can give a functional relationship between the states of the master and slavey=G x( ). In this Chapter, a special case of hybrid projective hyperchaotic generalized synchronization

( ) ( ) ( )

y=G x =g x t z t (3.1) is studied, where x and y are state variable vectors of the master and slave, respectively. z is state vector of a third hyperchaotic system, called functional system, since it is a constituent of function G.. While the entries of constant vector g can be either positive or negative, hybrid projective synchronization is named. When z is chaotic, chaotic generalized synchronization is named. Pragmatical asymptotical stability theorem is used, pragmatical synchronization is named. As a whole, pragmatical hybrid projective hyper-chaotic generalized synchronization (PHPHGS) is named.

The master system is

The control task is to force the slave state vector to track an n-dimensional desired vector

The controlling goal is that can be accomplished on the base of pragmatical asymptotical stability theorem by adaptive backstepping control.

3.3. Numerical results of PHPHGS by adaptive backstepping control

Take the Mathieu-Duffing system (2.3) as master system, where a, b, c, d, ,

e f are uncertain parameters and following Mathieu-Duffing system as slave system:

1 2

constants. When a₁=0.01, b₁=1.00063, c₁=5, d₁=0.66635, f₁=0.05, chaos occurs as are added to each equation of Eq.(3.8), respectively:

1 2 1

where

i i i i i i i i

e = −y g x z −g x z , (i=1, 2,3, 4) (3.12) Step1. We consider the asymptotical stability of e₁=0:

1 2 2 2 2 1 2 1 1 1 2 1

e = +e g x z −g x z −g x z +u (3.13) where

e

₂ is regarded as a controller. Choose a control Lyapunov function (CLF) of the form

1 2 1 2 1

v = e (3.14) Its time derivative along the solution of Eq.(3.13) is

1 1( 2 2 2 2 1 2 1 1 1 2 1) Choose a control Lyapunov function of the form

2 2 2 2 2

2 1 12( 2 ) 0

v = +v w +a +b +c +d > , (3.21) where a= −a aˆ,b= −b b^ˆ,c= −c cˆ,d = −d d^ˆ and ˆa,ˆb, ˆc, ˆd are estimated values of the unknown parameters a, b, c, d, respectively. We have

2 1 2 1 2

3 3

w = (3.26) e Choose a control Lyapunov function of the form

1 2

3 2 2 3 0

v = +v w > (3.27) Its time derivative through the third equation of Eq.(3.11) is

3 2 3( 4 4 4 4 3 4 3 3 3 4 3) Choose a Lyapunov function of the form

2 2 2

4 4 4 4

f ,while from Eqs(3.14),(3.21),(3.27),(3.34)

2 2 2 2

The Lyapunov asymptotical stability theorem can not be satisfied in this case. The common origin of error dynamics and parameter update dynamics cannot be determined to be asymptotically stable. By pragmatical asymptotical theorem (see Appendix ) D is a 10 -manifold, n = 10 and the number of error state variables p = 4.

Whene₁ = e₂ = e = ₃ e₄= 0 and

a

, b, c, d, e, f take arbitrary values , V = , 0 so X is a 6-dimational space, m = n – p = 10 – 4 = 6. m + 1≤ n are satisfied. By the pragmatical asymptotical stability theorem, not only error vector e tends to zero but also all estimated parameters approach their uncertain parameters. PHPHCGS of chaotic systems by adaptive backstepping control is accomplished. The equilibrium point e₁ =

e2 = e = ₃ e₄ = a = b = c = d = e = f = 0 is pragmatically asymptotically stable. The generalized synchronization is accomplished after 2000s with g = -0.1, ₁

g = 4,2 g = -0.1, and ₃ g = 0.2 while ₄ e₁(0)= −2.4001, e₂(0)= −85.9999,

3(0) 3.0001,

e = − and e₄(0)= −1.9999. The estimated parameters approach the uncertain parameters of the chaotic system as shown in Figs 3.2-3.8. The initial values of estimated parameters areaˆ(0)=bˆ(0)=cˆ(0)=dˆ(0)=eˆ(0)= 0.

Fig. 3.1 Chaotic phase portrait of new Duffing – Van der Pol system.

Fig. 3.2 The time histories of errors (e₁, e₂, e , ₃ e₄).

Fig. 3.3 The time histories of estimated parameter ˆa .

Fig. 3.4 The time histories of estimated parameter ˆb.

Fig. 3.5 The time histories of estimated parameter ˆc .

Fig. 3.6 The time histories of estimated parameter ˆd.

Fig. 3.7 The time histories of estimated parameter ˆe.

Fig. 3.8 The time histories of estimated parameter ˆf .

Chapter 4

Pragmatical Hybrid Projective and Symplectic Synchronization of Different Order Systems with New Control Lyapunov Function by Adaptive Backstepping

Control

4.1Preliminaries

In this Chapter, a new kind of synchronization and a new control Lyapunov function for backstepping are proposed. The symplectic* synchronization

( , , , )

y=H x y z t (4.1) is studied, where x and y are the state vectors of the ‘‘master’’ and of the ‘‘slave’’

respectively, and z is a given function vector of time, which may take various forms, either a regular or a chaotic function of time [28]. When

( , , , ) ( , , )

y=H x y z t =H x z t (4.2) it reduces to the generalized synchronization. Therefore symplectic [72] symchronization is an extension of generalized synchronization.

In Eq. (4.1), the final desired state y of the ‘‘slave’’ system not only depends upon the“master’’ system state x, but also depends upon the ‘‘slave’’ system state y itself.

Therefore the ‘‘slave’’ system is not a traditional pure slave completely obeying the

*The term ‘‘symplectic’’ comes from the Greek for ‘‘interwined’’. H. Weyl first introduced the term in 1939 in his book “The Classical Groups＂(P. 165 in both the first edition, 1939, and second edition, 1946, Princeton University Press)

‘‘master’’ system but plays a role to determine the final desired state of the ‘‘slave’’

system. In other words, it plays a ‘‘interwined’’ role, so we call this kind of synchronization ‘‘symplectic synchronization’’, and call the“master’’ system partner A, the ‘‘slave’’ system partner B.

The application of symplectic synchronization has great potential. For instance, if the symplectically synchronized chaotic signal is used as the signal carrier from a transmitter, secure communication is more difficult to be deciphered.

In this Chapter, a new control Lyapunov function ( ) exp( ^T ) 1

V e = ke e − (4.3) is proposed for backstepping control where e is error dynamics. Using the new control Lyapunov function, error tolerance can be decreased marvelously as comparing to that obtained by traditional control Lyapnov function

( ) ^T

V e =e e (4.4) This Chapter is organized as follows. In Section 2, by the GYC pragmatical asymptotical stability theorem with new control Lyapunov functions, a pragmatical hybrid projective and symplectic synchronization scheme is achieved. In Section 3, chaos in the Mathieu - Duffing system and Lusystem [36] with four dimensions are given. In Section 4, using the new control Lyapunov functions numerical results of PHPSS by adaptive backstepping control is achieved.

4.2. Symplectic synchronization scheme

Assume that there are two different nonlinear chaotic dynamical systems and that the partner A controls the partner B partially. The partner A is given by

( ) chaotic or regular function vector of time, h is a given vector function.

Our goal is to design the controller u(t) so that the state vector y of the partner B

A new control Lyapnuov function V(e) is chosen as a positive definite function of e: ( ) exp( ^T ) 1

V e = ke e − (4.12) where k is a positive constant. Its time derivative along any solution of Eq.(4.11)becomes

( ) 2 ^T{ ( ) ( ) H ( ) H ( ) H ( ) H}exp( ^T ) negative definite constant matrix, then V is a negative definite function of e. By Lyapunov theorem of asymptotical stability The symplectic synchronization is obtained [22-24].

4.3. Lü system

Lusystem can be described as follows [36]:

1 1 2 1 three states as the different order system for four states new Mathieu – Duffing system.

The additional fourth equation is chosen as

z₄ = + + (4.16) z₁ z₂ z₃ From Eq. (4.16) , it is obtained that

d ₄ ( _{2 1}) _{1 3 1 1 2 3} z a z z z z cz z z bz

dt = − − + + − (4.17) Augmented Lusystem with four states becomes:

1 1 2 1

4.4. Numerical results of CHPS by adaptive backstepping control

Take Mathieu-Duffing system (2.3) as master system, where a, b, c, d, ,

e f are uncertain parameters and following Mathieu-Duffing system as slave system:

1 2

1 2 2 2 2 2 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 2 1

Step1. We consider the asymptotical stability of e₁=0:

1 2 2 2 2 2 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 2 1

e = +e g x z y −g x z y −g a x z y +g a x z y −g x z y + (4.25) u where

e

₂ is regarded as a controller. Choose a control Lyapunov function of the form

2

1 exp( 1 1) 1 0

v = k e − > (4.26) Its time derivative along the solution of Eq.(4.25) is

1 1 1 2 2 2 2 2 1 2 1 1 1 1 1 2 1 1 1 1 1 1

Step2. When

e

₂ is considered as a controller,

α

₁

( e

₁

)

is an estimative function. Choose a control Lyapunov function of the form

2 1 2 2 2 2

2 1 exp( 2 2) 2( ) 1 0

v = +v k w + a +b +c +d − > , (4.33) where a= −a aˆ,b= −b b^ˆ,c= −c cˆ,d = −d d^ˆ and ˆa,ˆb, ˆc, ˆd are estimated values of the unknown parameters a, b, c, d, respectively. We have

2

2 2 Choose a control Lyapunov function of the form

2

3 2 exp( 3 3) 1 0

v = +v k w − > (4.39) Its time derivative through the third equation of Eq.(4.39) is

3 2 3 3 4 4 4 4 4 3 4 3 3 3 3 1 2 3 3 3 3 3 3

Choose a control Lyapunov function of the form

The Lyapunov asymptotical stability theorem can not be satisfied in this case. The

common origin of error dynamics and parameter update dynamics cannot be determined to be asymptotically stable. By GYC pragmatical asymptotical stability theorem (see Appendix ) D is a 10 -manifold, n = 10 and the number of error state variables p = 4.

Whene₁ = e₂ = e = ₃ e₄= 0 and a, b, c, d, e, f take arbitrary values , V = , 0 so X is of 6 dimensions, m = n – p = 10 – 4 = 6. m + 1≤ n are satisfied. By the GYC pragmatical asymptotical stability theorem, not only error vector e tends to zero but also all estimated parameters approach their uncertain parameters. PHPSS of chaotic systems by adaptive backstepping control is accomplished. The equilibrium point e₁ = e₂ = e ₃

= e₄ = a = b = c = d = e = f = 0 is asymptotically stable.

The numerical results are shown in Figs 4.2-4.6. the numerical data used are:

g = 0.0001 , 1 g = 0.0001₂ − , g = -0.0001 , ₃ g = -0.0001 ,₄ k₁=0.01, k₂ =0.01,

3 0.01,

k = and k₄ =0.01, e₁(0)= −2.00014 , (0) 9.99999e₂ = , e₃(0)= −2.00014 and

4(0) 9.99999

e = . Using the new control Lyapunov functions, the error tolerances are reduced marvelously to 10⁻¹⁷ of that using traditional control Lyapnov function

( ) ^T

V e =e e as shown in Figs.4.2-4.3. The estimated parameters approach the uncertain parameters of the chaotic system as shown in Figs. 4.4-4.6. The initial values of estimated parameters are ˆ(0)a = ˆ(0)b = ˆ(0)c = ˆ(0)d = ˆ(0)e = ˆ(0) 0f = . This property makes the proposed PHPSS is very effective.

Fig. 4.1 Chaotic phase protract of Lu system.

Fig. 4.2 Synchronization error for traditional control Lyapunov function ( )V e =e e^T from 600s ~ 1000s.

Fig. 4.3 Synchronization error for new control Lyapunov function ( ) exp(V e = ke e^T ) 1− from 600s ~ 1000s.

Fig. 4.4 The time histories of estimated parameter a and b.

Fig. 4.5 The time histories of estimated parameter c and d.

Fig. 4.6 The time histories of estimated parameter e and f .

Chapter 5

Chaos Control of a New Mathieu- Duffing System by GYC Partial Region Stability Theory

5.1 Preliminaries

In this Chapter, a new strategy to achieve chaos control by GYC partial region stability [33, 34] is proposed. By using the GYC partial region stability theory, the Lyapunov function is a simple linear homogeneous function of error states and the controllers are simpler than traditional controllers and so reduce the simulation error because they are in lower degree than that of traditional controllers.

This Chapter is organized as follows. In Section 2, chaos control scheme by GYC partial region stability theory is proposed. In Section 3, numerical Simulations of chaos control of new Mathieu – Duffing systems as simulated examples by GYC are achieved.

5.2 Chaos control scheme

The goal system which can be either chaotic or regular, is

( , )t

=

y g y (5.2)

where y=

[

y y1, , ,2 y_n

]

^T∈Rⁿ is a state vector, :g R₊×Rⁿ →Rⁿ is a vector function.

In order to make the chaos state x approaching the goal state y , define = −e x y as the state error. The chaos control is accomplished in the sense that:

lim lim( ) 0

t→∞e=t→∞ x y− = (5.3)

In this Chapter, we will use examples in which the e state is placed in the first quadrant of coordinate system and use on partial region stability theory. The Lyapunov function is a simple linear homogeneous function of states and the controllers are simpler because they are in lower degree than that of traditional controllers and so reduce the simulation error because they are in lower degree than that of traditional controllers.

5.3 Numerical simulations of chaos control by GYC

The following chaotic system is a Mathieu – Duffing system of which the old origin is translated to ( , , , ) (50, 50, 50, 50)x x x x_{1 2 3 4} = :

and the chaotic motion always happens in the first quadrant of coordinate system

1 2 3 4

( , , , )x x x x . This translated Mathieu – Duffing system is presented as simulated

examples where the initial conditions is x₁(0) 49,= x₂(0) 61,= x₃(0) 49,=

4(0) 61

x = . The chaotic motion is shown in Fig. 5.1.

In order to lead ( , , , )x x x x_{1 2 3 4} to the goal, we add control terms u1, u2 and u3 to each equation of Eq. (5.4), respectively.

1 2 1

CASE I. Control the chaotic motion to zero.

In this case we will control the chaotic motion of the Mathieu – Duffing system (5.4) to zero. The goal is y= . The state error is e x y x0 = − = and error dynamics becomes

In Fig. 5.1, we see that the error dynamics always exists in first quadrant.

By GYC partial region stability, one can easily choose a Lyapunov function in the form of a positive definite function in first quadrant as:

1 2 3 4

V = + + + (5.7) e e e e

Its time derivative through error dynamics (5.6) is

1 2 3 4

which is negative definite function. The numerical results are shown in Fig.5.2. After 30 sec, the motion trajectories approach the origin.

CASE II. Control the chaotic motion to a sine function.

In this case we will control the chaotic motion of the Mathieu – Duffing system (5.5) to sine function of time. The goal is y=msinwt. The error equation

2 2

In Fig. 5.3, the error dynamics always exists in first quadrant.

By GYC partial region stability, one can easily choose a Lyapunov function in the form of a positive definite function in first quadrant as:

1 2 3 4

which is negative definite function. The numerical results are shown in Fig.5.4 and Fig.

5.5, where m=0.5 and w₁ =w₂ =w₃=w₄ = . After 30 sec., the errors approach zero 2 and the motion trajectories approach to sine functions.

CASE III. Control the chaotic motion of a new Mathieu – Duffing system to chaotic

motion of a generalized Lorenz system.

In this case we will control chaotic motion of a new Mathieu – Duffing system (5.5) to that of a generalized Lorenz system. The goal system is generalized Lorenz system [22]: By Fig. 5.6, we know the error dynamics always exists in first quadrant.

By GYC partial region stability, one can easily choose a Lyapunov function in the form of a positive definite function in first quadrant as:

1 2 3 4

V = + + + (5.20) e e e e

Its time derivative is

( )

which is negative definite function. The numerical results are shown in Fig.5.7 and Fig.5.8 where a=0.2, b=0.2, and c=5.7. After 30 sec, the errors approach zero and the chaotic trajectories of the new Mathieu – Duffing system approach to that of the generalized Lorenz system.

Fig. 5.1 Phase portrait of error dynamics for Case I.

Fig.5.2 Time histories of x x x x for Case I. ₁, , ,_{2 3 4}

Fig. 5.3 Phase portrait of error dynamics for Case II.

Fig. 5.4 Time histories of errors for Case II.

Fig. 5.5 Time histories of x x x x for Case II. ₁, , ,_{2 3 4}

Fig. 5.6 Phase portrait of error dynamics for Case III.

Fig. 5.7 Time histories of errors for Case III.

Fig. 5.8 Time histories of x x x x z z z z for Case III. _{1, 2, 3, 4, 1, 2, 3, 4}

Chapter 6

Chaos Generalized Synchronization of New Mathieu- Duffing Systems and Chaotization by

GYC Partial Region Stability Theory

6.1 Preliminaries

Among many kinds of synchronizations, the generalized synchronization is investigated. This means that we can give a function relationship between the state vector

x of the master and the state vector y of slave: y=G x( ). In this chapter, a new chaos generalized synchronization strategy by GYC partial region stability theory is proposed.

This Chapter is organized as follow. In Section 2, chaos generalized synchronization strategy are proposed. In Section 3, numerical simulations of chaos generalized synchronization of Mathieu – Duffing systems as simulated examples by GYC are achieved. In Section 4, chaotization of a regular motion to the chaotic motion of a new Mathieu –Duffing systemis studied.

6.2 Chaos generalized synchronization strategy

Consider the following unidirectional coupled chaotic systems ( , )

wherex=

[

x x1, , ,2 x_n

]

^T ∈Rⁿ , y=

[

y y1, , ,2 y_n

]

^T∈Rⁿ denote two state vectors, f and h are nonlinear vector functions, and u=

[

u u1, , ,2 u_n

]

^T∈Rⁿ is a control input vector.

The generalized synchronization can be accomplished when t→ ∞ , the limit of the error vector e=

[

e e1, , ,2 e_n

]

^T approaches zero:

By using the GYC partial region stability theory, the Lyapunov function is easier to find, a homogeneous function of first degree can be used to construct a positive definite Lyapunov function and the controllers can be designed in lower degree.

6.3 Numerical simulations of chaos generalized synchronization by GYC theory

Two new Mathieu – Duffing systems, with the unidirectional coupling appear as

1 2

1 2 1 k =60 in order that the error dynamics always happens in first quadrant. 4

Our goal is y= +x k, i.e. the controlling goal is that

find that the error dynamics always exists in first quadrant as shown in Fig. 6.1. By GYC partial region asymptotical stability theorem, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

1 2 3 4

which is negative definite function in first quadrant. After 30 sec, four error states approach zero versus time as shown in Fig. 6.2. Time histories of states are shown in Fig.

6.3.

CASE II. The generalized synchronization error function is e_i = − +x_i y_i msinwt+ , k_i , (i=1, 2,3, 4).

Our goal is y_i = +x_i msinwt+k_i , i.e. lim _i lim( _{i i} sin _i) 0 in first quadrant as shown in Fig. 6.4. By GYC partial region asymptoical stability theorem, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

2

which is negative definite function in first quadrant. After 30 sec, four error states approach zero versus time as shown in Fig. 6.5. Time histories of x_i− + are shown y_i k_i in Fig. 6.6.

CASE III. The generalized synchronization error function is 1 ²

i 2 i i i we find that the error dynamics always exists in first quadrant as shown in Fig. 6.7. By GYC partial region asymptotial stability theorem, one can choose a Lyapunov function in

the form of a positive definite function in first quadrant:

which is negative definite function in first quadrant. After 30 sec, four error states approach zero versus time as shown in Fig. 6.8. Time histories of ¹_{2 i}x²− + are y_i k

z is the state vector of generalized Lorenz system.

The goal system for synchronization is generalized chaotic Lorenz system [12]:

1 1 2 1 4 we find the error dynamics always exists in first quadrant as shown in Fig. 6.10. By GYC partial region asymptotical stability theorem, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

1 2 3 4

2

which is negative definite function in first quadrant. After 30 sec, four error states approach zero versus time as shown in Fig. 6.11. Time histories of x_i − + are shown y_i k_i in Fig. 6.12.

6.4 Chaotization to a new Mathieu –Duffing system

In this section we will control a periodic motion of a periodic system to the chaotic motion of the new Mathieu – Duffing system. The periodic system is

1 2 3

In order to lead ( ,y y_{1 2}, y₃,y₄) to the goal, we add control terms u1, u2 and u3 to each equation of Eq. (6.32), respectively.

1 2 3 1

The goal system is new Mathieu – Duffing system:

1 2 that the error dynamics always happens in first quadrant.

Our goal is x= +y k, i.e. the controlling goal is that lim _i lim( _{i i i}) 0

t e t y x k

→∞ = →∞ − + = , ( 1, 2,3, 4)i= (6.36) The error dynamics becomes

2 2 the error dynamics always exists in first quadrant as shown in Fig. 6.14. By GYC partial region asymptotical stability theorem, one can choose a Lyapunov function in the form of a positive definite function in first quadrant:

1 2 3 4

which is negative definite function. After 30 sec, four error states approach zero versus

time as shown in Fig. 6.15. Time histories of states are shown in Fig. 6.16.

Fig. 6.1 Phase portrait of four errors dynamics for Case I.

Fig.6.2 Time histories of errors for Case I.

Fig. 6.3 Time histories of x x x x y y y y for Case I. _{1, 2, 3, 4, 1, 2, 3, 4}

Fig. 6.4 Phase portrait of error dynamics for Case II.

Fig. 6.5 Time histories of errors for Case II.

Fig. 6.6 Time histories of x_i− + and y_i k_i −msinωt for Case II.

Fig. 6.7 Phase portrait of error dynamics for Case III.

Fig. 6.8 Time histories of errors for Case III.

Fig. 6.9 Time histories of ¹_{2 i}x and ² y for Case III. _i

Fig. 6.10 Phase portrait of error dymanics for Case IV.

Fig. 6.11 Time histories of errors for Case IV.

Fig. 6.12 Time histories of x_i− + and y_i k_i − for Case IV. z_i

Fig. 6.13 Phase portraits of a new periodic system.

.

Fig. 6.14 Phase portraits of error dynamics for Section 6.4.

Fig. 6.15 Time histories of errors.

Fig. 6.16 Time histories of x x x x y y y y . _{1, 2, 3, 4, 1, 2, 3, 4}

Chapter 7

Chaos of a New Mathieu – Duffing System with Bessel Function Parameters

7.1 Preliminaries

The chaotic behaviors in a new Mathieu - Duffing system with Bessel function parameters is studied numerically by phase portraits, Poincaré maps and Lyapunov exponent diagram. It is found that chaos exists.

7.2 A new Mathieu – Duffing system

with Bessel function parameters

Mathieu system and Duffing system are two typical nonlinear nonautonomous systems:

obtain a new autonomous Mathieu – Duffing system:

1 2

A new Mathieu – Duffing system with Bessel function [75] parameters is obtained, where

2

where Γ is Gamma function, J±μ is Bessel function of the first kind, Y₀ is Bessel function of the second kind, I±μ is modified Bessel function of the first kind, Kμ is

modified Bessel function of the second kind, H₁¹is Bessel function of the third kind. The time histories of ( )a t , ( )b t , ( )c t , d t , ( )( ) e t , f t are shown in Figs 7.1-7.6. The ( ) numerical simulations are carried out by MATLAB with using the fractional operator in the Simulink environment.

7.3 Simulation results

The time history of four states, phase portraits, Poincaré maps, power spectrum, bifurcation and Lyapunov exponents of the new Mathieu – Duffing system Bessel function parameters are showed in Fig. 7.7~Fig. 7.12. Chaos exists for all cases.

For Figs. 7.7-7.12, we vary the system parameter d, other system parameters are fixed, which are given a t( )=J t₀( ) 15+ , b t( )=Y t₀( +0.01) 1+ ,

( ) 1( 0.01) 0.005

c t =K t+ + , d t( )=H₁¹(t+0.01) 23− , e t( )=K t₀( +0.01) 0.002+ , ( ) 1( ) 14

f t =J t + . The numerical simulations are carried out by FORTRAN with using the Runge-Kutta method for four dimensions in Fig.7.12. .

Fig. 7.1 The time history of a(t) with 50 sec.

Fig. 7.2 The time history of b(t) with 50 sec.

Fig. 7.3 The time history of c(t) with 50 sec.

Fig. 7.4 The time history of d(t) with 50 sec.

Fig. 7.5 The time history of e(t) with 50 sec.

Fig. 7.6 The time history of f(t) with 50 sec.

Fig. 7.7 The time history of the four states with 5000 sec.

Fig. 7.8 The phase portrait and Poincaré map of x x dimensions. ₁, ₂

Fig. 7.9 The phase portrait and Poincaré map of x x dimensions. ₃, ₄

Fig. 7.10 The chaotic power spectrum of x for Mathieu-Duffing system. ₁

Fig. 7.11 Bifurcation of chaotic Mathieu-Duffing system.

Fig. 7.12 Lyapunov exponents of chaotic Mathieu-Duffing system (+,0,-,-).

Chapter 8

Pragmatical Hybrid Projective Generalized Synchronization of New Mathieu- Duffing Systems with Bessel Function Parameters by Adaptive Control

and GYC Partial Region Stability Theory

8.1 Preliminaries

In this Chapter, our study is devoted to a pragmatical hybrid projective generalized synchronization of two chaotic systems of four-dimension, two new Mathieu-Duffing systems with Bessel function parameters. The parameters of master system of the new Mathieu-Duffing system are fully unknown or uncertain. Based on GYC pragmatical asymptotical stability theorem, GYC partial region stability theory, and adaptive control the master-slave systems are in pragmatical hybrid projective generalized synchronization (PHPGS). Both projective synchronization and projective anti-synchronization are

In this Chapter, our study is devoted to a pragmatical hybrid projective generalized synchronization of two chaotic systems of four-dimension, two new Mathieu-Duffing systems with Bessel function parameters. The parameters of master system of the new Mathieu-Duffing system are fully unknown or uncertain. Based on GYC pragmatical asymptotical stability theorem, GYC partial region stability theory, and adaptive control the master-slave systems are in pragmatical hybrid projective generalized synchronization (PHPGS). Both projective synchronization and projective anti-synchronization are

在文檔中 新Mathieu-Duffing 系統的渾沌現象與其應用適應逆步控制及部分區域穩定理論之實用混合投影渾沌同步及辛渾沌同步 (頁 20-0)