Numerical simulations of chaos control by GYC

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 obtained. Numerical simulations prove the effectiveness of the scheme.

8.2 Synchronization scheme

Among many kinds of synchronizations [18-24], the generalized synchronization is investigated [25-30]. This means that we can give a function relationship between the states of the master and slave: y = G(x) . In this Chapter, a special case of hybrid

projective generalized synchronizations g

y = G(x) = x(t) (8.1) is studied, where x and y are state variable vectors of the master and slave, respectively, and G is a given vector function. Since g is a constant rector with both positive and negative entries, hybrid projective synchronization is named. GYC pragmatical asymptotical stability theorem is used, pragmatical synchronization is named.

As a whole, pragmatical hybrid projective generalized synchronization (PHPGS) is named.

The control task is to force the slave state vector to track an n-dimensional desired vector can be accomplished on the base of GYC pragmatical asymptotical stability theorem and GYC partial region stability theory. By using the GYC partial region stability theory, the

Lyapunov function is easier to find as a homogeneous function of first degree of error states, which is a positive definite Lyapunov function of error states in first quadrant. The controllers can be designed in lower degree.

8.3. Numerical results of PHPGS by GYC partial region stability theory

Take new Mathieu-Duffing system with Bessel function parameters (7.3) as master system, where a, b, c, d, e, f are uncertain parameters and following new Mathieu-Duffing system as slave system:

1 2

k =k =k =k =100 , in order that the error dynamics always happens in first quadrant.

In order to lead( , , , )y y y y_{1 2 3 4} to(g x_{1 1}+k g x₁, _{2 2}+k g x₂, _{3 3}+k g x₃, _{4 4}+k₄) , u u u ₁, ,_{2 3} and u₄ are added to each equation of Eq.(8.7), respectively:

1 2 1 Fig. 8.1. By GYC partial region asymptotical stability theorem, one can choose a

Lyapunov function in the form of a positive definite function of e , ₁ e , ₂ e , ₃ e , ₄ a, b,

c, d, e, f in first quadrant:

1 2 3 4

quadrant.

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. PHPGS of chaotic systems by GYC partial region stability theory 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 8.3-8.5, The generalized synchronization is accomplished with g = 1.5, ₁ − g = 0.5,₂ g = 1.5,₃ − and g = 2₄ while

1(0) 105.000001,

e = e₂(0) 94.999999,= e₃(0) 105.000001= and e₄(0) 119.99999= . Four error states versus time are shown in Fig. 8.2. The estimated parameters approach the uncertain parameters of the chaotic system as shown in Figs. 8.3-5. The initial values of estimated parameters are ˆ(0) 15a = , ˆ(0) 1b = , ˆ(0) 0.005c = , ˆ(0)d = −23 ,

ˆ(0) 0.002

e = and ˆ (0) 14f = .

Fig. 8.1 Phase portraits of error dynamics.

Fig. 8.2 The time histories of errors (e₁, e₂, e , ₃ e₄).

Fig. 8.3 The time histories of a and b.

Fig. 8.4 The time histories of c and d.

Fig. 8.5 The time histories of e and f .

Chapter 9 Conclusions

Chaotic system features that it has complex dynamical behaviors and sensitive behavior dependence initial conditions. Because of this property, chaotic systems are thought difficult to be synchronized or controlled. In practice, some or all of the system parameters are uncertain. Additionally, these parameters change at every time. A lot of researchers have studied to solve this problem by different control theories. There are many control techniques which are presented to synchronize and control chaotic systems, such as backstepping design method [2], impulsive control method [3], invariant manifold method [4], adaptive control method [5], linear and nonlinear feedback control method [6], and active control approach [7], PC method [8], etc. In this thesis, we have studied the chaos of a new Mathieu – Duffing system by phase portraits, Poincaré maps, power spectrum and Lyapunov exponent diagram in Chapter 2.

In Chapter 3, by the pragmatical asymptotical stability theorem, the estimated parameters approach uncertain parameters can be answered strictly. In the current scheme of adaptive synchronization [12-15], the traditional Lyapunov stability theorem and Babalat lemma are used to prove that the error vector approaches zero, as time approaches infinity. But the question of that why the estimated parameters also approach uncertain parameters remains unanswered. By the pragmatical asymptotical stability theorem, the question can be answered strictly. A new Mathieu – Duffing system and a new Duffing – van der Pol system are used as simulated example. Pragmatical hybrid projective hyper chaotic generalized synchronization of chaotic systems by adaptive backstepping control is accomplished.

In Chapter 4, a new kind of symplectic synchronization and a new control Lyapunov

function are proposed. A new kind of symplectic synchronization plays a ‘‘interwined’’

role, so we call the“master’’ system partner A, the ‘‘slave’’ system partner B. Using the new control Lyapunov function

( ) exp( ^T ) 1

V e = ke e − (9.1) , the error tolerance introduced by using this new control Lyapunov function can be reduced marvelously to 10⁻¹⁷ of that using traditional control Lyapnov function

( ) ^T

V e =e e (9.2) In Chapter 5, by the GYC partial region stability theory, chaos control is achieved.

In GYC partial region stability theory, Lyapunov function is simpler a traditional Lyapunov function of error states, which is a linear homogenous function of error states.

The simulation error can be reduced by using the GYC partial region stability and simple controllers. A new Mathieu – Duffing system in the first quadrant is used as simulated examples which effectively confirm the scheme.

In Chapter 6, 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. Two new Mathieu – Duffing systems are in chaos generalized synchronization successfully.

In Chapter 7, the chaotic behaviors of a new Mathieu – Duffing systems Bessel function parameters are first proposed. The chaotic behaviors of a new Mathieu – Duffing systems with Bessel function parameters is studied numerically by phase portraits, Poincaré maps, bifurcation diagram and Lyapunov exponent diagram.

In Chapter 8, by the GYC pragmatical asymptotical stability theorem and GYC partial region stability theory, the error vector tends to zero and the estimated parameters approach uncertain values is guaranteed and the controllers of are simpler than traditional controllers and so reduce the simulation error. Pragmatical hybrid projective generalized

synchronization of new Mathieu-Duffing systems with Bessel function parameters by adaptive control is achieved. The GYC pragmatical asymptotical stability theorem and GYC partial region stability theory are powerful to synchronize and control chaotic systems. The security of communication is greatly increased.

Appendix A

GYC Pragmatical asymptotical theorem

The stability for many problems in real dynamical systems is actual asymptotical stability, although it may not be mathematical asymptotical stability. The mathematical asymptotical stability demands that trajectories from all initial states in the neighborhood of zero solution must approach the origin as t→∞. If there are only a small part or even one of the initial states from which the trajectories or trajectory do not approach the origin as t→∞, the zero solution is not mathematically asymptotically stable. However, when the probability of occurrence of an event is zero, it means the event does not occur actually. If the probability of occurrence of the event that the trajectries from the initial states are that they do not approach zero when t→∞, is zero, the stability of zero solution is actual asymptotical stability though it is not mathematical asymptotical stability. In order to analyze the asymptotical stability of the equilibrium point of such systems, the pragmatical asymptotical stability theorem is used.

Let X and Y be two manifolds of dimensions m and n(m<n), respectively, and ϕ be a differentiable map from X to Y; then ϕ(X) is a subset of Lebesque measure 0 of Y [74] . For an autonomous system

)

x be an equilibrium point for the system (A.1), then

0

Definition : The equilibrium point for the dynamic system is pragmatically asymptotically stable provided that with initial points on C which is a subset of Lebesque measure 0 of D, the behaviors of the corresponding trajectories cannot be determined, while with initial points on D−C, the corresponding trajectories behave as that agree with traditional asymptotical stability [17, 18].

Theorem: Let V =[x₁,x₂, x_n]^T :D→R₊ positive definite, analytic on D, where

1, ,2 _n

x x x are all space coordinates such that the derivative of V through differential equation, V , is negative semi-definite.

Let X be the m-manifold consisting of point set for which ∀x≠0, 0V(x)= and D is an n-manifold. If m+1<n, then the equilibrium point of the system is pragmatically asymptotically stable.

Proof :Since every point of X can be passed by a trajectory of Eq.(A.1), which is one dimensional, the collection of these trajectories, C, is a (m+1)-manifold [17, 18]. If

) 1

(m+ ＜n, then the collection C is a subset of Lebesque measure 0 of D. By the above definition, the equilibrium point of the system is pragmatically asymptotically stable. □

If an initial point is ergodicly chosen in D, the probability of that the initial point falls on the collection C is zero. Here, equal probability is assumed for every point chosen as an initial point in the neighborhood of the equilibrium point. Hence, the event

If an initial point is ergodicly chosen in D, the probability of that the initial point falls on the collection C is zero. Here, equal probability is assumed for every point chosen as an initial point in the neighborhood of the equilibrium point. Hence, the event

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 48-0)