Chaos Control Scheme

Consider the following chaotic systems ( , )t

=

x& f x (4-1)

where x=

[

x x1, , ,2 L x_n

]

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

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

=

y& g y (4-2)

where y=

[

y y1, , ,2 L y_n

]

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

In order to make the chaotic state x approaching the goal state y, define error

= −

e x y as the state error. The chaos control is accomplished in the sense that :

lim lim( ) 0

t→∞e=t→∞ x y− = (4-3)

In this Chapter, we will use examples in which the error dynamics happens in the

31

first quadrant of coordinate system and use the partial region stability theory. The Lyapunov function is a simple linear homogeneous function of error states and the controllers are simpler because they are in lower degree than that of traditional controllers.

4.3 Numerical Simulations

The following chaotic system

1 2

is the inertial tachometer system of which old origin is translated to

1 2 3 4

( , , , ) (20, 20, 20, 20)x x x x = in order that the error dynamics always happens in first quadrant. This system is presented as simulated examples where initial conditions are (x₁₀,x₂₀,x₃₀,x₄₀) (20, 20, 22, 22)= and the parameters are m₁=9 , m₂ =1 ,

10.5

A= , η =1, l=0.3, k=0.5, g =9.81.

In order to lead the states (x₁,x₂,x₃,x₄) to the goal, we add control terms u1, u2, u3 and u4 to each equation of Eq. (4-4), respectively.

32

In this case we will control the chaotic motion of the inertial tachometer system (2.3) to zero. The goal is y_i =0, (i=1, 2,3, 4). The state error is e_i =x_i −y_i =x_i,

lim _i lim( _i 0) 0

t e t x

→∞ = →∞ − = , (i=1, 2,3, 4). (4-6) The error dynamics becomes

1 1 2 1

In Fig. 4.1, we can 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:

4

Its time derivative through error dynamics (4-7) is

33

& & & & &

(4-9)

which are added at 50s.

We obtain

which is negative definite function in first quadrant. The time histories of error states are shown in Fig. 4.2. After 50 sec, the trajectories approach the origin.

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

In this case we will control the chaotic motion of the inertial tachometer system (2-3) to regular function of time. The goal is y_i =F_isinw t_i , (i=1, 2,3, 4). The error The error dynamics is

34

In Fig. 4.3, 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:

4

Its time derivative is

1 2 3 4

& & & & &

t

which are added at 50s.

35

which is negative definite function in first quadrant. The numerical results are shown in Fig. 4.4 and Fig. 4.5. After 50 sec., the errors approach zero and the chaotic trajectories approach to regular functions of time.

CASE III. Control the chaotic motion of the inertial tachometer system to chaotic motion of the new Mathieu-Van der pol system.

In this case we will control chaotic motion of the inertial tachometer system (2-3) to that of the new Mathieu-Van der pol system. The goal system for control is new Mathieu-Van der pol system and initial states are (0.1, -0.5, 0.1, -0.5), system parameters a₁=10, b₁=3, c₁ =0.4, d₁ =70, e₁=1, f₁=5, g₁ =0.1. The error dynamics become

1 1 1 2 1 2

By Fig. 4.6, we know that the error dynamics always exists in first quadrant.

36

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

4

Its time derivative is

1 2 3 4

& & & & &

2

which are added at 50s.

We obtain

which is negative definite function in first quadrant. The numerical results are shown in Fig. 4.7 and Fig. 4.8. After 50 sec., the errors approach zero and the chaotic trajectories of an inertial tachometer system approach to that of the new Mathieu-Van der pol system.

37

Fig. 4.1 Phase portraits of error dynamics for Case I.

Fig. 4.2 Time histories of errors for Case I.

38

Fig. 4.3 Phase portraits of error dynamics for Case II.

Fig. 4.4 Time histories of errors for Case II.

39

Fig. 4.5 Time histories of x1, x2, x3, x4 and Fsinw t_i for Case II.

Fig. 4.6 Phase portraits of error dynamics for Case III.

40

Fig. 4.7 Time histories of errors for Case III.

41

Fig. 4.8 Time histories of x1, x2, x3, x4 for Case III.

Chapter 5

Boids Control of Chaos for an Inertial Tachometer System

5.1 Preliminaries

The aggregate motion of a flock of birds or a herd of land animals is a beautiful and familiar part of the natural world. They exhibit complex and emergent behaviors such as flocking behavior, separation behavior, and obstacle avoiding behavior. This Chapter explores an approach based on simulation as an alternative to scripting the paths of each bird individually. Flock centering and separation, obstacle avoidance are studied. A nonautonomous inertial tachometer system is used for simulation example.

5.2 Boids Nonlinear Control

Many nonlinear systems which are known to present chaotic behavior are modeled by a set of nonlinear nonautonomous differential equations:

1 2

42

We will assume for simplicity that all boids are identical and each boid is coupled locally only to those neighbor boids whose trajectories lie inside a prescribed sphere S_α of radius ε : the dynamics of the locally coupled chaotic nonlinear networks, namely, the dynamics of boids nonlinear networks is defined by

1 2 1 2

Case I Flock Centering: Boids attempt to move toward the average position of nearby flockmates.

The center of nearby flockmates is defined by

( )

where N_α indicates the number of nearby flockmates. The boids can move toward the center x_i^αby using chaotic synchronization [36]. Therefore, flock centering is implemented here by imposing the control dynamics

1 2

43

where d_i^α＞0.

Case II Separation of Flocks: Boids keep a distance from different kinds of flocks.

A flock may attempt to go away from other kinds of flocks. If a flock gets close enough to a different groups of flocks, that is, if the distance between the centers of two flocks becomes less than ε_g＞0, boids attempt to scatter. Separation of flocks is implemented by the dynamics of chaotic desynchronization

1 2

Case III Obstacle Avoidance: Boids attempt to dodge static obstacles.

Assume that a static obstacle is defined by the equation

1 2

If a boid gets close enough to a static obstacle, that is, if the distance between a boid and a static obstacle is less than ε , the boids must attempt to dodge the static ₀ obstacle. Obstacle avoidance can be implemented by switching over to a new vector field:

44

5.3 Chaos of an Inertial Tachometer

In this section, an inertial tachometer system is studied. The physical model of the inertial tachometer system is shown in Fig.5.1. There exists viscous damping in bent rod bearing (O point). The mass of bent rod is neglected and the balls m₁ and

m2 are considered as two particles.

We can write the kinetic and potential energies of the system as follow：

2 2 2 2 2 2 2 2 2 2 2

J ：the moment of inertia of the shaft about vertical center axis, l ： the length of rod,

ϕ：the angle between the shaft and the rod, θ^&：the angular velocity of the shaft,

g ：gravity acceleration,

The Lagrangian is L T= − Π , the corresponding Lagrange equations are

2 2 2 where k is damping coefficient in bent rod bearing.

The state equation can be written as：

2

45

We assume that the inertial tachometer is subjected to an external vertical vibration to basement Asin( )η . The Lagrange equation now are given in a t noninertial vibrating reference frame, which is fixed with the basement. Due to the inertial force appearing in the noninertial frame, the gravity acceleration in the noninertial frame becomes g+Aη²sin( )ηt . Let ϕ=x₁ , ψ =x₂ , ω =x₃ , the equation (5-13) is rewritten in the form

1 2 (0, 0, 2). Its phase portraits as shown in Fig. 5.2 and Fig. 5.3. The Lyapunov exponents and the bifurcation diagram of the inertial tachometer are shown in Fig. 5.4 and Fig. 5.5 for A between 9.1 and 10.9.

5.4 Numerical Simulations of Boids Control

The inertial tachometer system is the master system:

1 2

46

The slave system is

1 2

Case I Flock Centering: Boids attempt to move toward the average position of nearby flockmates.

Flock centering is implemented here by imposing the control dynamics:

1 2

The slave system is rewritten as follows:

1 2 3

where d=0.000001.The simulations of flocking behavior of tachometer systems are shown in Figs. 5.6~5.8. The flocking of two tachometer systems are illustrated in Fig.

5.6. The distance between two systems is given in Fig. 5.7. The synchronization behavior of two tachometer systems is given in Fig. 5.8.

Case II Separation of Flocks: Boids keep a distance from different kinds of flocks.

Separation of flocks is implemented by the dynamics of chaotic

47

The slave system is rewritten as follows:

1 2 3

where d=-0.00002. The simulations of flocking behavior of tachometer systems are shown in Figs. 5.9~5.11. The separation of two tachometer systems is illustrated in Fig. 5.9. The distance between two systems is given in Fig. 5.10. The desynchronization behavior of two tachometer systems is given in Fig. 5.11.

Case III Obstacle Avoidance: Boids attempt to dodge static obstacles.

Obstacle avoidance can be implemented by switching over to a new vector field:

1 2

48

and its normal vector n=( , , )n n n_{x y z} at the point (x, y, z) by

1 1

( , , ) (2(n n n_{x y z} = x−x), 2(y−y),0) (5-25) Therefore, the “sphere” and the “cylinder” obstacles are specified by the parameters:

1 1 1 1

( , , , )x y z r and ( , , )x y r_{2 2 2} respectively.

CASE III-1: Obstacle Avoidance-sphere

The inertial tachometer system is rewritten as follows:

1 1 2 1 1 1 The simulations of the obstacle avoidance behavior for sphere are illustrated in Figs. 5.12~5.13.

CASE III-2: Obstacle Avoidance-cylinder

The inertial tachometer system is rewritten as follows:

1 1 2 1 1 2

r = , The simulations of the obstacle avoidance behavior for cylinder are illustrated in Figs.5.14~5.15.

49

Fig. 5.1 Mechanical model of an inertial tachometer.

Fig . 5.2 Chaotic phase portrait for inertial tachometer system.

50

Fig . 5.3 Chaotic phase portrait for inertial tachometer system.

Fig. 5.4 Lyapunov exponents for A between 9.1 and 10.9.

51

Fig. 5.5 Bifurcation diagram of x₁ for A between 9.1 and 10.9.

Fig. 5.6 Flocking of two inertial tachometer systems.

52

Fig. 5.7 Distance between two inertial tachometer systems.

Fig. 5.8 Synchronization of two inertial tachometer systems.

53

Fig .5.9 Separation of two inertial tachometer systems.

Fig. 5.10 Distance between two inertial tachometer systems.

54

Fig. 5.11 Desynchronization of two inertial tachometer systems.

Fig. 5.12 Obstacle avoidance for inertial tachometer system (sphere).

55

Fig. 5.13 Obstacle avoidance for inertial tachometer system (sphere).

Fig. 5.14 Obstacle avoidance for inertial tachometer system (cylinder).

56

Fig. 5.15 Obstacle avoidance for inertial tachometer system (cylinder).

57

Chapter 6

Hyperchaos of a Lorenz System with Bessel Function Parameters

6.1 Preliminaries

The chaotic behaviors of a Lorenz system with Bessel function parameters is firstly studied numerically by time histories of states, phase portraits, Poincaré maps, bifurcation diagram, Lyapunov exponents and parameter diagram. It is found that hyperchaos and chaos exist. The hyperchaos is identified by the existence of two positive Lyapunov exponents and gives more security for secret communication.

6.2 Lorenz System with Bessel Function Parameters

The Lorenz system which is equivalent to a four-dimensional autonomous system.

σ , γ , b are given as：

58 in Figs 6.1-6.2. The numerical simulations are carried out by MATLAB using the fractional operator in the Simulink environment.

6.3 Numerical Simulations

This system exhibits periodic motion when the parameters of system (6-1) are 10 J t0( )

σ = + , γ =20.2+Y t₀( +0.01) , b=8 / 3 and the initial condition is ( , , )x y z =(0.1, 10, 0.5) . When the parameters are σ =10+J t₀( ) ,

25 Y t0( 0.01)

γ = + + , b=8 / 3, the motion becomes chaotic. The time histories of three states, phase portraits, Poincaré maps, and bifurcation diagrams of the system are shown in Fig. 6.3~Fig 6.8.

Lyapunov exponents and parametric diagram are also given to certify the existence of hyperchaos. Let us assume Lyapunov exponents λ ( 1,2,3,4)_i i= satisfying λ₁>λ₂ >λ₃, and λ₄ =0. Then the dynamics of system (6-1) can be characterized as follows:

(1) When λ_1,2,3< and 0 λ₄ =0, system (6-2) is periodic.

(2) When λ₁ >0, λ_2,3< , and 0 λ₄ =0, system (6-2) exhibits chaotic motion.

(3) When λ_1,2 >0, λ₃ < , and 0 λ₄ =0, system (6-2) exhibits hyperchaotic motion.

Four cases are studied as follows.

Case I

Fix k₁, b, vary k₂. The Lyapunov exponents of the system (6-1) for k₁=1, and b=8 / 3 are shown in Fig. 6.9. The parametric diagram of system (6-1) for

59

varying k₁ and k₂ with b=8 / 3 is shown in Fig. 6.10. The white area corresponds to periodic motion. By simulation, system is periodic when 0.01≤k₂≤20.62. The blue area corresponds to chaotic motion. And the green area corresponds to hyperchaotic motion, which is identified by the existence of two positive Lyapunov exponents, as clearly shown in Fig. 6.9. As k₂ increases to 20.63≤k₂ ≤40, the system displays complex behavior, with an interweaving between chaotic and hyperchaotic motions. The hyperchaotic motion becomes more and more as k₂ increases. Just like Monet’s picture , Fig 6.10 gives a beautiful scene. White area is the bank of a river, blue area is the water of the river and green area is the duckweed in the river.

Case II

Fix b=8 / 3, vary k₁, k₂. k₁ increases intermittently for increment of 10. And k2 varies slowly for increment of 0.01. Some typical values of k₁ and k₂ that generate hyperchaos with two positive Lyapunov exponents are shown in Tables 1～3, respectively. Comparing Table 1～3, a particular phenomenon appears when k₁ increases. As k₁ increases, the value of Lyapunov exponent λ becomes larger. It ₂ means that larger k₁ can arouse hyperchaotic motion. In other words, hyperchaos is aroused with enlarged Bessel function of first kind.

Table 1 Typical values of parameter k₂ that generate hyperchaos for ^k1=1

60

35.74 0.95094 0.00126 -13.95231 0

Table 2 Typical values of parameter k₂ that generate hyperchaos for ^{k1 =10} a function of b to classify the chaotic or periodic motions. With increasing b, the motion of system (6-1) becomes periodic when 0.01≤ ≤b 0.5. Periodic motions occur again with b≥3.2. As b increases to 0.51≤ ≤b 3.19 , system displays chaotic behavior. In this case, hyperchaotic motion was not found.

61

Fig. 6.1 The time history ( )σ t .

Fig . 6.2 The time history of Y t₀( +0.01).

62

Fig . 6.3 The time histories of the three states.

Fig. 6.4 The phase portrait and Poincaré map for x , y states.

63

Fig . 6.5 The time histories of the x , y , z states.

Fig. 6.6 The phase portrait of x , y , z states.

64

Fig. 6.7 The phase portrait and Poincaré map for x , y states.

Fig. 6.8 The bifurcation diagram for σ =10+J t₀( ), and b=8 / 3.

65

(a )

(b) Amplified diagram for λ ₂

Fig. 6.9 Lyapunov exponents of system (2-2) for varying k₂.

66

Fig. 6.10 The parametric diagram of system (2-1) for varying k₁ and k₂.

F ig.6.11 Lyapunov exponents of system (2-2) for varying b, withk₁ =30andk₂ =28.

67

Chapter 7

Symplectic Symchronization of Different Order Nonautonomous Systems via Nonlinear Control

7.1 Preliminaries

In this chapter, a new symplectic synchronization*

( , , )

y=F x y t (7-1) is studied, where x , y are state vectors of the“master＂and of the“slave＂, respectively, F x y t( , , ) is a given function of x , y and time.

When ( , , )F x y t = ( , )F x t , Eq. (7-1) reduces to the generalized synchronization ( , )

y=F x t . Therefore the generalized synchronization is a special case of symplectic synchronization.

In Eq. (7-1), 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 yitself. 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. In other words, it plays an“interwined＂role, so we call this kind of synchronization“symplectic synchronization＂, and call the“master＂

system partner A, the“slave＂system partner B.

There exists great potential of the application of the symplectic synchronization.

For instance, when the symplectically synchronized chaotic signal is used as a signal carrier, the secure communication is more difficult to be deciphered. There are many control techniques to synchronize chaotic systems, such as linear error feedback

*The term “symplectic＂comes the Greek for“interwined＂. H. Weyl first introduced in 1939 in this book“The Classical

68

Groups＂(P. 165 in both the first edition, 1939, and second edition, 1946, Princeton University Press).

control, adaptive control, active control , fuzzy control, impulsive control [6-15].

This chapter proposes a new symplectic synchronization algorithm based on nonlinear control and Barbalat Lemma [34], which noticeably expanded the application ranges of generalized synchronization.

This chapter is organized as follow. In Section 2, symplectic synchronization scheme is proposed. In Section 3, Duffing system, Van der Pol system and Chen-Lee system [35] are used as simulated examples.

7.2 Symplectic Synchronization Scheme of Different Order Nonautonomous Chaotic Systems

Consider the nonautonomous master system :

1( , )

x&= f x t (7-2)

where x=[ ,..., ]x₁ x_n ^T∈ℜ is the state vector of partner A, ⁿ f₁( )⋅ is a continuous vector function.

The nonautonomous slave system is given by the following equation:

( ) ( ) ( , )2

69 synchronization errors are defined as

( , , )

From Eq. (7-5), it is obtained that

( , , )

Let the controller u is designed as

1 2

Construct a Lyapunov error function of the following form

1 1 2

( ) 2 2

V t = e eT = e (7-11) Evaluating the time derivative of ( )V t along the trajectory of Eq. (7-10) and

70

using the Lipschitz condition, we have

2 2 is uniformly continuous. According to the Barbalat Lemma [34], if ( )f t is uniformly continuous, and (7-10) is asymptotically stable. The partner A and the partner B are in symplectic synchronization.

7.3 Numerical Results

Two illustrative examples are given to demonstrate the validity of the proposed scheme.

Case I Symplectic synchronization of Duffing system and van der Pol system.

Consider the following Duffing system :

1 2

71

β = , 1 P₁=0.8, ω₁=1, and the initial condition is ( , ) (0.5, 0.2)x x_{1 2} = . Its phase portrait and time histories are shown in Fig. 7.1 and Fig. 7.2.

The van der Pol system is adopted as the partner B, which is

u= u u is the controller. This system exhibits chaos without controller when the parameters of system are ε =5, P₂ =1.25, ω₂ =4.2, and the initial condition

72 lim _i lim[ _{i i}( , , )] 0

t e t y F x y t

→∞ = →∞ − = , i=1,2 (7-16) With the parameters and initial conditions above, the controller u

1 2

is designed according to Eq. (7-8). Fig. 7.5 and Fig. 7.6 show the time histories of error functions e t₁( ), e t₂( ), respectively. Exactly, partner A (7-14) and partner B (7-15) achieve the symplectic synchronization.

Case II Symplectic synchronization of Chen-Lee system [35] and Duffing system.

Consider the following partner A, Chen-Lee system :

1 2 3 1

73

and the initial condition is ( , , ) (0.3, 0.02, 0.2)x x x_{1 2 3} = − . Its phase portraits and time histories as shown in Fig. 7.7 ～ Fig. 7.9.

The Duffing system is adopted as partner B :

1 2 1 With the parameters and initial conditions above, the controller u

74 is designed according to Eq. (7-8). Fig. 7.10 and Fig. 7.11 show the time histories of

error functions e t₁( ), e t₂( ), respectively. Exactly, partner A (7-18) and partner B (7-19) are in symplectic synchronization.

75

Fig. 7.1 Phase portrait of Duffing system.

Fig. 7.2 Time histories of two states of Duffing system.

76

Fig. 7.3 Phase portrait of van der Pol system.

Fig. 7.4 Time histories of the two states of van der Pol system.

77

Fig. 7.5 Time history of error e t₁( ) for Case I.

Fig. 7.6 Time history of error e t₂( ) for Case I.

78

Fig. 7.7 Phase portrait of van der Pol system.

Fig. 7.8 Phase portraits of Chen-Lee system.

79

Fig. 7.9 Time histories of the three states of Chen-Lee system.

Fig. 7.10 Time history of error e t₁( ) for Case II.

80

Fig. 7.11 Time history of error e t₂( ) for Case II.

81

Chapter 8 Conclusions

Chaos and boids control, generalized, symplectic synchronization and htperchaos of chaotic systems are studied in this thesis.

In Chapter 2, the chaotic behavior in an inertial tachometer system is studied by phase portraits, time history, Poincaré maps, Lyapunov exponent, bifurcation diagrams and parametric diagram.

In Chapter 3 and Chapter 4, a new strategy to achieve chaos generalized synchronization and chaos control by GYC partial region stability theory are 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 more simple and have less simulation error because they are in lower order than that of traditional controllers.

In Chapter 5, boids control that is an interesting strategy for control is presented.

We have investigated several chaotic nonlinear networks controlled by several boids rules. They exhibited complex and emergent behaviors. The “synchronization”

phenomenon can only be achieved with the proposed model. In this Chapter, the chaotic boids are controlled by using three state variables, and all boids are assumed to be identical for simplicity.

In Chapter 6, Lorenz system with Bessel function parameters is studied firstly.

The results are verified by time histories of states, phase portraits, Poincaré maps, bifurcation analysis, Lyapunov exponents and parametric diagram. Abundant hyperchaos is found for this system. Especially enlarging the parameter with Bessel function, the hyperchaos is more obvious.

82

In Chapter 7, a new symplectic synchronization problems of nonautonomous chaotic systems are investigated based on Barbalat lemma [34]. Traditional generalized synchronizations are special cases of the symplectic synchronization. A sufficient condition is derived to ensure the symplectic synchronization between two different systems. The simulation results show that the proposed scheme can achieve not only the symplectic synchronization of chaotic systems with same order, but also the symplectic synchronization between chaotic systems with different orders.

Symplectic synchronization may be applied to the design of secret communication with more security than generalized synchronization.

In Appendix A, GYC (Ge-Yao-Chen) partial region stability theory is given.

83

Appendix A

GYC Partial Region Stability Theory

Consider the differential equations of disturbed motion of a nonautonomous system in the normal form

( , , , ),1 ( 1, , )

s

s n

dx X t x x s n

dt = L = L (A-1) where the function X_s is defined on the intersection of the partial region Ω (shown in Fig. A-1) and that X_s is smooth enough to ensure the existence, uniqueness of the solution of the initial value problem. When X_s does not contain t explicitly, the system is autonomous.

Obviously, x_s =0 (s= L1, n) is a solution of Eq.( A-1). We are interested to the asymptotical stability of this zero solution on partial region Ω (including the boundary) of the neighborhood of the origin which in general may consist of several

Obviously, x_s =0 (s= L1, n) is a solution of Eq.( A-1). We are interested to the asymptotical stability of this zero solution on partial region Ω (including the boundary) of the neighborhood of the origin which in general may consist of several

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