Numerical Simulations of Boids Control - Boids Control of Chaos for an Inertial Tachometer Syst

Chapter 5 Boids Control of Chaos for an Inertial Tachometer System

5.4 Numerical Simulations of Boids Control

The inertial tachometer system is the master system:

1 2

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

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

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 r₂ 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.

Fig. 5.1 Mechanical model of an inertial tachometer.

Fig . 5.2 Chaotic phase portrait for inertial tachometer system.

Fig . 5.3 Chaotic phase portrait for inertial tachometer system.

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

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

Fig. 5.6 Flocking of two inertial tachometer systems.

Fig. 5.7 Distance between two inertial tachometer systems.

Fig. 5.8 Synchronization of two inertial tachometer systems.

Fig .5.9 Separation of two inertial tachometer systems.

Fig. 5.10 Distance between two inertial tachometer systems.

Fig. 5.11 Desynchronization of two inertial tachometer systems.

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

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

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

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

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

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

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.

Fig. 6.1 The time history ( )σ t .

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

Fig . 6.3 The time histories of the three states.

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

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

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

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.

(a )

(b) Amplified diagram for λ ₂

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

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.

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

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

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

β = , 1 P₁=0.8, ω₁=1, and the initial condition is ( , ) (0.5, 0.2)x x₁ 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

and the initial condition is ( , , ) (0.3, 0.02, 0.2)x x x₁ 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.

Fig. 7.1 Phase portrait of Duffing system.

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

Fig. 7.3 Phase portrait of van der Pol system.

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

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

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

Fig. 7.7 Phase portrait of van der Pol system.

Fig. 7.8 Phase portraits of Chen-Lee system.

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.

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

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.

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.

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 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 subregions (Fig. A.1).

84 is satisfied for the solutions of Eq.(A-27) on Ω , then the disturbed motion

0 ( 1, )

xs = s= Ln is stable on the partial region Ω. Definition 2:

If the undisturbed motion is stable on the partial region Ω, and there exists a

' 0 is satisfied for the solutions of Eq.(A-1) on Ω , then the undisturbed motion

0 ( 1, )

xs = s= Ln is asymptotically stable on the partial region Ω.

The intersection of Ω and region defined by Eq.(A-2) is called the region of attraction. are single-valued and have continuous partial derivatives and become zero when

1 _n 0

x =L=x = . Definition 3:

If there exists t₀ >0 and a sufficiently small h>0, so that on partial region Ω1 and t≥t₀, V ≥0 (or ≤0), then V is a positive (or negative) semidefinite, in

Definition 6: Bounded function V

If there exist t₀ >0, h>0, so that on the partial region Ω₁, we have

( , , , )1 _n

V t x K x <L (B.9) where L is a positive constant, then V is said to be bounded on Ω₁.

Definition 7: Function with infinitesimal upper bound

If V is bounded, and for any λ >0, there exists μ >0, so that on Ω₁ when then V admits an infinitesimal upper bound on Ω₁.

Theorem 1 [28, 29]

If there can be found for the differential equations of the disturbed motion (Eq.(A-27)) a definite function V t x( , , , )₁ K x_n on the partial region, and for which the derivative with respect to time based on these equations as given by the following :

86 is a semidefinite function on the paritial region whose sense is opposite to that of V, or if it becomes zero identically, then the undisturbed motion is stable on the partial region.

Proof:

Let us assume for the sake of definiteness that V is a positive definite function.

Consequently, there exists a sufficiently large number t₀ and a sufficiently small number h < H, such that on the intersection Ω₁ of partial region Ω and and t≥t₀, the following inequality is satisfied

1 1

( , , , )_n ( , , )_n

V t x K x ≥W x K x (A-13) where W is a certain positive definite function which does not depend on t. Besides that, Eq. (A-7) may assume only negative or zero value in this region.

Let ε be an arbitrarily small positive number. We shall suppose that in any case ε <h. Let us consider the aggregation of all possible values of the quantities

1, , _n and let us designate by l>0 the precise lower limit of the function W under this condition. by virtue of Eq. (B.5), we shall have

( , , , )1 _n

V t x K x ≥l for ( , , )x₁K x_n on ω . (A-15) ₂ We shall now consider the quantities x_s as functions of time which satisfy the differential equations of disturbed motion. We shall assume that the initial values x_s₀ of these functions for t=t₀ lie on the intersection Ω₂of Ω₁and the region

87 obviously possible. We shall suppose that in any case the number δ is smaller than

ε .Then the inequality

2 ,

s s

x <ε

∑

(A-18) being satisfied at the initial instant will be satisfied, in the very least, for a sufficiently small t−t₀, since the functions x t_s( ) very continuously with time. We shall show that these inequalities will be satisfied for all values t>t₀. Indeed, if these inequalities were not satisfied at some time, there would have to exist such an instant t=T for which this inequality would become an equality. In other words, we would have

and consequently, on the basis of Eq. (A-9) ( , ( ), , ( ))1 _n

V T x T K x T ≥l (A-20) On the other hand, since ε <h, the inequality (Eq.(A-4)) is satisfied in the entire interval of time [t0, T], and consequently, in this entire time interval dV 0

dt ≤ . This yields

1 0 10 0

( , ( ), , ( ))_n ( , , , _n ),

V T x T K x T ≤V t x K x (A-21) which contradicts Eq. (A-12) on the basis of Eq. (A-11). Thus, the inequality (Eq.(A-1)) must be satisfied for all values of t>t₀, hence follows that the motion is stable.

Finally, we must point out that from the view-point of mathenatics, the stability on partial region in general does not be related logically to the stability on whole region. If an undisturbed solution is stable on a partial region, it may be either stable

or unstable on the whole region and vice versa. From the viewpoint of dynamics, we wre not interesting to the solution starting from Ω₂ and going out of Ω.

Theorem 2 [28, 29]

If in satisfying the conditions of theorem 1, the derivative ^dV

dt is a definite function on the partial region with opposite sign to that of V and the function V itself permits an infinitesimal upper limit, then the undisturbed motion is asymptotically stable on the partial region.

Proof:

Let us suppose that V is a positive definite function on the partial region and that consequently, ^dV

dt is negative definite. Thus on the intersection Ω₁ of Ω and the region defined by Eq. (A-4) and t≥t₀ there will be satisfied not only the inequality (Eq.(A-5)), but the following inequality as will:

1( ,1 _n),

dV W x x

dt ≤ − K (A-22) where W₁ is a positive definite function on the partial region independent of t.

Let us consider the quantities x_s as functions of time which satisfy the differential equations of disturbed motion assuming that the initial values x_s₀ =x t_s( )₀ of these quantities satisfy the inequalities (Eq. (A-10)). Since the undisturbed motion is stable in any case, the magnitude δ may be selected so small that for all values of t≥t0 the quantities x_s remain within Ω₁. Then, on the basis of Eq. (A-13) the derivative of function V t x t( , ( ), , ( ))₁ K x t_n will be negative at all times and, consequently, this function will approach a certain limit, as t increases without limit, remaining larger than this limit at all times. We shall show that this limit is equal to some positive quantity different from zero. Then for all values of t≥t₀ the following inequality will be satisfied:

89 ( , ( ), , ( ))1 _n

V t x t K x t >α (A-23) where α >0.

Since V permits an infinitesimal upper limit, it follows from this inequality that

2( ) , ( 1, , ),

s s

x t ≥λ s= n

∑

^K (B.24) where λ is a certain sufficiently small positive number. Indeed, if such a number λ did not exist, that is , if the quantity _s( )

∑

x t were smaller than any preassigned number no matter how small, then the magnitude V t x t( , ( ), , ( ))₁ K x t_n , as follows from the definition of an infinitesimal upper limit, would also be arbitrarily small, which contradicts (A-14).

If for all values of t≥t₀ the inequality (Eq. (A-15)) is satisfied, then Eq. (A-13) shows that the following inequality will be satisfied at all times:

1, dV l

dt ≤ − (A-25) where l₁ is positive number different from zero which constitutes the precise lower limit of the function W t x t₁( , ( ), , ( ))₁ K x t_n under condition (Eq. (A-15)). Consequently,

which is, obviously, in contradiction with Eq.(A-14). The contradiction thus obtained shows that the function V t x t( , ( ), , ( ))₁ K x t_n approached zero as t increase without limit. Consequently, the same will be true for the function W x t( ( ), , ( ))₁ K x t_n as well, from which it follows directly that

lim ( ) 0, (_s 1, , ),

t x t s n

→∞ = = K (A-26) which proves the theorem.

subregion 2

subregion 3 subregion 1

Ω Ω

Ω

X₁ O

Ω1

Fig. A.1. Partial regions Ω and Ω₁.

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