Pragmatical generalized synchronization of chaotic systems with uncertain parameters by adaptive control

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www.elsevier.com/locate/physd

Pragmatical generalized synchronization of chaotic systems with uncertain

parameters by adaptive control

Zheng-Ming Ge

∗

, Cheng-Hsiung Yang

Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan, ROC Received 4 September 2006; received in revised form 5 March 2007; accepted 30 March 2007

Available online 20 April 2007 Communicated by R. Roy

Abstract

A new kind of generalized synchronization of two chaotic systems with uncertain parameters is proposed. Based on a pragmatical asymptotical stability theorem and an assumption of equal probability for ergodic initial conditions, an adaptive control law is derived so that it can be proved strictly that the common null solution of error dynamics and of parameter dynamics is actually asymptotically stable, i.e. these two identical systems are in generalized synchronization and the estimated parameters approach the uncertain values. It is called pragmatical generalized synchronization. Finally, two numerical examples are studied for two Quantum-CNN oscillator chaotic systems to show the effectiveness of the proposed generalized synchronization strategy with a double Duffing chaotic system as a goal system.

c

Keywords:Quantum Cellular Neural Network (Quantum-CNN); Chaos; Pragmatical generalized synchronization; Pragmatic asymptotical stability theorem; Adaptive control

1. Introduction

The synchronization phenomenon has the following feature: the trajectories of the drive and response systems are identical notwithstanding starting from different initial conditions. However, slight errors of initial conditions, for chaotic dynamical systems, will lead to completely different trajectories [1–4]. Therefore, how controlling two chaotic systems to be synchronized is an attractive objective [5–8]. Many approaches have been presented for the synchronization of chaotic systems such as linear and nonlinear feedback control [9,10]. Most of them are based on the exact knowledge of the system structure and parameters. But in practice, some or all of the system parameters are uncertain. Moreover, these parameters change from time to time. A lot of works have proceeded to solve this problem by adaptive synchronization [11,12]. In the current scheme of adaptive synchronization [13–15], the traditional Lyapunov stability theorem and Babalat lemma are used to prove that the error

∗_{Corresponding author. Tel.: +886 3 5712121 55119; fax: +886 3 5720634.}

E-mail address:[email protected](Z.-M. Ge).

vector approaches zero as time approaches infinity. But the question of why the estimated parameters also approach uncertain values has remained without answer. Based on a pragmatical asymptotical stability theorem and an assumption of equal probability for ergodic initial conditions [16,17], the question is answered.

Among many kinds of synchronizations [18–24], the generalized synchronization is investigated [25–30]. This means that there exists a given functional relationship between the states of the master and that of the slave y = G(x). In this paper, a special kind of generalized synchronizations

y = G(x) = x + F(t) (1)

is studied, where x, y are the state vectors of the master and of the slave, respectively. F(t) is a given vector function of time which may take various forms, either regular or chaotic functions of time. When F(t) = 0, it reduces to a complete synchronization [31,32].

As two numerical examples, two identical Quantum Cellular Neural Network (Quantum-CNN) chaotic systems [33] and a double Duffing chaotic system are used as the master system, slave system, and goal system, respectively. The goal system

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gives a chaotic F(t). Quantum-CNN oscillator equations are derived from a Schr¨odinger equation taking into account quantum dot cellular automata structures to which in the last decade a wide interest has been devoted, with particular attention towards quantum computing.

This paper is organized as follows. In Section 2, by the pragmatical asymptotical stability theorem, a new pragmatical generalized synchronization scheme by adaptive control is given. In Section 3, adaptive controllers are designed for the pragmatical generalized synchronization of two Quantum-CNN chaotic oscillators with a double Duffing chaotic system as a goal system in two examples. Numerical simulations are also given in Section3. Finally, conclusions are drawn in Section4. 2. Pragmatical generalized synchronization scheme, by adaptive control

There are two identical nonlinear dynamical systems, and the master system controls the slave system. The master system is given by

˙

x = Ax + f(x, B) (2)

where x = [x1, x2, . . . , xn]T ∈ Rn denotes a state vector,

A is an n × n uncertain constant coefficient matrix, f is a nonlinear vector function, and B is a vector of uncertain constant coefficients in f .

The slave system is given by ˙

y = ˆAy + f(y, ˆB) + u(t) (3) where y = [y1, y2, . . . , yn]T ∈ Rndenotes a state vector, ˆAis

an n × n estimated coefficient matrix, ˆBis a vector of estimated coefficients in f , and u(t) = [u1(t), u2(t), . . . , un(t)]T ∈ Rn

is a control input vector.

Our goal is to design a controller u(t) so that the state vector of the slave system(3) asymptotically approaches the state vector of the master system(2)plus a given chaotic vector function F(t) = [F1(t), F2(t), . . . , Fn(t)]T. This is a special

kind of generalized synchronization; y is a given function of x:

y = G(x) = x + F(t). (4)

The synchronization can be accomplished when t → ∞; the limit of the error vector e(t) = [e1, e2, . . . , en]Tapproaches

zero: lim

t →∞e =0 (5)

where

e = x − y + F(t). (6)

From Eq.(6)we have ˙

e = ˙x − ˙y + ˙F(t) (7) ˙

e = Ax − ˆAy + f(x, B) − f (y, ˆB) + ˙F(t) − u(t). (8) A Lyapunov function V(e, ˜Ac, ˜Bc) is chosen as a positive

definite function V(e, ˜Ac, ˜Bc) = 1 2e T_{e +}1 2 ˜ AT_cA˜c+ 1 2 ˜ B_cTB˜c (9)

where Ã = A − Â, ˜B = B − ˆB, Ãc and ˜Bc are two column

matrices whose elements are all the elements of matrix ˜Aand of matrix ˜B, respectively.

Its derivative along any solution of the differential equation system consisting of Eq.(8)and update parameter differential equations for ˜Acand ˜Bcis

˙

V(e, ˜Ac, ˜Bc) = eT[Ax − ˆAy + B f(x) − ˆB f (y)

+ ˙F(t) − u(t)] + ˜AcA˙˜c+ ˜BcB˙˜c (10)

where u(t), A˙˜c, and B˙˜c are chosen so that ˙V = eTC e, C

is a diagonal negative definite matrix, and ˙V is a negative semi-definite function of e and parameter differences ˜Ac and

˜

Bc. In the current scheme of adaptive synchronization [13–

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 why the estimated parameters also approach uncertain parameters remains unanswered. By the pragmatical asymptotical stability theorem, the question can be answered strictly.

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 the zero solution must approach the origin as t → ∞. If there are only a small part or even a few of the initial states from which the trajectories 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 trajectories from the initial states are such that they do not approach zero when t → ∞, is zero, the stability of the 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 ; thenϕ(X) is a subset of Lebesque measure 0 of Y [34]. For an autonomous system

dx

dt = f(x1, . . . , xn) (11) where x = [x1, . . . , xn]T is a state vector, the function f =

[f1, . . . , fn]Tis defined on D ⊂ Rn and kxk ≤ H > 0. Let

x =0 be an equilibrium point for the system(11). Then

f(0) = 0. (12)

Definition. The equilibrium point for the system (11) 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

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corresponding trajectories behave as if they agree with traditional asymptotical stability [16,17].

Theorem. Let V = [x1, . . . , xn]T : D → R+ be positive

definite and analytic on D, such that the derivative of V through Eq.(11), ˙V , is negative semi-definite.

Let X be the m-manifold consisting of the point set for which ∀x 6= 0, ˙V(x) = 0 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.(11), which is one dimensional, the collection of these trajectories, C, is an(m + 1)-manifold [16,17].

If m + 1< 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 the initial point falling 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 that the initial point is chosen from collection C does not occur actually. Therefore, under the equal probability assumption, pragmatical asymptotical stability becomes actual asymptotical stability. When the initial point falls on D − C, ˙V(x) < 0, the corresponding trajectories behave as if they agree with traditional asymptotical stability because by the existence and uniqueness of the solution of the initial-value problem, these trajectories never meet C.

In Eq. (9) V is a positive definite function of n variables, i.e. p error state variables and n − p = m differences between unknown and estimated parameters, while ˙V = eTC e is a negative semi-definite function of n variables. Since the number of error state variables is always more than one, p > 1, m +1< n is always satisfied; by the pragmatical asymptotical stability theorem we have

lim

t →∞e =0 (13)

and the estimated parameters approach the uncertain parame-ters. The pragmatical generalized synchronization is obtained. Therefore, the equilibrium point of the system is pragmatically asymptotically stable. Under the equal probability assumption, it is actually asymptotically stable for both error state variables and parameter variables.

3. Numerical results of pragmatical generalized chaos synchronization of two Quantum-CNN oscillators by adaptive control

CaseI. The chaotic states of a goal system, a double Duffing chaotic system, used as F(t).

For a two-cell Quantum-CNN, the following differential equations are used [33] as the master system:

                         d dtx1= −2a1 q 1 − x₁2sin x2 d dtx2= −ω1(x1−x3) + 2a1 x1 q 1 − x₁2 cos x2 d dtx3= −2a2 q 1 − x₃2sin x4 d dtx4= −ω2(x3−x1) + 2a2 x3 q 1 − x₃2 cos x4 (14)

where x1, x3 are polarizations, x2, x4 are quantum phase

displacements, a1 and a2 are proportional to the inter-dot

energy inside each cell and ω1 and ω2 are parameters that

weigh effects on the cell of the difference of the polarization of neighboring cells, like the cloning templates in traditional CNNs. Let a1=4.9, a2=4.9, ω1=3.03, ω2=1.83.

A slave system is described by                          d dty1= −2 â1 q 1 − y₁2sin y2 d dty2= − ˆω1(y1−y3) + 2â1 y1 q 1 − y₁2 cos y2 d dty3= −2 â2 q 1 − y₃2sin y4 d dty4= − ˆω2(y3−y1) + 2â2 y3 q 1 − y₃2 cos y4. (15) In order to lead (y1, y2, y3, y4) to (x1+F1(t), x2+F2(t),

x3+ F3(t), x4 +F4(t)), we add u1, u2, u3, and u4 to each

equation of Eq.(15), respectively:                          d dty1= −2 â1 q 1 − y₁2sin y2+u1 d dty2= − ˆω1(y1−y3) + 2â1 y1 q 1 − y₁2 cos y2+u2 d dty3= −2 â2 q 1 − y₃2sin y4+u3 d dty4= − ˆω2(y3−y1) + 2â2 y3 q 1 − y₃2 cos y4+u4. (16)

Subtracting Eq. (16) from Eq. (14), we obtain an error dynamics. The initial values of the master system and the slave system are taken as x1(0) = 0.8, x2(0) = −0.77,

x3(0) = −0.72, x4(0) = 0.57, y1(0) = 0.1, y2(0) = 0.28,

y3(0) = 0.42, and y4(0) = −0.72, respectively.

The goal system for generalized synchronization is a double Duffing chaotic system

       ˙ z1=z2 ˙ z2=z1−z3₁−δ1z2+ f1cosψ1t ˙ z3=z4 ˙ z4=z3−z33−δ2z4+ f2cosψ2t (17) where δ1 = 13.5, δ2 = 12.5, f1 = −24.9, f2 = −33.1, Ψ1 = 10.9, Ψ2 = 19.9, z1(0) = 0.75, z2(0) = −0.3, z3(0) = −0.4, and z4(0) = 0.5.

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We have lim

t →∞ei =t →∞lim(xi−yi+zi) = 0, i = 1, 2, 3, 4 (18)

where ˙e = ˙x − ˙y + ˙z, and ˙ e1= −2a1 q 1 − x₁2sin x2+2 â1 q 1 − y₁2sin y2−u1+ ˙z1 ˙ e2 = −ω1(x1−x3) + ˆω1(y1−y3) +2a1 x1 q 1 − x₁2 cos x2−2 â1 y1 q 1 − y₁2 cos y2−u2+ ˙z2 ˙ e3= −2a2 q 1 − x₃2sin x4+2 â2 q 1 − y₃2sin y4−u3+ ˙z3 ˙ e4 = −ω2(x3−x1) + ˆω2(y3−y1) + 2a2 x3 q 1 − x₃2 cos x4 −2 â2 y3 q 1 − y₃2 cos y4−u4+ ˙z4 (19) where e1=x1−y1+z1, e2=x2−y2+z2, e3=x3−y3+z3, and e4=x4−y4+z4.

Choose a Lyapunov function in the form of a positive definite function: V(e1, e2, e3, e4, ã1, ã2, ˜ω1, ˜ω2) = 1 2(e 2 1+e 2 2+e 2 3+e 2 4+ ã 2 1+ ã 2 2+ ˜ω 2 1+ ˜ω 2 2) (20) where ã1=a1− â1, ã2=a2− â2, ˜ω1=ω1− ˆω1, ˜ω2=ω2− ˆω2

and ˆa1, ˆa2, ˆω1, ˆω2are estimates of uncertain parameters a1, a2,

ω1, andω2, respectively.

Its time derivative is ˙ V = e1[−2a1 q 1 − x₁2sin x2+2 â1 q 1 − y₁2sin y2−u1+z2] +e2 h −ω₁(x₁−x3) + ˆω1(y1−y3) +2a1 x1 q 1 − x₁2 cos x2−2 â1 y1 q 1 − y₁2 cos y2−u2 + z1−z31−δ1z2+ f1cosψ1t i +e3[−2a2 q 1 − x₃2sin x4+2 â2 q 1 − y₃2sin y4 −u3+z4] +e4 h −ω₂(x₃−x1) + ˆω2(y3−y1) +2a2 x3 q 1 − x₃2 cos x4−2 â2 y3 q 1 − y₃2 cos y4 −u4+z3−z33−δ2z4 + f2cosψ2t i

+ ã1(−˙â1) + ã2(−˙â2)

+ ˜ω₁(− ˙ˆω₁) + ˜ω₂(− ˙ˆω₂). (21) Choose u1 = −2a1 q 1 − x₁2sin x2+2 â1 q 1 − y₁2sin y2+ â1e1 +aˆ1z2 a1 + ã₁2 u2 = 2a1 x1 q 1 − x₁2 cos x2−2 â1 y1 q 1 − y₁2 cos y2+z1−z3₁ +f1cosψ1t + ˆω1e2−ω1(x1−x3) + ˆω1(y1−y3) −ωˆ1δ1 ω1 z2+ ˜ω21 (22) u3 = −2a2 q 1 − x₃2sin x4+2 â2 q 1 − y₃2sin y4 + â2e3+ ˆ a2z4 a2 + ã2₂ u4 = 2a2 x3 q 1 − x₃2 cos x4−2 â2 y3 q 1 − y₃2 cos y4+z3−z33 +f2cosψ2t + ˆω2e4−ω2(x3−x1) + ˆω2(y3−y1) −ωˆ2δ2 ω2 z4+ ˜ω22 ˙ã1= − ˙â1= − e1z2 a1 + ã1e1−e21 ˙˜ ω1= − ˙ˆω1= δ1 ω1 e2z2+ ˜ω1e2−e22 ˙ã2= − ˙â2= − e3z4 a2 + ã2e3−e23 ˙˜ ω2= − ˙ˆω2= δ 2 ω2 e4z4+ ˜ω2e4−e24. (23)

The initial values of estimates for uncertain parameters are ˆ

a1(0) = ˆa2(0) = ˆω1(0) = ˆω2(0) = 0.

Substituting Eqs.(22)and(23)into Eq.(21), we obtain ˙

V = −a1e21−ω1e22−a2e23−ω2e24≤0 (24)

which is a negative semi-definite function of e1, e2, e3, e4, ˜a1,

˜

a2, ˜ω1, and ˜ω2. The Lyapunov asymptotical stability theorem is

not satisfied. We cannot obtain that the common origin of error dynamics(19)and parameter dynamics(23)is asymptotically stable. Now, D is an 8-manifold, n = 8 and the number of error state variables p = 4. When e1 = e2 = e3 =

e4 = 0 and ˜a1, ˜a2, ˜ω1, ˜ω2 take arbitrary values, ˙V = 0,

so X is a 4-manifold, m = n − p = 8 − 4 = 4. m +1 < n is satisfied. By the pragmatical asymptotical stability theorem, error vector e approaches zero and the estimated parameters also approach the uncertain parameters. The pragmatical generalized synchronization is obtained. The equilibrium point e1=e2=e3=e4= ˜a1= ˜a2= ˜ω1= ˜ω2=

0 is pragmatically asymptotically stable. Under the assumption of equal probability, it is actually asymptotically stable. The numerical results are shown inFig. 1. After 10 s, the generalized synchronization is accomplished.

Case II. The cubics of chaotic states of the goal system, a double Duffing chaotic system, used as F(t).

We demand lim t →∞ei = lim t →∞(xi −yi +z3i) = 0, i = 1, 2, 3, 4 (25) and then

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˙ e = ˙x − ˙y +3z2z.˙ (26) ˙ e1= −2a1 q 1 − x₁2sin x2+2 â1 q 1 − y₁2sin y2−u1+3z12z˙1 ˙ e2 = −ω1(x1−x3) − ˆω1(y1−y3) + 2a1 x1 q 1 − x₁2 cos x2 −2 â1 y1 q 1 − y₁2 cos y2−u2+3z22˙z2 (27) ˙ e3= −2a2 q 1 − x₃2sin x4+2 â2 q 1 − y₃2sin y4−u3+3z32z˙3 ˙ e4 = −ω2(x3−x1) + ˆω2(y3−y1) + 2a2 x3 q 1 − x₃2 cos x4 −2 â2 y3 q 1 − y₃2 cos y4−u4+3z24˙z4 where e1=x1−y1+z3₁, e2=x2−y2+z3₂, e3=x3−y3+z3₃, and e4=x4−y4+z3₄.

Choose a Lyapunov function in the form of a positive definite function: V(e1, e2, e3, e4, ã1, ã2, ˜ω1, ˜ω2) =1 2(e 2 1+e 2 2+e 2 3+e 2 4+ ã 2 1+ ã 2 2+ ˜ω 2 1+ ˜ω 2 2) (28) where ã1=a1− â1, ã2=a2− â2, ˜ω1=ω1− ˆω1, ˜ω2=ω2− ˆω2

and ˆa1, ˆa2, ˆω1, ˆω2are estimates of uncertain parameters a1, a2,

ω1, andω2, respectively.

Its time derivative is ˙ V = e1[−2a1 q 1 − x₁2sin x2+2 â1 q 1 − y₁2sin y2 −u1+3z21z2] +e2 " −ω₁(x₁−x3) + ˆω1(y1−y3) +3z2₂(z1−z31−δ1z2+ f1cosψ1t) + 2a1 x1 q 1 − x₁2 cos x2−2 â1 y1 q 1 − y₁2 cos y2−u2   +e3[−2a2 q 1 − x2₃sin x4+2 â2 q 1 − y₃2sin y4 −u3+3z23z4] +e4 " −ω₂(x₃−x1) + ˆω2(y3−y1) +3z2₄(z3−z33−δ2z4+ f2cosψ2t) + 2a2 x3 q 1 − x₃2 cos x4−2 â2 y3 q 1 − y₃2 cos y4−u4  

+ ã1(−˙â1) + ã2(−˙â2) + ˜ω1(− ˙ˆω1) + ˜ω2(− ˙ˆω2). (29)

Choose u1 = −2a1 q 1 − x₁2sin x2+2 â1 q 1 − y₁2sin y2 + â1e1+ 3 â1z2₁z2 a1 + ã2₁ u2 = 2a1 x1 q 1 − x₁2 cos x2−2 â1 y1 q 1 − y₁2 cos y2 +3z2₂(z1−z13+ f1cosψ1t) + ˆω₁e2−ω1(x1−x3) + ˆω1(y1−y3) −3δ1 ω1 ˆ ω1z3₂+ ˜ω21 (30) u3 = −2a2 q 1 − x₃2sin x4+2 â2 q 1 − y₃2sin y4 + â2e3+ 3 â2z2₃z4 a2 + ã₂2 u4 = 2a2 x3 q 1 − x₃2 cos x4−2 â2 y3 q 1 − y₃2 cos y4 +3z2₄(z3−z3₃+ f2cosψ2t) + ˆω2e4−ω2(x3−x1) + ˆω₂(y₃−y1) − 3δ2 ω2 ˆ ω2z34+ ˜ω 2 2 ˙ã1= − ˙â1= − 3e1z2₁z2 a1 + ã1e1−e12 ˙˜ ω1= − ˙ˆω1= 3δ1 ω1 e2z32+ ˜ω1e2−e22 ˙ã2= − ˙â2= − 3e3z2₃z4 a2 + ã2e3−e32 ˙˜ ω2= − ˙ˆω2= 3δ2 ω2 e4z34+ ˜ω2e4−e24. (31)

The initial values of estimates for uncertain parameters are ˆ

a1(0) = ˆa2(0) = ˆω1(0) = ˆω2(0) = 0. Substituting Eqs.(30)

and(31)into Eq.(29), it can be rewritten as ˙

V = −a1e12−ω1e22−a2e23−ω2e24≤0 (32)

which is a negative semi-definite function of e1, e2, e3,

e4, ˜a1, ˜a2, ˜ω1, and ˜ω2. The Lyapunov asymptotical stability

theorem is not satisfied. We cannot obtain that the common origin of error dynamics (27) and parameter dynamics (31)

is asymptotically stable. In our case, ˙V = 0 when e1 =

e2 = e3 = e4 = 0, and ˜a1, ˜a2, ˜ω1, and ˜ω2 take arbitrary

values. n = 8, m = 4, m + 1 < n is satisfied. By the pragmatical asymptotical stability theorem, the equilibrium point e1 = e2 = e3 = e4 = ˜a1 = ˜a2 = ˜ω1 = ˜ω2 = 0

is pragmatically asymptotically stable. Under the assumption of equal probability, it is actually asymptotically stable. The error vector e approaches zero and the estimated parameters approach the uncertain parameters. The numerical results are shown inFig. 2. After 10 s, the generalized synchronization is accomplished.

4. Conclusions

In this paper pragmatical generalized synchronization of adaptive control is studied. The pragmatical asymptotical stability theorem fills the vacancy between the actual asymptotical stability and mathematical asymptotical stability; the conditions of the Lyapunov function for pragmatical asymptotical stability are lower than those for traditional asymptotical stability. By using this theorem, with the same conditions for the Lyapunov function, V > 0, ˙V ≤ 0, as in

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Fig. 1. Time histories of states, state errors, z1, z2, z3, z4, ˆa1, ˆa2, ˆw1, and ˆw2for Case I with a1=4.9, a2=4.9, ω1=3.03, ω2=1.83.

the current scheme of adaptive synchronization, we not only obtain the generalized synchronization of chaotic systems but also prove that the estimated parameters approach the uncertain values. Two Quantum-CNN chaotic systems and a double Duffing chaotic system are used as the master system, slave system, and goal system, respectively, in two cases: the chaotic states of a goal system, a double Duffing chaotic system, used as F(t) and the cubics of chaotic states of the same goal system used as F(t). These generalized synchronizations of chaotic

systems by adaptive control can be used to increase the security of communication.

Acknowledgment

This research was supported by the National Science Council, Republic of China, under Grant Number NSC 95-2221-E-009-175.

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Fig. 2. Time histories of states, state errors, z1, z2, z3, z4, ˆa1, ˆa2, ˆw1, and ˆw2for Case II a1=4.9, a2=4.9, ω1=3.03, ω2=1.83.

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