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    DOI: 10.1142/S0217984910023037

    Modern Physics Letters B, Vol. 24, No. 10 (2010) 979994c World Scientific Publishing Company

    ADAPTIVE FUZZY SYNCHRONIZATION OF TWO DIFFERENT

    CHAOTIC SYSTEMS WITH STOCHASTIC

    UNKNOWN PARAMETERS

    W. J. YOO, D. H. JI and S. C. WON

    Department of Electronic and Electrical Engineering,

    Pohang University of Science and Technology,

    San 31 Hyoja-Dong, Pohang, Gyungbuk, 790-784, Republic of [email protected]@postech.ac.kr

    Received 22 June 2009Revised 21 August 2009

    In this paper, we present a method for synchronizing two different chaotic systems thathave unknown parameters that are affected by stochastic variations generated by theWiener process. The parameters are expressed by the sum of their mean values andthe white Gaussian noise multiplied by the diffusion matrices. To describe the unknownnonlinear function yielded by Itos lemma due to the unknown diffusion matrices, afuzzy logic system is employed. Using adaptive fuzzy control, the response system issynchronized with the drive system within an arbitrarily small error bound. Numericalsimulations show the effectiveness of the proposed method.

    Keywords: Adaptive fuzzy; synchronization; stochastic chaos.

    1. Introduction

    Chaotic systems occur in many real-world scientific and engineering problems. Such

    systems are very sensitive to the initial conditions and to variations of parameters,and this sensitivity can cause unpredictable phenomena. As a result the challenge

    of synchronizing chaotic systems has been intensively studied. Since chaotic syn-

    chronization was introduced by Pecora and Carroll,1 many types of synchronization

    technique have been developed.28

    Recently, the problem of synchronizing two different chaotic systems has been

    studied.914 Nonlinear feedback control schemes have been used for synchroniza-

    tion of two different chaotic systems.9,10 Active control approaches have been pre-

    sented11,12 for synchronization between Lorenz15 and Chen,16 or Lorenz and Lu

    systems.17 Adaptive control schemes have been studied for synchronization of differ-ent chaotic systems with unknown parameters, which were estimated by adaptation

    laws.13,14

    However, all these approaches deal with deterministic chaotic systems. In prac-

    tical situations, chaotic systems often show stochastic characteristics because the

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    http://dx.doi.org/10.1142/S0217984910023037http://dx.doi.org/10.1142/S0217984910023037
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    980 W. J. Yoo, D. H. Ji & S. C. Won

    systems are interferred by random noise so the chaos synchronization is affected sig-

    nificantly. In this paper, we focus on the stochastic variation of system parameters

    due to random noise. Recently, Salarieh and Alasty have proposed the synchroniza-

    tion of chaotic systems with stochastic unknown parameters.18 The time-varying

    unknown parameters are assumed to be disturbed by white Gaussian noise (WGN)

    which is generated by the Wiener process. The parameters are expressed by the

    sum of the their mean values and the WGNs multiplied by the diffusion matrices.

    However, Salarieh and Alasty assumed that the diffusion matrices are known.

    In practice, it is not easy to find the diffusion matrices. If there is no stochastic

    information about the system parameters, unknown nonlinear terms are generated

    in derivation of stability condition using Itos lemma.19 In this paper, we propose

    an adaptive fuzzy logic scheme to compensate for the unknown nonlinear terms

    using control input.

    As an extension of Ref. 18, this paper addresses an adaptive fuzzy synchro-

    nization based on the drive-response framework for two different chaotic systems

    with unknown parameters which are perturbed by WGN, which is assumed to be

    generated by the Wiener process. The model proposed by Salarieh and Alasty18

    is adopted to describe the stochastic variation of unknown parameters, but it is

    assumed that the information about their mean and diffusion matrices as well as

    their bound is fully unknown. The unknown nonlinearities are approximated by the

    fuzzy logic system (FLS),20 in which parameters are adjusted by an adaptation lawsatisfying the Lyapunov criterion. It is shown that the mean square of synchroniza-

    tion error between the drive and the response system converges into any arbitrarily

    given small bound. Simulation results are provided to verify the effectiveness of the

    proposed method.

    2. Fuzzy Logic Systems and the Universal Approximation Theorem

    Consider an n input, single-output FLS20 with M fuzzy IF-THEN rules:

    Rule r: If x1 isAr1 and andxn isArn

    Then z= zr, (1)

    where 1 r M, x= [x1, . . . , xn]T XRn is the input of the FLS and zR

    is the output of the FLS, with a compact set X. zr is the fuzzy singleton for the

    output ofrth rule and Ar1, . . . , Arn are fuzzy sets with the Gaussian membership

    functions rj (xj) = exp[(xj crj)2/rj ] where crj is the center and rj is the

    width of the Gaussian membership function. Using this singleton fuzzy model, the

    output of the fuzzy system can be described as

    z=

    Mr=1zr(

    nj=1rj(xj))M

    r=1(n

    j=1rj (xj))=T(x) , (2)

    where = [z1, . . . ,zM]T is a parameter vector and(x) = [1(x), . . . , M(x)]T is the

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    Adaptive Fuzzy Synchronization of Two Different Chaotic Systems 981

    fuzzy basis function vector defined by

    r(x) =n

    j=1rj (xj)Mi=1(

    nj=1ij(xj))

    , r= 1, 2, . . . , M . (3)

    Now, we introduce the following theorem:

    Universal Approximation Theorem.20 Suppose that the input universe of dis-

    courseXis a compact set inRn. Then, for any given real continuous functiong(x)

    onX and arbitrary >0, there exists a fuzzy systemf(x) =T(x) such that

    supxX

    |g(x) f(x)|< . (4)

    That is, fuzzy systems that are composed of a product inference engine, a single-ton fuzzifier, a center average defuzzifier and Gaussian membership functions are

    universal approximators.

    In this paper, a singleton FLS is used to describe the unknown nonlinear function

    that the diffusion matrices cause in the stability analysis of stochastic systems. The

    fuzzy parameter vector can be adjusted using an adaptation law to satisfy the

    Lyapunov stability criterion.

    3. Problem Formulation

    We consider the following chaotic drive (5) and response (6) systems18:

    x= f(x) +F(x) , (5)

    y= g(y) +G(y)+H(y)u (6)

    wherex Rn is the state of the drive system; y Rn is the state of the response

    system;u Rn is the control vector of the response system; f() Rn and g()

    Rn are nonlinear function vectors; F() Rnp, G() Rnq, H() Rnn are

    nonlinear function matrices, each with a Euclidean norm bounded by an unknown

    positive constant; Rp

    is the system parameter vector of the drive system;and Rq is the system parameter vector of the response system. The system

    parameter vectors, and in the model proposed by Salarieh and Alasty18 are:

    =+ v , (7)

    = + w , (8)

    where is the unknown mean value of ; is the unknown mean value of ;

    and are the unknown diffusion matrices; andv andw are independent zero-mean

    vector Wiener processes of which the elements satisfy

    E{dv}= 0, E{dw}= 0, E{dvidwj}= 0, i, j ,

    E{dvidvj}=

    dt, if i= j

    0, otherwise , E{dwidwj}=

    dt, if i= j

    0, otherwise , (9)

    dtdt= 0 .

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    982 W. J. Yoo, D. H. Ji & S. C. Won

    , , and are assumed to be bounded but in this paper the upper bounds need

    not be known.Using Eqs. (7) and (8), the dynamic equations (5) and (6) can be rewritten as

    Ito differential equation19:

    dx= [f(x) +F(x)]dt+F(x)dv , (10)

    dy= [g(y) +G(y)+H(y)u]dt+G(y)dw . (11)

    By subtracting Eq. (11) from Eq. (10), the error dynamics is described by

    de= [f(x) g(y) +F(x) G(y) H(y)u]dt+F(x)dv G(y)dw , (12)

    where e = x y. To synchronize the response system with the drive system, weneed to refer to Itos lemma.

    Lemma 1. (Itos lemma)19 Assume that ann-dimensional vector stochastic pro-

    cessX satisfies the Ito differential equation(12). Let(X, t) be an arbitrary func-

    tion of X and t. Then, the differentiation of the function is given by the Taylor

    expansion of(X, t), i.e.

    d(X, t) =

    tdt+

    n

    i=1

    XidXi+

    1

    2

    n

    i=1

    n

    j=1

    2

    XiXjdXidXj, (13)

    whereXi is the ith element of X.

    For stochastic systems, the objective of synchronization is to design a controller

    to make the mean square of the synchronization error approach the origin, i.e.

    e(t) 0 as t , where e = E{eT(t)e(t)}1/2. The adaptive fuzzy synchro-

    nization scheme is presented in the next section.

    4. Adaptive Fuzzy Synchronization

    The adaptive fuzzy synchronization controller consists of the nonlinear feedback

    controller and the adaptive controller. In the adaptive controller part, the fuzzy

    parameter vector is updated to approximate the unknown nonlinear function (13)

    of Lemma 1. The approximation error for the nonlinear function by the FLS is

    compensated by a state error feedback controller of which the gain is tuned by an

    adaptation law. Also, the mean and are estimated by the adaptive parameters.

    The main result is summarized by the following theorem.

    Theorem 1. Consider the drive and the response systems(10) and(11). The syn-

    chronization error would converge to a setS={e|E{eTe}< } with the arbitrarilygiven error tolerance ast if the control and adaptation laws are given by

    u= H1(y)[f(x) g(y) +Ke +F(x) G(y)+eT(x, y)] , (14)

    K=k1+k2, (15)

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    Adaptive Fuzzy Synchronization of Two Different Chaotic Systems 983

    = 1F(x)Te , (16)

    =2G(y)Te , (17)

    k2 =3eTe, if E{eTe}> , (18)

    = 4eTe(x, y), if E{eTe}> , (19)

    k2 =3, if E{eTe} , (20)

    = 0, if E{eTe} , (21)

    wherek1 is a positive constant for the convergence of mean squared error (M SE);

    k2 is the adaptive gain to compensate for the fuzzy approximation error; RM

    is the fuzzy parameter vector; (, ) RM is the fuzzy basis function vector; M

    is the number of rules for the FLS; Rp is the estimate of ; Rq is the

    estimate of;1,2, 3 and4 are positive constants; is a positive constant such

    that < .

    Proof. Let us define the following function:

    V1 = 1

    2e(t)2 +

    1

    212 +

    1

    222 , (22)

    where = and = . Differentiating both sides of Eq. (22) yields:

    dV1 = 1

    2E

    d(eTe) +

    2

    1T dt+

    2

    2T dt

    . (23)

    Using Itos lemma, the error dynamics (12) and the property (9), d(eTe) is

    d(eTe) = 2eTde+n

    i,j=1

    deidej,

    = 2eTde+pi=1

    (TFT(x)F(x))iidt+qi=1

    (TGT(y)G(y))iidt , (24)

    where the subscript ii means the ith diagonal element. Substituting Eq. (12) into

    Eq. (24) and then substituting the result into Eq. (23) yields

    dV1 =E

    eT([f(x) g(y) +F(x) G(y) H(y)u]dt+F(x)dv G(y)dw)

    +

    1

    2

    pi=1

    (T

    FT

    (x)F(x))iidt+

    1

    2

    qi=1

    (T

    GT

    (y)G(y))iidt

    + 1

    1T dt+

    1

    2T dt

    . (25)

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    984 W. J. Yoo, D. H. Ji & S. C. Won

    Because of the properties of the Wiener process Eq. (9),

    dV1 =E

    eT([f(x) g(y) +F(x) G(y) H(y)u]dt)

    +1

    2

    pi=1

    (TFT(x)F(x))iidt+1

    2

    qi=1

    (TGT(y)G(y))iidt

    + 1

    1T dt+

    1

    2T dt

    . (26)

    Thus,

    V1 =E

    eT([f(x) g(y) +F(x) G(y) H(y)u])

    +1

    2

    pi=1

    (TFT(x)F(x))ii+1

    2

    qi=1

    (TGT(y)G(y))ii

    + 1

    1T +

    1

    2T

    . (27)

    Then define a nonlinear function N(x, y) as

    N(x, y) = (eTe)1

    12

    pi=1

    (TFT(x)F(x))ii+ 12

    qi=1

    (TGT(y)G(y))ii

    . (28)

    From the universal approximation theorem, there exists an FLST(x, y) such that

    N(x, y) =T(x, y) +(x, y) , (29)

    where(x, y) is the approximation error which is bounded by an unknown constant

    as

    |(x, y)|< . (30)

    Let denote the optimal fuzzy parameter vector, i.e.

    = arg minRM

    supx,y

    |N(x, y) T(x, y)|

    . (31)

    Then define = andk2 = k2 and a Lyapunov function candidate:

    V =V1+ 1

    23k2

    2 + 1

    242 . (32)

    Differentiating Eq. (32) yields:

    V = V1+E

    1

    4T

    +

    1

    3k2

    k2

    =E

    eT([f(x) g(y) +F(x) G(y) H(y)u])

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    Adaptive Fuzzy Synchronization of Two Different Chaotic Systems 985

    +1

    2

    p

    i=1

    (TFT(x)F(x))ii+1

    2

    q

    i=1

    (TGT(y)G(y))ii

    + 1

    1T +

    1

    2T +

    1

    4T +

    1

    3k2

    k2

    . (33)

    Because = , = , = and k2 = k2, substituting Eqs. (14), (28),

    (29), (30), (31) into Eq. (33) yields

    V =E

    eTeN(x, y) eTeT(x, y) k1e

    Te k2eTe

    + 11

    T(1FT(x)e ) + 1

    2T(2G

    T(y)e ) 14

    T 13

    k2k2

    =E

    eTeT(x, y) +eTe(x, y) eTeT(x, y) k1e

    Te k2eTe

    + 1

    1T(1F

    T(x)e ) + 1

    2T(2G

    T(y)e ) 1

    4T

    1

    3k2k2

    < E

    k1e

    Te+ 1

    1T(1F

    T(x)e ) + 1

    2T(2G

    T(y)e )

    + 1

    4T(4e

    Te(x, y) ) + 1

    3k2(3e

    Te k2)

    , (34)

    where is the minimum fuzzy approximation error when = .

    From the adaptation laws (16)(19),

    V < E{k1eTe} 0 . (35)

    Becausek1 is a positive constant, V

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    986 W. J. Yoo, D. H. Ji & S. C. Won

    we obtain the following inequality:

    V < E

    k1eTe+L k2e

    Te eTeT(x, y) 13

    k2k2

    < E

    k1+L k2

    T(x, y) 1

    3k2k2

    . (36)

    Because the adaptation law (19) updates in the positive direction, the FLS

    T(x, y) is not negative if(0) 0. So, if the error signal k2 is redefined as

    k2 =L k2, (37)

    where L =L/ k1, then

    V < E

    L

    k1

    k2

    1

    3k2k2

    =E

    (L k2)

    1

    3k2k2

    =E

    1

    3k2(3 k2)

    . (38)

    The adaptation law (20) yields that V < 0. Therefore, e(t) remains in thesetS.

    Remark 1. For smaller, 3 and 4, which are the learning rates of the adaptive

    parameters, should be larger to maintain a good transient performance. However,

    the larger learning rates may cause larger control inputs. Hence, there is a tradeoff

    between error tolerance and control effort.

    5. Numerical ExamplesTo verify the effectiveness of the proposed method, three examples are presented

    in this section. The first example presents the stochastic chaos synchronization

    between the unified chaotic system21 and the Rossler system.22 The second example

    presents the synchronization between the hyper Lorenz23 and the hyper Lu.24 The

    third example presents the synchronization of the 3D hysteresis multiscroll chaotic

    system2527 and 1D saturated multiscroll chaotic system.28

    Example 1. (Unified chaotic system Rossler system synchronization)

    Consider the unified chaotic system (the drive system) of the form21:

    x1x2

    x3

    =

    0x1x3

    x1x2

    +

    x2 x1 0 0 00 x1 x2 0

    0 0 0 x3

    1234

    . (39)

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    Adaptive Fuzzy Synchronization of Two Different Chaotic Systems 987

    For the numerical simulation, the values ofi fori= 1, 2, 3, 4 are

    1234

    =

    25a+ 10

    35a 28

    29a 1

    (8 +a)/3

    , (40)

    wherea [0, 1] is the constant parameter. When 0 a

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    988 W. J. Yoo, D. H. Ji & S. C. Won

    (a) The Lorenz attractor (a= 0). (b) The Chen attractor (a= 1.0).

    (c) The Lu attractor (a= 0.8). (d) Rossler system without control.

    Fig. 1. State trajectories of (a)(c) the unified chaotic system and (d) the Rossler system.

    The hyper-Lu system (the response system) has the form24

    y1y2y3y4

    =

    y4y1y3y1y2y1y3

    +

    y2 y1 0 0 0

    0 y2 0 0

    0 0 y3 0

    0 0 0 y4

    1234

    +

    1 0 0 0

    0 1 0 0

    0 0 1 0

    0 0 0 1

    u1u2u3u4

    .(44)

    The trajectories of both systems vary depending on the value of their system param-

    eters. The hyper-Lorenz system (43) shows hyperchaotic behavior when 1 = 10,

    2 = 28, 3 = 8/3 and4 = 0.1 (Fig.6(a)). The hyper-Lu sysetm (44) shows hyper-

    chaotic behavior when 1= 36, 2 = 20, 3 = 3,0.35 4 1.3 (Fig.6(b)). For

    the numerical simulation, we chose = [10, 28, 8/3, 0.1]T, = [36, 20, 3, 1]T, = 0.01I3 and = 0.01I3. The initial conditions were x(0) = [1, 1, 1, 1]T and

    y(0) = [5, 2, 5, 2]T.

    For the FLS, cxij andcyij were chosen in [30, 30] forj = 1, 2, 3 and in [0, 60]

    forj = 4, with uniform distance. The variances were chosen as xij =yij = 15 for

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    Adaptive Fuzzy Synchronization of Two Different Chaotic Systems 989

    (a) Lorenz attractor (a= 0). (b) Chen attractor (a= 1).

    (c) Lu attractor (a= 0.8).

    Fig. 2. (Color online) Synchronized trajectories of the unified chaotic system and the Rosslersystem.

    Fig. 3. (Color online) The synchronization errors between the Lorenz (a = 0) and the Rosslersystems.

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    990 W. J. Yoo, D. H. Ji & S. C. Won

    Fig. 4. (Color online) The synchronization errors between the Chen (a = 1) and the Rosslersystems.

    Fig. 5. (Color online) The synchronization errors between the Lu (a = 0.8) and the Rosslersystems.

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    Adaptive Fuzzy Synchronization of Two Different Chaotic Systems 991

    (a) (x1, x2, x3) space. (b) (x1, x2, x4) space. (c) (x1, x3, x4) space. (d) (x2, x3, x4) space.

    (e) (y1, y2, y3) space. (f) (y1, y2, y4) space. (g) (y1, y3, y4) space. (h) (y2, y3, y4) space.

    Fig. 6. State trajectories of (a)(d) the hyper Lorenz and (e)(h) the hyper Lu systems.

    Fig. 7. (Color online) The synchronization errors between the hyper Lorenz and the Lu systems.

    j= 1, 2, 3, 4. The same control parameters in Example 1 were also used. As shown

    in Fig. 7, the hyper-Lu system was also well synchronized with the hyper-Lorenzsystem and the synchronization errors converged to the set specified by .

    Example 3. (Multiscroll chaotic systems synchronization)

    Consider the 3D hysteresis multiscroll chaotic system (the drive system) of the

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    992 W. J. Yoo, D. H. Ji & S. C. Won

    (a) 3D hysteresis multiscroll chaotic system. (b) 1D saturated multiscroll chaotic system.

    Fig. 8. State trajectories of the multiscroll chaotic systems.

    form25 x1x2

    x3

    =

    x2 H2x3 H3

    0

    +

    0 0 00 0 0

    H1 x1 H2 x2 H3 x3

    12

    3

    , (45)

    whereHj is the hysteresis series function defined as

    Hj =H(xj , pj , qj) =

    pj, if xj qj 1,

    for j= 1, 2, 3 , (46)

    in which pj and qj are positive integers.

    For [1,2,3]T = [0.8, 0.72, 0.66]T, p1 = q1 = 2, p2 = 3, q2 = 4, p3 = q3 = 1,

    system (45) has a 3D 5 8 3-grid scroll chaotic attractor (Fig.8(a)).

    The 1D saturated multiscroll chaotic system (the response system) is28

    y1y2y3

    = y2y30

    + 0 0 0 00 0 0 0y1 y2 y3 S(y1)

    1234

    + 1 0 00 1 0

    0 0 1

    u1u2u3

    , (47)

    whereS(y) is the saturated function series of the form:

    S(y) =S(y, K1, h1, a1, b1) =

    (2b1+ 1)K1, if y > b1h1+ 1 ,

    K1(y ih1) + 2iK1, if |y ih1| 1 ,

    a1 i b1,

    (2i+ 1)K1, if 1< y ih1 < h1 1 ,a1 i b1 1 ,

    (2a1+ 1)K1, if x

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    Adaptive Fuzzy Synchronization of Two Different Chaotic Systems 993

    Fig. 9. (Color online) The synchronization errors between the 3D hysteresis multiscroll chaoticsystem and the 1D saturated multiscroll chaotic system.

    For 1 = 2= 3 = 4 = 0.7,K1= 9,h1 = 18,a1=b1 = 2, system (47) has a

    1D 6-scroll chaotic attractor (Fig.8(b)).

    For the numerical simulation, the diffusion matrices were chosen, = 0.3I3 and

    = 0.3I4. The initial conditions were x(0) = [0.5, 1, 1]T and y(0) = [5, 3, 10]T.

    All the control parameters were set to the same values used in Example 1. For the

    FLS, cxij and cyij were chosen in the interval [6, 6] with uniform distance, and

    the variances were chosen asxij =yij = 3 for j = 1, 2, 3.

    We can see that the response system was also synchronized with the drive system

    and the synchronization errors converged to the set specified by (Fig.9).

    6. Conclusion

    In this paper, we consider the adaptive fuzzy synchronization of two different chaotic

    systems with stochastic unknown parameters based on the drive-response frame-

    work. The system parameters are disturbed by the WGN, which is generated by

    the Wiener processes. It is assumed that the WGNs are multiplied by the diffusion

    matrices and added to the mean values of the system parameters. However, the

    proposed method does not require any information about the value of the means orthe diffusion matrices. An FLS is used to describe the unknown nonlinear function

    induced by the diffusion matrices of the system parameters. The FLS approxima-

    tion error is compensated for by the state error feedback control law. Based on

    Lyapunov stability analysis, adaptation laws are derived for the fuzzy parameter

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    994 W. J. Yoo, D. H. Ji & S. C. Won

    vector, the synchronization error feedback gain and the mean values of the system

    parameters. Numerical examples demonstrate that the drive and response systems

    are well synchronized and the mean square of the synchronization error converges

    to the neighborhood of the origin within any arbitrarily chosen bound.

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