Adaptive reduced-order anti-synchronization of chaotic systems with ...
Commun Nonlinear Sci Numer Simulat 15 (2010) 3022–3034
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Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters M. Mossa Al-sawalha a,*, M.S.M. Noorani b a b
Faculty of Science, Mathematics Department, University of Hail, Saudi Arabia Center for Modelling & Data Analysis, School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
a r t i c l e
i n f o
Article history: Received 17 February 2009 Received in revised form 31 October 2009 Accepted 1 November 2009 Available online 10 November 2009 Keywords: Reduced-order Anti-synchronization Adaptive control Unknown parameters
a b s t r a c t In this paper, we investigate the reduced-order anti-synchronization of uncertain chaotic systems. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third-order chaotic system can be anti-synchronized with the canonical projection of a fourth-order chaotic system even though their parameters are unknown. The techniques are successfully applied to two examples: hyperchaotic Lorenz system (fourth-order) and Lorenz system (third-order); Lü hyperchaotic system (fourth-order) and Chen system (third-order). Theoretical analysis and numerical simulations are shown to verify the results. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Our natural world consists of physical systems that are undoubtedly nonlinear. As one understand this fact, one will realize that chaos is inevitable and inherently present in our lives, even though it may not be seen with the naked eye. It has been repeatedly demonstrated by scientists in the last recent decades that nonlinear systems, which models our real world, can display a variety of behaviors including chaos and hyperchaos. The most important characteristic of chaotic dynamics is its critical sensitivity to initial conditions, which is responsible for initially neighboring trajectories separating from each other exponentially in the course of time. This behavior made chaos undesirable and unwanted in many cases of research as it reduces their predictability over long time scales. But this special attribute may be a valuable advantage in certain areas of research. The ability of chaotic dynamics to amplify small perturbations improves their utility for reaching specific desired states with very high flexibility and low energy cost is indeed advantageous. In other words, we could try controlling chaos for the benefit of our needs. Synchronization or antisynchronization of different chaotic or hyperchaotic systems is one of the few main control methods which are popularly discussed recently. This is generally due to its prospective applications especially in chemical reactions, power converters, biological systems, information processing, secure communications, etc. [1]. As the problems of synchronization and anti-synchronization of chaos are interesting, nontraditional, and indeed very challenging [2–7], a widely variety of approaches have been proposed for chaos synchronization and anti-synchronization such as adaptive control [8–21], linear and nonlinear feedback control [22–32], active control [33–39], complete synchronization [40] and reduced-order synchronization [41,42].
* Corresponding author. E-mail address: [email protected] (M.M. Al-sawalha). 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.11.001
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The Synchronization of chaotic systems with different order or reduced-order synchronization has received less attention. To the author’s best knowledge, there are only a few results in the literature about the synchronization between chaotic systems whose order is different [41–44]. Moreover, in many real systems, the synchronization is carried out even though the oscillators have different order. In the case of thalamic neurons, for instance, such a problem is reasonable if their order is different from the one of the hippocampal neurons. Another example happens in the synchronization between heart and lung. One can observe that both, circulatory and respiratory systems, behave in synchronous way. However, one can expect that the model of the circulatory system is strictly different from the respiratory one, which can involve different order. In addition, the synchronization of strictly different chaotic systems is interesting by itself. Reduced-order synchronization is the problem of synchronizing a slave system with projections of a master system. It should be noted that the reduced-order synchronization is not the partial synchronization problem. On the one hand, the partial synchronization is for coupling two chaotic systems which have an equal order. A main feature of the partial synchronization is that, at least, one state of the slave system is not synchronous in some sense. On the other hand, in reduced-order synchronization, all states of the slave system are synchronous, in some sense. The main feature of the reduced-order synchronization is that the order of the slave system is less than the master one. Consequently, the reduced-order synchronization of systems with uncertainties may play an important role in many fields, including chaotic secure communication. The epitome of this paper centers on chaos anti-synchronization between two chaotic systems whose order is different, since some of the aforementioned methods and many other existing methods are mainly concern to synchronized or antisynchronized between two chaotic system with same order. This feature limits the complexity of the chaotic dynamics. It is believed that the anti-synchronization between two chaotic systems whose order is different have much wider application. The rest of the paper is organized as follows. Section 2 briefly describes the problem formulation and systems description. In Section 3 and 4 we present the adaptive anti-synchronization strategies with a parameter update law for hyperchaotic Lorenz (fourth-order) and Lorenz (third-order) systems and hyperchaotic Lü (fourth-order) and Chen (third-order) systems to perform the anti-synchronization, respectively. A conclusion is given at the end.
2. Problem formulation and systems description 2.1. Problem formulation Consider the chaotic system described by
x_ ¼ f ðxÞ þ FðxÞa;
ð1Þ
m
m
m
where x 2 R is the state vector of the system (1), f : R ! R is a continuous vector function including nonlinear terms, F : Rm ! Rmk , and a 2 Rk is the parameter vector of system. Eq. (1) is considered as the master system. Similarly, the slave system is given by
y_ ¼ gðyÞ þ GðyÞb þ U;
ð2Þ
n
n
n
n
n‘
‘
where y 2 R is the state vector, g : R ! R is a continuous vector function, G : R ! R , and b 2 R is the parameter vector. The purpose of chaos anti-synchronization is to design a controller U ðU 2 Rn Þ, which is able to anti-synchronize the state of the master and the slave systems. When order n ¼ m; ‘ ¼ k and the functions f ¼ g; F ¼ G, the slave system is identical to the master system, and the CAS problem has been well studied. When two systems satisfy the condition n < m (of course f – g; and F – G), that is, the order of the slave oscillator is lower than that of the master system, the anti-synchronization is only attained in reduce-order. Actually, reduced-order anti-synchronization is the problem of controlling a slave system to be the projection of the master system. Therefore, we can divide the master system into two parts. The projection:
x_ ı ¼ fı ðxÞ þ F ı ðxÞa; n
m
ð3Þ n
m
nk
where xı 2 R ; f ı : R ! R ; and F ı : R ! R
. The rest:
x_ | ¼ f| ðxÞ þ F | ðxÞa; u
m
ð4Þ u
m
uk
where x| 2 R ; f | : R ! R ; F | : R ! R and orders n; u satisfy n þ u ¼ m. With a suitable controller, the reduced-order anti-synchronization between two different systems can be achieved, i.e.,
lim kyðtÞ þ xı ðtÞk ¼ 0;
t!1
ð5Þ
where the error vector is defined by eðtÞ ¼ yðtÞ þ xı ðtÞ, where eðtÞ 2 Rn , we add Eq. (3) to Eq. (1) and
e_ ¼ gðyÞ þ GðyÞb þ U þ fı ðxÞ þ F ı ðxÞa ¼ qðe; xÞ þ Gðe; xÞb þ F ı ðxÞa þ U; n
m
ð6Þ
n
where q : R R ! R is a continuous vector function. In practical situations, the parameters belonging to the master and the slave systems are always unknown. Therefore, by using the adaptive control and the parameter identification techniques, the adaptive nonlinear controller [42] can be decided as
^ F ı ðxÞa ^; U ¼ Q ðe; xÞ Gðe; xÞb
ð7Þ
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^ are the estimated vector of unknown parameters, and the updating laws of the estimated ^ and b where Q : Rn Rm ! Rn ; a parameters [42] are given by
^_ ¼ GT ðyÞeT ; b a^_ ¼ F T ðxÞeT :
ð8Þ
ı
~ , where a ~T b ~ ¼ b b. ^ With the choice of the updat~ ¼aa ~T a ^ and b ~þb Assume a positive Lyapunov function V ¼ 12 eT e þ a ing laws above and reasonable control function Q ðe; xÞ, the time derivative of V along the solution in Eq. (6) will be smaller than zero. In other words, the error vector will approach to zero as time goes infinite and from Lyapunov stability theory [45], the states of the slave system and projected master system are asymptotically anti-synchronized. 2.2. Systems description The hyperchaotic Lorenz system [46,47] is described by
x_ ¼ aðy xÞ þ w; y_ ¼ xz þ rx y; z_ ¼ xy bz;
ð9Þ
_ ¼ xz þ dw; w where x, y, z, and w are state variables, and a; b; c, and d are real constants. When a ¼ 36; b ¼ 3; c ¼ 20 and d ¼ 1:3, system (9) has hyperchaotic attractor. The (positive) Lyapunov exponents for the hyperchaotic Lorenz system are k1 ¼ 0:3985 and k2 ¼ 0:2481. The projections of the hyperchaotic Lorenz system system attractor is shown in Fig. 1(a). The Lorenz system is known to be a simplified model of several physical systems. At the origin, it was derived from a model of the earths atmospheric convection flow heated from below and cooled from above [48]. Furthermore, it has been reported that Lorenz equations may describe such different systems as laser devices, disk dynamos and several problems related to convection. The Lorenz system is described by
x_ ¼ aðy xÞ; y_ ¼ cx xz y; z_ ¼ xy bz;
ð10Þ
where x, y and z are, respectively, proportional to the convective velocity, the temperature difference between descending and ascending flows, and the mean convective heat flow, and a, b and the so-called bifurcation parameter c are real
(a)
(c)
(b)
(d)
Fig. 1. (a) Hyperchaotic Lorenz attractor; (b) Lorenz attractor; (c) hyperchaotic Lü attractor; and (d) Chen attractor.
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constants. The (positive) Lyapunov exponent for Lorenz system is about k1 ¼ 0:9056. Throughout this paper, we set a ¼ 10; b ¼ 8=3 and c ¼ 28 so the system exhibits chaotic behavior. The projections of the Lorenz chaotic attractor is shown in Fig. 1(b). The hyperchaotic Lü system [49] is described by
x_ ¼ aðy xÞ þ w; y_ ¼ xz þ cy; z_ ¼ xy bz;
ð11Þ
_ ¼ xz þ rw; w where x, y, z, and w are state variables, and a, b, c, and r are real constants. When a ¼ 36; b ¼ 3; c ¼ 20; 1:03 6 r 6 0:46, system (11) has periodic orbit, when a ¼ 36; b ¼ 3; c ¼ 20; 0:46 < r 6 0:35, system (11) has chaotic attractor, when a ¼ 36; b ¼ 3; c ¼ 20; 0:35 < r 6 1:3, system (11) has a hyperchaotic attractor. The (positive) Lyapunov exponents for the hyperchaotic Lü system are k1 ¼ 1:5046 and k2 ¼ 0:1643 when r ¼ 1:3. The projections of the hyperchaotic Lü system system attractor is shown in Fig. 1(c). Chen dynamical system [50] is described by the following system of differential equations:
x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy; z_ ¼ xy bz;
ð12Þ
where x, y and z are state variables, a, b and c are positive parameters. Bifurcation studies shows that with the parameters a ¼ 35 and c ¼ 28, system (12) exhibits chaotic behavior when b ¼ 3. It is interesting to note that the (positive) Lyapunov exponent for the Chen system is about k1 ¼ 2:0272 [51]. The projections of the chaotic Chen attractor is shown in Fig. 1(d). 3. Adaptive anti-synchronization between hyperchaotic Lorenz and Lorenz systems with unknown parameters In order to observe the anti-synchronization behavior between hyperchaotic Lorenz system and Lorenz system, we assume that projection of the hyperchaotic Lorenz systems is the master system which will be denoted by the subscript 1, therefore, the master system is the projection part of Eq. (9), and it can be presented in the form of
x_ 1 ¼ a1 ðy1 x1 Þ þ w1 ; y_ 1 ¼ x1 z1 þ r 1 x1 y1 ; z_ 1 ¼ x1 y1 b1 z1 ;
ð13Þ
and we assume that the Lorenz system is the response system which will be denoted by the subscript 2, and it can be presented in the form of
x_ 2 ¼ a2 ðy2 x2 Þ þ u1 ; y_ 2 ¼ c2 x2 x2 z2 y2 þ u2 ; z_ 2 ¼ x2 y2 b2 z2 þ u3 ;
ð14Þ
where u1 ; u2 ; u3 are three control functions to be designed and all parameters a1 ; r1 ; b1 ; d1 ; a2 ; c2 ; b2 are unknown. We add Eq. (14) to Eq. (13) and get
e_1 ¼ a2 ðy2 x2 Þ þ a1 ðy1 x1 Þ þ w1 þ u1 ; e_2 ¼ c2 x2 x2 z2 y2 x1 z1 þ r1 x1 y1 þ u2 ; e_3 ¼ x2 y2 b2 z2 þ x1 y1 b1 z1 þ u3 ;
ð15Þ
where e1 ¼ x2 þ x1 ; e2 ¼ y2 þ y1 ; e3 ¼ z2 þ z1 , our goal is to find proper control functions ui ði ¼ 1; 2; 3Þ and parameter update rule, such that system equation (14) asymptotically anti-synchronizes system equation (13). i.e. limt!1 kek ¼ 0, where e ¼ ½e1 ; e2 ; e3 T . For this end, we propose the following adaptive control laws for system equation (15)
^2 ðy2 x2 Þ a ^1 ðy1 x1 Þ w1 e1 ; u1 ¼ a u2 ¼ ^c2 x2 þ x2 z2 þ y2 þ x1 z1 ^r1 x1 þ y1 e2 ; ^ 2 z 2 x1 y þ b ^ 1 z1 e 3 ; u3 ¼ x2 y2 þ b 1 and parameter update rules
ð16Þ
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^_ 1 ¼ ðy1 x1 Þe1 ; a ^_ 1 ¼ z1 e3 ; b ^r_ ¼ x e ; 1
1 2
ð17Þ
^_ 2 ¼ ðy2 x2 Þe1 ; a ^_ 2 ¼ z2 e3 ; b ^c_ ¼ x e ; 2
2 2
^1 ; ^r 1 ; a ^2 ; ^c2 are estimates of a1 ; b1 ; r1 ; a2 ; b2 ; c2 , respectively. ^1 ; b ^2 ; b where a Theorem 1. For any initial conditions the Lorenz system is anti-synchronize with projection of hyperchaotic Lorenz system asymptotically anti-synchronize by using adaptive control law equation (16) and parameter update rule equation (17). Proof. Applying control law equation (16) to Eq. (15) yields the resulting the closed-loop error dynamical system as follows:
~2 ðy2 x2 Þ þ a ~1 ðy1 x1 Þ e1 ; e_1 ¼ a e_2 ¼ ~c2 x2 þ ~r1 x1 e2 ; ~2 z2 b ~1 z1 e3 ; e_3 ¼ b
ð18Þ
~1 ¼ b1 b ^1 ; ~r 1 ¼ r 1 ^r 1 ; a ~ 2 ¼ b2 b ^2 ; ~c2 ¼ c2 ^c2 . Consider the following Lyapunov func~1 ¼ a1 a ^1 ; b ~2 ¼ a2 a ^2 ; b where a tion candidate
V¼
1 T ~2 þ ~r 2 þ a ~2 þ ~c2 > 0; ~21 þ b ~22 þ b e eþa 1 1 2 2 2
ð19Þ
then the time derivative of V along the solution of error dynamical system equation (18) gives
~_ 1 þ ~r 1~r_ 1 þ a ~_ 2 þ ~c2 ~c_ 2 ~1 b ~2 b ~_ 1 þ b ~_ 2 þ b ~1 a ~2 a V_ ¼ eT e_ þ a
~ 3 e3 þ a ~1 ðz1 e3 Þ ~2 ðy2 x2 Þ þ a ~1 ðy1 x1 Þ e1 Þ þ e2 ð~c2 x2 þ ~r1 x1 e2 Þ þ e3 be ~1 ððy1 x1 Þe1 Þ þ b ¼ e 1 ða ~2 ðz2 e3 Þ þ ~c2 ðx2 e2 Þ ¼ eT e 6 0: ~2 ððy2 x2 Þe1 Þ þ b þ ~r 1 ðx1 e2 Þ þ a
ð20Þ
Since V is positive definite and V_ is negative definite in the neighborhood of zero solution of system equation (15), it follows ^1 ; ^r1 ; a ^2 ; ^c2 2 L1 , from Eq. (18), we have e_ 1 ; e_ 2 ; e_ 3 2 L1 , since V_ ¼ eT e then we obtain ^1 ; b ^2 ; b that e1 ; e2 ; e3 2 L1 , a
Z 0
t
kek2 dt 6
Z
t
eT e dt ¼
0
Z
t
V_ dt ¼ Vð0Þ VðtÞ 6 Vð0Þ
0
Thus, e_ 1 ; e_ 2 ; e_ 3 2 L2 , by Barbalat’s lemma, we have limt!1 keðtÞk ¼ 0. Therefore, the response system equation (14) can antisynchronize the drive system equation (13) asymptotically. h 3.1. Numerical simulations To verify and demonstrate the effectiveness of the proposed method, we discuss the simulation results for the anti-synchronization problem between the projective of the hyperchaotic Lorenz and the Lorenz systems. In the numerical simulations, the fourth-order Runge–Kutta method is used to solve both systems with time step size 0.001. Assume that the initial conditions, ðx1 ð0Þ; y1 ð0Þ; z1 ð0Þ; w1 ð0ÞÞ ¼ ð0:1; 0:2; 0:6; 0:4Þ, and ðx2 ð0Þ; y2 ð0Þ; z2 ð0ÞÞ ¼ ð2; 4; 13Þ are employed. Hence the error system has the initial values e1 ð0Þ ¼ 2:1; e2 ð0Þ ¼ 4:2 and e3 ð0Þ ¼ 13:6. The unknown parameters are chosen as a1 ¼ 10; b1 ¼ 8=3; r1 ¼ 28 and a2 ¼ 10; b2 ¼ 8=3; c2 ¼ 28 in simulations so that the both systems exhibits a chaotic behavior anti-synchronization of the systems (13) and (14) via adaptive control law (16) and (17) with the initial estimated param^1 ð0Þ ¼ 2; a ^2 ð0Þ ¼ 5 and ^c2 ð0Þ ¼ 30 are shown in Figs. 2–4. Figs. 2 and 3 display ^ð0Þ2 ¼ 25; b ^1 ð0Þ ¼ 0:2; ^r1 ð0Þ ¼ 10 ¼ b eters a the state response and the anti-synchronization errors of systems (13) and (14). Fig. 4 shows that the estimates ^1 ðtÞ; ^r1 ðtÞ; a ^2 ðtÞ and ^c2 ðtÞ of the unknown parameters converges to a1 ¼ 10; b1 ¼ 8=3; r 1 ¼ 28 and a2 ^2 ðtÞ; b ^1 ðtÞ; b a ¼ 10; b2 ¼ 8=3; c2 ¼ 28 as t ! 1. Fig. 5 show that the Lorenz system is control to the projective of the hyperchaotic Lorenz system. 4. Adaptive anti-synchronization between hyperchaotic Lü and Chen systems with unknown parameters In this section we will study the anti-synchronization problem between the hyperchaotic Lü system (11) and the Chen system (12). Similarly, we regard the mater as the projection part of hyperchaotic Lü system which consider as the master system and denoted with subscript 1 and it is described by
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(a)
(b)
(c)
Fig. 2. State trajectories of drive system (13) and response system (14): (a) Signals x1 and x2 ; (b) signals y1 and y2 ; and (c) signals z1 and z2 .
x_ 1 ¼ a1 ðy1 x1 Þ þ w1 ; y_ 1 ¼ x1 z1 þ c1 y1 ; z_ 1 ¼ x1 y1 b1 z1 ;
ð21Þ
and we assume that the Chen system is the response system which is denoted by the subscript 2, and it can be presented in the form of
x_ 2 ¼ a2 ðy2 x2 Þ þ u1 ; y_ 2 ¼ ðc2 a2 Þx x2 z2 þ c2 y2 þ u2 ; z_ 2 ¼ x2 y2 b2 z2 þ u3 ;
ð22Þ
where u1 ; u2 ; u3 are three control functions to be designed and all parameters a1 ; b1 ; c1 ; a2 ; c2 ; b2 are unknown. In order to determine the control functions to realize the anti-synchronization between system equations (21) and (22). We add Eq. (22) to Eq. (21) and get
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(a)
(b)
(c)
Fig. 3. The error signals e1 ; e2 ; e3 of the hyperchaotic Lorenz and Lorenz systems under the controller (16) and the parameters update law (17).
e_ 1 ¼ a2 ðy2 x2 Þ þ a1 ðy1 x1 Þ þ w1 þ u1 ; e_ 2 ¼ ðc2 a2 Þx x2 z2 þ c2 y2 x1 z1 þ c1 y1 þ u2 ; e_ 3 ¼ x2 y2 b2 z2 þ x1 y1 b1 z1 þ u3 ;
ð23Þ
where e1 ¼ x1 þ x2 ; e2 ¼ y1 þ y2 ; e3 ¼ z1 þ z2 , our goal is to find proper control functions ui ði ¼ 1; 2; 3Þ and parameter update rule, such that system equation (22) anti-synchronizes asymptotically with system equation (21). i.e. limt!1 kek ¼ 0 where e ¼ ½e1 ; e2 ; e3 T . For this end we propose the following adaptive control law for system equation (23)
^2 ðy2 x2 Þ a ^1 ðy1 x1 Þ w1 e1 ; u1 ¼ a ^2 Þx þ x2 z2 ^c2 y2 þ x1 z1 ^c1 y1 e2 ; u2 ¼ ð^c2 a ^2 z2 x1 y þ b ^1 z1 e3 ; u3 ¼ x2 y þ b 2
and parameter update rules
1
ð24Þ
M.M. Al-sawalha, M.S.M. Noorani / Commun Nonlinear Sci Numer Simulat 15 (2010) 3022–3034
(a)
(b)
Fig. 4. Changing parameters a1 ; b1 ; r 1 and a2 ; b2 ; c2 of the hyperchaotic Lorenz and Lorenz systems with time t.
Fig. 5. Hyperchaotic Lorenz system (solid line) and the controlled Lorenz system (dotted line).
^_ 1 ¼ ðy1 x1 Þe1 ; a ^_ 1 ¼ z1 e3 ; b ^c_ ¼ y e ; 1
1 2
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^_ 2 ¼ ðy2 x2 Þe1 x2 e2 a ^_ 2 ¼ z2 e3 ; b ^c_ ¼ ðx þ y Þe ; 2
2
2
2
^1 ; ^c1 and a ^2 ; ^c2 are the estimates of a1 ; b1 ; c1 ; a2 ; b2 ; c2 , respectively. ^1 ; b ^2 ; b where a Theorem 2. For any initial conditions the Chen system asymptotically anti-synchronize with the projection of hyperchaotic Lü system asymptotically by using adaptive control law equation (24) and parameter update rule equation (25). Proof. Applying control law equation (24) to Eq. (23) yields the resulting error dynamics as follows:
~2 ðy2 x2 Þ þ a ~1 ðy1 x1 Þ e1 ; e_ 1 ¼ a ~2 Þx þ ~c2 y2 þ ~c1 y1 e2 ; _e2 ¼ ð~c2 a ~2 z2 b ~1 z1 e3 ; e_ 3 ¼ b
(a)
(b)
(c)
Fig. 6. State trajectories of drive system (21) and response system (22): (a) Signals x1 and x2 ; (b) signals y1 and y2 ; and (c) signals z1 and z2 .
ð26Þ
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~1 ¼ b b ^1 ; ~c1 ¼ c1 ^c1 ; a ~2 ¼ b2 b ^2 ; ~c2 ¼ c2 ^c2 . ~1 ¼ a1 a ^1 ; b ~ 2 ¼ a2 a ^2 ; b where a Consider the following Lyapunov function candidate:
V¼
1 T ~2 þ ~c2 þ a ~2 þ ~c2 > 0: ~21 þ b ~22 þ b e eþa 1 1 2 2 2
ð27Þ
The time derivative of V is as follows:
~_ 1 þ ~c1 ~c_ 1 þ a ~_ 2 þ ~c2 ~c_ 2 ~1 b ~2 b ~_ 1 þ b ~_ 2 þ b ~2 a ~1 a V_ ¼ eT e_ þ a ^_ 1 þ ~c1 ^c_ 1 þ a ^_ 2 þ ~c2 ^c_ 2 : ~1 b ~2 b ~1 a ^_ 1 þ b ^_ 2 þ b ~ 2 a ¼ e1 e_ 1 þ e2 e_ 2 þ e3 e_ 3 þ a
ð28Þ
Inserting (25) and (26) into (28) yields the follows:
V_ ¼ e21 e22 e23 6 0:
ð29Þ
(a)
(b)
(c)
Fig. 7. The error signals e1 ; e2 ; e3 between the hyperchaotic Lü system and Chen system under the controller (24.) and the parameters update law (25).
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Since V is positive definite and V_ is negative definite in the neighborhood of zero solution of system equation (23), it follows ^1 ; ^c1 ; a ^2 ; ^c2 2 L1 , from Eq. (26), we have e_ 1 ; e_ 2 ; e_ 3 2 L1 , since V_ ¼ eT e then we obtain ^1 ; b ^2 ; b that e1 ; e2 ; e3 2 L1 , a
Z 0
t
kek2 dt 6
Z
t
eT e dt ¼
Z
0
t
V_ dt ¼ Vð0Þ VðtÞ 6 Vð0Þ
0
Thus, e_ 1 ; e_ 2 ; e_ 3 2 L2 , by Barbalats lemma, we have limt!1 keðtÞk ¼ 0. Therefore, response system equation (22) can anti-synchronize drive system equation (21) asymptotically. h 4.1. Numerical simulations To verify and demonstrate the effectiveness of the proposed method, we discuss the simulation results for the anti-synchronization problem between the projective of the hyperchaotic Lü and Chen systems. In the numerical simulations, the fourth-order Runge–Kutta method is used to solve the systems with time step size 0.001. We assume that the initial condition, ðx1 ð0Þ; y1 ð0Þ; z1 ð0Þ; w1 ð0ÞÞ ¼ ð5; 8; 1; 3Þ, and ðx2 ð0Þ; y2 ð0Þ; z2 ð0ÞÞ ¼ ð10; 0; 37Þ. Hence the error system has the initial values e1 ð0Þ ¼ 5; e2 ð0Þ ¼ 8; e3 ð0Þ ¼ 36. The unknown parameters are chosen as a1 ¼ 36; b1 ¼ 3; c1 ¼ 20 and a2 ¼ 35; b2 ¼ 3; c2 ¼ 28. anti-synchronization of the systems (21) and (22) via adaptive control law (24) and (25) with the initial estimated parameters a1 ð0Þ ¼ 0:2; b1 ð0Þ ¼ 2; c1 ð0Þ ¼ 10 and a2 ð0Þ ¼ 25; b2 ð0Þ ¼ 5; c2 ð0Þ ¼ 30 are shown in Figs. 6–9. Figs. 6 and 7 display the state response and the anti-synchronization errors of systems (21) and (22). Fig. 6 shows that ^1 ðtÞ; ^c1 ðtÞ and a ^2 ðtÞ; ^c2 ðtÞ of the unknown parameters converges to a1 ¼ 36; b1 ¼ 3; c1 ¼ ^2 ðtÞ; b ^1 ðtÞ; b the estimates a 20 and a2 ¼ 35; b2 ¼ 3; c2 ¼ 28 as t ! 1. Fig. 9 show that the Chen system is control to be the projective of the hyperchaotic Lü system. 5. Concluding remark In this article, reduced-order anti-synchronization problem of uncertain chaotic systems has been studied. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third-order chaotic system can be anti-synchronized with the canonical projection of a fourth-order chaotic system even though their param-
(a)
(b)
Fig. 8. Changing parameters a1 ; b1 ; c1 and a2 ; b2 ; c2 of the drive system (21) and the response system (22) with time t.
M.M. Al-sawalha, M.S.M. Noorani / Commun Nonlinear Sci Numer Simulat 15 (2010) 3022–3034
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Fig. 9. Hyperchaotic Lü system (solid line) and the controlled Chen system (dotted line).
eters are unknown. The techniques are successfully applied to two examples: hyperchaotic Lorenz system (fourth-order) and Lorenz system (third-order); Lü hyperchaotic system (fourth-order) and Chen system (third-order). Theoretical analysis and numerical simulations are shown to verify the results. Acknowledgments This work is financially supported by the Malaysian Ministry of Higher Education Grant: UKM-ST-06-FRGS0008-2008. Also the authors are grateful to the anonymous referees for their helpful comments on the earlier draft of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
Chen G, Dong X. From chaos to order. Singapore: World Scientific; 1998. Boccaletti S, Kurths J, Osipov G, Valladares DL, Zhou CS. The synchronization of chaotic systems. Phys Rep 2002;366:1–101. Nijmeijer H. A dynamical control view on synchronization. Physica D 2001;154:219–28. Boukabou A. On the control and synchronization design for autonomous chaotic systems. Nonlinear Dyn Syst Theory 2008;8(2):151–67. Yang XS. A framework for synchronization theory. Chaos Soliton Fract 2000;11:1365–8. Luo ACJ. A theory for synchronization of dynamical systems. Commun Nonlinear Sci Numer Simulat 2009;14:1901–51. Rafikov M, Balthazar J. On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commun Nonlinear Sci Numer Simulat 2008;13:1246–55. Bowong S. Adaptive synchronization between two different chaotic dynamical systems. Commun Nonlinear Sci Numer Simulat 2007;12:976–85. Chen X, Lü J. Adaptive synchronization of different chaotic systems with full unknown parameters. Phys Lett A 2007;364:123–8. Wang Z. Anti-synchronization in two non-identical hyperchaotic systems with known or unknown parameters. Commun Nonlinear Sci Numer Simulat 2009;14:1901–51. Li Ruihong, Xu Wei, Li Shuang. Anti-synchronization on autonomous and non-autonomous chaotic systems via adaptive feedback control. Chaos Soliton Fract 2007. doi:10.1016/j.chaos.2007.09.032. Wu X, Guan Z, Wu Z. Adaptive synchronization between two different hyperchaotic systems. Nonlinear Anal TMA 2008;68(5):1346–51. Sun Y, Cao J. Adaptive synchronization between two different noise-perturbed chaotic systems with fully unknown parameters. Physica A 2007;376:253–65. Yassen MT. Adaptive control and synchronization of a modified Chua’s circuit system. Appl Math Comput 2003;135:113–28. Park J. Adaptive synchronization of hyperchaotic Chen system with uncertain parameters. Chaos Soliton Fract 2005;26:959–64. Xian W, Hongmin Z. Synchronization of two hyperchaotic systems via adaptive control, Chaos Soliton Fract. doi:10.1016/j.chaos.2007.06.100. Shihua C, Lü J. Parameters identification and synchronization of chaotic systems based upon adaptive control. Phys Lett A 2002;299:353–8. Lu J, Wu X, Han X, Lü J. Adaptive feedback synchronization of a unified chaotic system. Phys Lett A 2004;329:327–33. Huang J. Chaos synchronization between two novel different hyperchaotic systems with unknown parameters. Nonlinear Anal TMA 2008;69(11):4174–81. Al-Sawalha MM, Noorani MSM. Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters. Commun Nonlinear Sci Numer Simulat 2010;15:1036–47.
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[21] Al-Sawalha MM, Noorani MSM. Anti-synchronization of chaotic systems with uncertain parameters via adaptive control. Phys Lett A 2009. doi:10.1016/j.physleta.2009.06.008. [22] Lòpez-Mancilla D, Cruz-Hernández C. Output synchronization of chaotic systems under nonvanishing perturbations. Chaos Soliton Fract 2008;37:1172–86. [23] Lòpez-Gutiérrez RM, Posadas-Castillo C, Lòpez-Mancilla D, Cruz-Hernández C. Communicating via robust synchronization of chaotic lasers and sliding mode control. Chaos Soliton Fract 2009. doi:10.1016/j.chaos.2008.11.019. [24] Yan JJ, Yang YS, Chiang TY, Chen CY. Robust synchronization of unified chaotic systems via sliding mode control. Chaos Soliton Fract 2007;34:947–54. [25] Lü J, Lu J. Controlling uncertain Lü system using linear feedback. Chaos Soliton Fract 2003;17:127–33. [26] Chen H. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos Soliton Fract 2005;23:1245–51. [27] Huang L, Feng R, Wang M. Synchronization of chaotic systems via nonlinear control. Phys Lett A 2004;320:271–5. [28] Zhang Q, Lu J. Chaos synchronization of a new chaotic system via nonlinear control. Chaos Soliton Fract 2008;37:175–9. [29] Park J. Chaos synchronization of a chaotic system via nonlinear control. Chaos Soliton Fract 2005;25:579–84. [30] Al-Sawalha MM, Noorani MSM. On anti-synchronization of chaotic systems via nonlinear control. Chaos Soliton Fract 2008. doi:10.1016/ j.chaos.2008.11.011. [31] Al-Sawalha MM, Noorani MSM. Anti-synchronization of two hyperchaotic systems via Nonlinear control. Commun Nonlinear Sci Numer Simulat 2009;14:3402–4311. [32] Yassen MT. On hyperchaos synchronization of a hyperchaotic Lü system. Nonlinear Anal TMA 2008;68(11):3592–600. [33] Bai E, Lonngen K. Synchronization of two Lorenz systems using active control. Chaos Soliton Fract 1997;8:51–8. [34] Li G, Zhou S. Anti-synchronization in different chaotic systems. Chaos Soliton Fract 2007;32:516–20. [35] El-Dessoky M. Synchronization and anti-synchronization of a hyperchaotic Chen system. Chaos Soliton Fract 2007. doi:10.1016/j.chaos. 2007.06.053. [36] Al-Sawalha MM, Noorani MSM. Chaos anti-synchronization between two novel different hyperchaotic systems. Chin Phys Lett 2008;25(8):2743–6. [37] Al-Sawalha MM, Noorani MSM. Active anti-synchronization between identical and distinctive hyperchaotic systems. Open Syst Inform Dyn 2008;15(4):371–82. [38] Ho MC, Hung YC, Chou CH. Phase and anti-phase synchronization of two chaotic systems by using active control. Phys Lett A 2002;296:43–8. [39] Yassen MT. Chaos synchronization between two different chaotic systems using active control. Chaos Soliton Fract 2005;23:31–140.. [40] Cao J, Wang Z, Sun Y. Synchronization in an array of linearly stochastically coupled networks with time delays. Physica A 2007;385:718–28. [41] Bowong S. Stability analysis for the synchronization of chaotic systems with different order: application to secure communications. Phys Lett A 2004;326:102–13. [42] Ho M, Hung Y, Liua Z, Jiang I. Reduced-order synchronization of chaotic systems with parameters unknown. Phys Lett A 2006;348:251–9. [43] Femat R, Perales G. Synchronization of chaotic systems with different order. Phys Rev E 2002;65:036226–33. [44] Wang C, Ge SS. Synchronization of two uncertain chaotic Systems via adaptive backstepping. Int J Bifurcat Chaos 2001;11(6):1743–51. [45] Khalil HK. Nonlinear systems, 3rd ed. Englewood Cliffs (NJ): Prentice-Hall; 2002. [46] Jia Q. Hyperchaos generated from the Lorenz chaotic system and its control. Phys Lett A 2007;366:217–22. [47] Jia Q. Projective synchronization of a new hyperchaotic Lorenz system. Phys Lett A 2007;370:40–5. [48] Lorenz EN. Deterministic nonperiodic flow. J Atmos Sci 1963;20(2):130–41. [49] Chen A, Lu J, Lü J, Yu S. Generating hyperchaotic Lü attractor via state feedback control. Physica A 2006;364:103–10. [50] Chen G, Ueta T. Yet another chaotic attractor. Int J Bifurcat Chaos 1999;9(7):1465–76. [51] Sprott JC. Chaos and time-series analysis. Oxford: Oxford University Press; 2003.
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters M. Mossa Al-sawalha a,*, M.S.M. Noorani b a b
Faculty of Science, Mathematics Department, University of Hail, Saudi Arabia Center for Modelling & Data Analysis, School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
a r t i c l e
i n f o
Article history: Received 17 February 2009 Received in revised form 31 October 2009 Accepted 1 November 2009 Available online 10 November 2009 Keywords: Reduced-order Anti-synchronization Adaptive control Unknown parameters
a b s t r a c t In this paper, we investigate the reduced-order anti-synchronization of uncertain chaotic systems. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third-order chaotic system can be anti-synchronized with the canonical projection of a fourth-order chaotic system even though their parameters are unknown. The techniques are successfully applied to two examples: hyperchaotic Lorenz system (fourth-order) and Lorenz system (third-order); Lü hyperchaotic system (fourth-order) and Chen system (third-order). Theoretical analysis and numerical simulations are shown to verify the results. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Our natural world consists of physical systems that are undoubtedly nonlinear. As one understand this fact, one will realize that chaos is inevitable and inherently present in our lives, even though it may not be seen with the naked eye. It has been repeatedly demonstrated by scientists in the last recent decades that nonlinear systems, which models our real world, can display a variety of behaviors including chaos and hyperchaos. The most important characteristic of chaotic dynamics is its critical sensitivity to initial conditions, which is responsible for initially neighboring trajectories separating from each other exponentially in the course of time. This behavior made chaos undesirable and unwanted in many cases of research as it reduces their predictability over long time scales. But this special attribute may be a valuable advantage in certain areas of research. The ability of chaotic dynamics to amplify small perturbations improves their utility for reaching specific desired states with very high flexibility and low energy cost is indeed advantageous. In other words, we could try controlling chaos for the benefit of our needs. Synchronization or antisynchronization of different chaotic or hyperchaotic systems is one of the few main control methods which are popularly discussed recently. This is generally due to its prospective applications especially in chemical reactions, power converters, biological systems, information processing, secure communications, etc. [1]. As the problems of synchronization and anti-synchronization of chaos are interesting, nontraditional, and indeed very challenging [2–7], a widely variety of approaches have been proposed for chaos synchronization and anti-synchronization such as adaptive control [8–21], linear and nonlinear feedback control [22–32], active control [33–39], complete synchronization [40] and reduced-order synchronization [41,42].
* Corresponding author. E-mail address: [email protected] (M.M. Al-sawalha). 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.11.001
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The Synchronization of chaotic systems with different order or reduced-order synchronization has received less attention. To the author’s best knowledge, there are only a few results in the literature about the synchronization between chaotic systems whose order is different [41–44]. Moreover, in many real systems, the synchronization is carried out even though the oscillators have different order. In the case of thalamic neurons, for instance, such a problem is reasonable if their order is different from the one of the hippocampal neurons. Another example happens in the synchronization between heart and lung. One can observe that both, circulatory and respiratory systems, behave in synchronous way. However, one can expect that the model of the circulatory system is strictly different from the respiratory one, which can involve different order. In addition, the synchronization of strictly different chaotic systems is interesting by itself. Reduced-order synchronization is the problem of synchronizing a slave system with projections of a master system. It should be noted that the reduced-order synchronization is not the partial synchronization problem. On the one hand, the partial synchronization is for coupling two chaotic systems which have an equal order. A main feature of the partial synchronization is that, at least, one state of the slave system is not synchronous in some sense. On the other hand, in reduced-order synchronization, all states of the slave system are synchronous, in some sense. The main feature of the reduced-order synchronization is that the order of the slave system is less than the master one. Consequently, the reduced-order synchronization of systems with uncertainties may play an important role in many fields, including chaotic secure communication. The epitome of this paper centers on chaos anti-synchronization between two chaotic systems whose order is different, since some of the aforementioned methods and many other existing methods are mainly concern to synchronized or antisynchronized between two chaotic system with same order. This feature limits the complexity of the chaotic dynamics. It is believed that the anti-synchronization between two chaotic systems whose order is different have much wider application. The rest of the paper is organized as follows. Section 2 briefly describes the problem formulation and systems description. In Section 3 and 4 we present the adaptive anti-synchronization strategies with a parameter update law for hyperchaotic Lorenz (fourth-order) and Lorenz (third-order) systems and hyperchaotic Lü (fourth-order) and Chen (third-order) systems to perform the anti-synchronization, respectively. A conclusion is given at the end.
2. Problem formulation and systems description 2.1. Problem formulation Consider the chaotic system described by
x_ ¼ f ðxÞ þ FðxÞa;
ð1Þ
m
m
m
where x 2 R is the state vector of the system (1), f : R ! R is a continuous vector function including nonlinear terms, F : Rm ! Rmk , and a 2 Rk is the parameter vector of system. Eq. (1) is considered as the master system. Similarly, the slave system is given by
y_ ¼ gðyÞ þ GðyÞb þ U;
ð2Þ
n
n
n
n
n‘
‘
where y 2 R is the state vector, g : R ! R is a continuous vector function, G : R ! R , and b 2 R is the parameter vector. The purpose of chaos anti-synchronization is to design a controller U ðU 2 Rn Þ, which is able to anti-synchronize the state of the master and the slave systems. When order n ¼ m; ‘ ¼ k and the functions f ¼ g; F ¼ G, the slave system is identical to the master system, and the CAS problem has been well studied. When two systems satisfy the condition n < m (of course f – g; and F – G), that is, the order of the slave oscillator is lower than that of the master system, the anti-synchronization is only attained in reduce-order. Actually, reduced-order anti-synchronization is the problem of controlling a slave system to be the projection of the master system. Therefore, we can divide the master system into two parts. The projection:
x_ ı ¼ fı ðxÞ þ F ı ðxÞa; n
m
ð3Þ n
m
nk
where xı 2 R ; f ı : R ! R ; and F ı : R ! R
. The rest:
x_ | ¼ f| ðxÞ þ F | ðxÞa; u
m
ð4Þ u
m
uk
where x| 2 R ; f | : R ! R ; F | : R ! R and orders n; u satisfy n þ u ¼ m. With a suitable controller, the reduced-order anti-synchronization between two different systems can be achieved, i.e.,
lim kyðtÞ þ xı ðtÞk ¼ 0;
t!1
ð5Þ
where the error vector is defined by eðtÞ ¼ yðtÞ þ xı ðtÞ, where eðtÞ 2 Rn , we add Eq. (3) to Eq. (1) and
e_ ¼ gðyÞ þ GðyÞb þ U þ fı ðxÞ þ F ı ðxÞa ¼ qðe; xÞ þ Gðe; xÞb þ F ı ðxÞa þ U; n
m
ð6Þ
n
where q : R R ! R is a continuous vector function. In practical situations, the parameters belonging to the master and the slave systems are always unknown. Therefore, by using the adaptive control and the parameter identification techniques, the adaptive nonlinear controller [42] can be decided as
^ F ı ðxÞa ^; U ¼ Q ðe; xÞ Gðe; xÞb
ð7Þ
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^ are the estimated vector of unknown parameters, and the updating laws of the estimated ^ and b where Q : Rn Rm ! Rn ; a parameters [42] are given by
^_ ¼ GT ðyÞeT ; b a^_ ¼ F T ðxÞeT :
ð8Þ
ı
~ , where a ~T b ~ ¼ b b. ^ With the choice of the updat~ ¼aa ~T a ^ and b ~þb Assume a positive Lyapunov function V ¼ 12 eT e þ a ing laws above and reasonable control function Q ðe; xÞ, the time derivative of V along the solution in Eq. (6) will be smaller than zero. In other words, the error vector will approach to zero as time goes infinite and from Lyapunov stability theory [45], the states of the slave system and projected master system are asymptotically anti-synchronized. 2.2. Systems description The hyperchaotic Lorenz system [46,47] is described by
x_ ¼ aðy xÞ þ w; y_ ¼ xz þ rx y; z_ ¼ xy bz;
ð9Þ
_ ¼ xz þ dw; w where x, y, z, and w are state variables, and a; b; c, and d are real constants. When a ¼ 36; b ¼ 3; c ¼ 20 and d ¼ 1:3, system (9) has hyperchaotic attractor. The (positive) Lyapunov exponents for the hyperchaotic Lorenz system are k1 ¼ 0:3985 and k2 ¼ 0:2481. The projections of the hyperchaotic Lorenz system system attractor is shown in Fig. 1(a). The Lorenz system is known to be a simplified model of several physical systems. At the origin, it was derived from a model of the earths atmospheric convection flow heated from below and cooled from above [48]. Furthermore, it has been reported that Lorenz equations may describe such different systems as laser devices, disk dynamos and several problems related to convection. The Lorenz system is described by
x_ ¼ aðy xÞ; y_ ¼ cx xz y; z_ ¼ xy bz;
ð10Þ
where x, y and z are, respectively, proportional to the convective velocity, the temperature difference between descending and ascending flows, and the mean convective heat flow, and a, b and the so-called bifurcation parameter c are real
(a)
(c)
(b)
(d)
Fig. 1. (a) Hyperchaotic Lorenz attractor; (b) Lorenz attractor; (c) hyperchaotic Lü attractor; and (d) Chen attractor.
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constants. The (positive) Lyapunov exponent for Lorenz system is about k1 ¼ 0:9056. Throughout this paper, we set a ¼ 10; b ¼ 8=3 and c ¼ 28 so the system exhibits chaotic behavior. The projections of the Lorenz chaotic attractor is shown in Fig. 1(b). The hyperchaotic Lü system [49] is described by
x_ ¼ aðy xÞ þ w; y_ ¼ xz þ cy; z_ ¼ xy bz;
ð11Þ
_ ¼ xz þ rw; w where x, y, z, and w are state variables, and a, b, c, and r are real constants. When a ¼ 36; b ¼ 3; c ¼ 20; 1:03 6 r 6 0:46, system (11) has periodic orbit, when a ¼ 36; b ¼ 3; c ¼ 20; 0:46 < r 6 0:35, system (11) has chaotic attractor, when a ¼ 36; b ¼ 3; c ¼ 20; 0:35 < r 6 1:3, system (11) has a hyperchaotic attractor. The (positive) Lyapunov exponents for the hyperchaotic Lü system are k1 ¼ 1:5046 and k2 ¼ 0:1643 when r ¼ 1:3. The projections of the hyperchaotic Lü system system attractor is shown in Fig. 1(c). Chen dynamical system [50] is described by the following system of differential equations:
x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy; z_ ¼ xy bz;
ð12Þ
where x, y and z are state variables, a, b and c are positive parameters. Bifurcation studies shows that with the parameters a ¼ 35 and c ¼ 28, system (12) exhibits chaotic behavior when b ¼ 3. It is interesting to note that the (positive) Lyapunov exponent for the Chen system is about k1 ¼ 2:0272 [51]. The projections of the chaotic Chen attractor is shown in Fig. 1(d). 3. Adaptive anti-synchronization between hyperchaotic Lorenz and Lorenz systems with unknown parameters In order to observe the anti-synchronization behavior between hyperchaotic Lorenz system and Lorenz system, we assume that projection of the hyperchaotic Lorenz systems is the master system which will be denoted by the subscript 1, therefore, the master system is the projection part of Eq. (9), and it can be presented in the form of
x_ 1 ¼ a1 ðy1 x1 Þ þ w1 ; y_ 1 ¼ x1 z1 þ r 1 x1 y1 ; z_ 1 ¼ x1 y1 b1 z1 ;
ð13Þ
and we assume that the Lorenz system is the response system which will be denoted by the subscript 2, and it can be presented in the form of
x_ 2 ¼ a2 ðy2 x2 Þ þ u1 ; y_ 2 ¼ c2 x2 x2 z2 y2 þ u2 ; z_ 2 ¼ x2 y2 b2 z2 þ u3 ;
ð14Þ
where u1 ; u2 ; u3 are three control functions to be designed and all parameters a1 ; r1 ; b1 ; d1 ; a2 ; c2 ; b2 are unknown. We add Eq. (14) to Eq. (13) and get
e_1 ¼ a2 ðy2 x2 Þ þ a1 ðy1 x1 Þ þ w1 þ u1 ; e_2 ¼ c2 x2 x2 z2 y2 x1 z1 þ r1 x1 y1 þ u2 ; e_3 ¼ x2 y2 b2 z2 þ x1 y1 b1 z1 þ u3 ;
ð15Þ
where e1 ¼ x2 þ x1 ; e2 ¼ y2 þ y1 ; e3 ¼ z2 þ z1 , our goal is to find proper control functions ui ði ¼ 1; 2; 3Þ and parameter update rule, such that system equation (14) asymptotically anti-synchronizes system equation (13). i.e. limt!1 kek ¼ 0, where e ¼ ½e1 ; e2 ; e3 T . For this end, we propose the following adaptive control laws for system equation (15)
^2 ðy2 x2 Þ a ^1 ðy1 x1 Þ w1 e1 ; u1 ¼ a u2 ¼ ^c2 x2 þ x2 z2 þ y2 þ x1 z1 ^r1 x1 þ y1 e2 ; ^ 2 z 2 x1 y þ b ^ 1 z1 e 3 ; u3 ¼ x2 y2 þ b 1 and parameter update rules
ð16Þ
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^_ 1 ¼ ðy1 x1 Þe1 ; a ^_ 1 ¼ z1 e3 ; b ^r_ ¼ x e ; 1
1 2
ð17Þ
^_ 2 ¼ ðy2 x2 Þe1 ; a ^_ 2 ¼ z2 e3 ; b ^c_ ¼ x e ; 2
2 2
^1 ; ^r 1 ; a ^2 ; ^c2 are estimates of a1 ; b1 ; r1 ; a2 ; b2 ; c2 , respectively. ^1 ; b ^2 ; b where a Theorem 1. For any initial conditions the Lorenz system is anti-synchronize with projection of hyperchaotic Lorenz system asymptotically anti-synchronize by using adaptive control law equation (16) and parameter update rule equation (17). Proof. Applying control law equation (16) to Eq. (15) yields the resulting the closed-loop error dynamical system as follows:
~2 ðy2 x2 Þ þ a ~1 ðy1 x1 Þ e1 ; e_1 ¼ a e_2 ¼ ~c2 x2 þ ~r1 x1 e2 ; ~2 z2 b ~1 z1 e3 ; e_3 ¼ b
ð18Þ
~1 ¼ b1 b ^1 ; ~r 1 ¼ r 1 ^r 1 ; a ~ 2 ¼ b2 b ^2 ; ~c2 ¼ c2 ^c2 . Consider the following Lyapunov func~1 ¼ a1 a ^1 ; b ~2 ¼ a2 a ^2 ; b where a tion candidate
V¼
1 T ~2 þ ~r 2 þ a ~2 þ ~c2 > 0; ~21 þ b ~22 þ b e eþa 1 1 2 2 2
ð19Þ
then the time derivative of V along the solution of error dynamical system equation (18) gives
~_ 1 þ ~r 1~r_ 1 þ a ~_ 2 þ ~c2 ~c_ 2 ~1 b ~2 b ~_ 1 þ b ~_ 2 þ b ~1 a ~2 a V_ ¼ eT e_ þ a
~ 3 e3 þ a ~1 ðz1 e3 Þ ~2 ðy2 x2 Þ þ a ~1 ðy1 x1 Þ e1 Þ þ e2 ð~c2 x2 þ ~r1 x1 e2 Þ þ e3 be ~1 ððy1 x1 Þe1 Þ þ b ¼ e 1 ða ~2 ðz2 e3 Þ þ ~c2 ðx2 e2 Þ ¼ eT e 6 0: ~2 ððy2 x2 Þe1 Þ þ b þ ~r 1 ðx1 e2 Þ þ a
ð20Þ
Since V is positive definite and V_ is negative definite in the neighborhood of zero solution of system equation (15), it follows ^1 ; ^r1 ; a ^2 ; ^c2 2 L1 , from Eq. (18), we have e_ 1 ; e_ 2 ; e_ 3 2 L1 , since V_ ¼ eT e then we obtain ^1 ; b ^2 ; b that e1 ; e2 ; e3 2 L1 , a
Z 0
t
kek2 dt 6
Z
t
eT e dt ¼
0
Z
t
V_ dt ¼ Vð0Þ VðtÞ 6 Vð0Þ
0
Thus, e_ 1 ; e_ 2 ; e_ 3 2 L2 , by Barbalat’s lemma, we have limt!1 keðtÞk ¼ 0. Therefore, the response system equation (14) can antisynchronize the drive system equation (13) asymptotically. h 3.1. Numerical simulations To verify and demonstrate the effectiveness of the proposed method, we discuss the simulation results for the anti-synchronization problem between the projective of the hyperchaotic Lorenz and the Lorenz systems. In the numerical simulations, the fourth-order Runge–Kutta method is used to solve both systems with time step size 0.001. Assume that the initial conditions, ðx1 ð0Þ; y1 ð0Þ; z1 ð0Þ; w1 ð0ÞÞ ¼ ð0:1; 0:2; 0:6; 0:4Þ, and ðx2 ð0Þ; y2 ð0Þ; z2 ð0ÞÞ ¼ ð2; 4; 13Þ are employed. Hence the error system has the initial values e1 ð0Þ ¼ 2:1; e2 ð0Þ ¼ 4:2 and e3 ð0Þ ¼ 13:6. The unknown parameters are chosen as a1 ¼ 10; b1 ¼ 8=3; r1 ¼ 28 and a2 ¼ 10; b2 ¼ 8=3; c2 ¼ 28 in simulations so that the both systems exhibits a chaotic behavior anti-synchronization of the systems (13) and (14) via adaptive control law (16) and (17) with the initial estimated param^1 ð0Þ ¼ 2; a ^2 ð0Þ ¼ 5 and ^c2 ð0Þ ¼ 30 are shown in Figs. 2–4. Figs. 2 and 3 display ^ð0Þ2 ¼ 25; b ^1 ð0Þ ¼ 0:2; ^r1 ð0Þ ¼ 10 ¼ b eters a the state response and the anti-synchronization errors of systems (13) and (14). Fig. 4 shows that the estimates ^1 ðtÞ; ^r1 ðtÞ; a ^2 ðtÞ and ^c2 ðtÞ of the unknown parameters converges to a1 ¼ 10; b1 ¼ 8=3; r 1 ¼ 28 and a2 ^2 ðtÞ; b ^1 ðtÞ; b a ¼ 10; b2 ¼ 8=3; c2 ¼ 28 as t ! 1. Fig. 5 show that the Lorenz system is control to the projective of the hyperchaotic Lorenz system. 4. Adaptive anti-synchronization between hyperchaotic Lü and Chen systems with unknown parameters In this section we will study the anti-synchronization problem between the hyperchaotic Lü system (11) and the Chen system (12). Similarly, we regard the mater as the projection part of hyperchaotic Lü system which consider as the master system and denoted with subscript 1 and it is described by
M.M. Al-sawalha, M.S.M. Noorani / Commun Nonlinear Sci Numer Simulat 15 (2010) 3022–3034
3027
(a)
(b)
(c)
Fig. 2. State trajectories of drive system (13) and response system (14): (a) Signals x1 and x2 ; (b) signals y1 and y2 ; and (c) signals z1 and z2 .
x_ 1 ¼ a1 ðy1 x1 Þ þ w1 ; y_ 1 ¼ x1 z1 þ c1 y1 ; z_ 1 ¼ x1 y1 b1 z1 ;
ð21Þ
and we assume that the Chen system is the response system which is denoted by the subscript 2, and it can be presented in the form of
x_ 2 ¼ a2 ðy2 x2 Þ þ u1 ; y_ 2 ¼ ðc2 a2 Þx x2 z2 þ c2 y2 þ u2 ; z_ 2 ¼ x2 y2 b2 z2 þ u3 ;
ð22Þ
where u1 ; u2 ; u3 are three control functions to be designed and all parameters a1 ; b1 ; c1 ; a2 ; c2 ; b2 are unknown. In order to determine the control functions to realize the anti-synchronization between system equations (21) and (22). We add Eq. (22) to Eq. (21) and get
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(a)
(b)
(c)
Fig. 3. The error signals e1 ; e2 ; e3 of the hyperchaotic Lorenz and Lorenz systems under the controller (16) and the parameters update law (17).
e_ 1 ¼ a2 ðy2 x2 Þ þ a1 ðy1 x1 Þ þ w1 þ u1 ; e_ 2 ¼ ðc2 a2 Þx x2 z2 þ c2 y2 x1 z1 þ c1 y1 þ u2 ; e_ 3 ¼ x2 y2 b2 z2 þ x1 y1 b1 z1 þ u3 ;
ð23Þ
where e1 ¼ x1 þ x2 ; e2 ¼ y1 þ y2 ; e3 ¼ z1 þ z2 , our goal is to find proper control functions ui ði ¼ 1; 2; 3Þ and parameter update rule, such that system equation (22) anti-synchronizes asymptotically with system equation (21). i.e. limt!1 kek ¼ 0 where e ¼ ½e1 ; e2 ; e3 T . For this end we propose the following adaptive control law for system equation (23)
^2 ðy2 x2 Þ a ^1 ðy1 x1 Þ w1 e1 ; u1 ¼ a ^2 Þx þ x2 z2 ^c2 y2 þ x1 z1 ^c1 y1 e2 ; u2 ¼ ð^c2 a ^2 z2 x1 y þ b ^1 z1 e3 ; u3 ¼ x2 y þ b 2
and parameter update rules
1
ð24Þ
M.M. Al-sawalha, M.S.M. Noorani / Commun Nonlinear Sci Numer Simulat 15 (2010) 3022–3034
(a)
(b)
Fig. 4. Changing parameters a1 ; b1 ; r 1 and a2 ; b2 ; c2 of the hyperchaotic Lorenz and Lorenz systems with time t.
Fig. 5. Hyperchaotic Lorenz system (solid line) and the controlled Lorenz system (dotted line).
^_ 1 ¼ ðy1 x1 Þe1 ; a ^_ 1 ¼ z1 e3 ; b ^c_ ¼ y e ; 1
1 2
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^_ 2 ¼ ðy2 x2 Þe1 x2 e2 a ^_ 2 ¼ z2 e3 ; b ^c_ ¼ ðx þ y Þe ; 2
2
2
2
^1 ; ^c1 and a ^2 ; ^c2 are the estimates of a1 ; b1 ; c1 ; a2 ; b2 ; c2 , respectively. ^1 ; b ^2 ; b where a Theorem 2. For any initial conditions the Chen system asymptotically anti-synchronize with the projection of hyperchaotic Lü system asymptotically by using adaptive control law equation (24) and parameter update rule equation (25). Proof. Applying control law equation (24) to Eq. (23) yields the resulting error dynamics as follows:
~2 ðy2 x2 Þ þ a ~1 ðy1 x1 Þ e1 ; e_ 1 ¼ a ~2 Þx þ ~c2 y2 þ ~c1 y1 e2 ; _e2 ¼ ð~c2 a ~2 z2 b ~1 z1 e3 ; e_ 3 ¼ b
(a)
(b)
(c)
Fig. 6. State trajectories of drive system (21) and response system (22): (a) Signals x1 and x2 ; (b) signals y1 and y2 ; and (c) signals z1 and z2 .
ð26Þ
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~1 ¼ b b ^1 ; ~c1 ¼ c1 ^c1 ; a ~2 ¼ b2 b ^2 ; ~c2 ¼ c2 ^c2 . ~1 ¼ a1 a ^1 ; b ~ 2 ¼ a2 a ^2 ; b where a Consider the following Lyapunov function candidate:
V¼
1 T ~2 þ ~c2 þ a ~2 þ ~c2 > 0: ~21 þ b ~22 þ b e eþa 1 1 2 2 2
ð27Þ
The time derivative of V is as follows:
~_ 1 þ ~c1 ~c_ 1 þ a ~_ 2 þ ~c2 ~c_ 2 ~1 b ~2 b ~_ 1 þ b ~_ 2 þ b ~2 a ~1 a V_ ¼ eT e_ þ a ^_ 1 þ ~c1 ^c_ 1 þ a ^_ 2 þ ~c2 ^c_ 2 : ~1 b ~2 b ~1 a ^_ 1 þ b ^_ 2 þ b ~ 2 a ¼ e1 e_ 1 þ e2 e_ 2 þ e3 e_ 3 þ a
ð28Þ
Inserting (25) and (26) into (28) yields the follows:
V_ ¼ e21 e22 e23 6 0:
ð29Þ
(a)
(b)
(c)
Fig. 7. The error signals e1 ; e2 ; e3 between the hyperchaotic Lü system and Chen system under the controller (24.) and the parameters update law (25).
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Since V is positive definite and V_ is negative definite in the neighborhood of zero solution of system equation (23), it follows ^1 ; ^c1 ; a ^2 ; ^c2 2 L1 , from Eq. (26), we have e_ 1 ; e_ 2 ; e_ 3 2 L1 , since V_ ¼ eT e then we obtain ^1 ; b ^2 ; b that e1 ; e2 ; e3 2 L1 , a
Z 0
t
kek2 dt 6
Z
t
eT e dt ¼
Z
0
t
V_ dt ¼ Vð0Þ VðtÞ 6 Vð0Þ
0
Thus, e_ 1 ; e_ 2 ; e_ 3 2 L2 , by Barbalats lemma, we have limt!1 keðtÞk ¼ 0. Therefore, response system equation (22) can anti-synchronize drive system equation (21) asymptotically. h 4.1. Numerical simulations To verify and demonstrate the effectiveness of the proposed method, we discuss the simulation results for the anti-synchronization problem between the projective of the hyperchaotic Lü and Chen systems. In the numerical simulations, the fourth-order Runge–Kutta method is used to solve the systems with time step size 0.001. We assume that the initial condition, ðx1 ð0Þ; y1 ð0Þ; z1 ð0Þ; w1 ð0ÞÞ ¼ ð5; 8; 1; 3Þ, and ðx2 ð0Þ; y2 ð0Þ; z2 ð0ÞÞ ¼ ð10; 0; 37Þ. Hence the error system has the initial values e1 ð0Þ ¼ 5; e2 ð0Þ ¼ 8; e3 ð0Þ ¼ 36. The unknown parameters are chosen as a1 ¼ 36; b1 ¼ 3; c1 ¼ 20 and a2 ¼ 35; b2 ¼ 3; c2 ¼ 28. anti-synchronization of the systems (21) and (22) via adaptive control law (24) and (25) with the initial estimated parameters a1 ð0Þ ¼ 0:2; b1 ð0Þ ¼ 2; c1 ð0Þ ¼ 10 and a2 ð0Þ ¼ 25; b2 ð0Þ ¼ 5; c2 ð0Þ ¼ 30 are shown in Figs. 6–9. Figs. 6 and 7 display the state response and the anti-synchronization errors of systems (21) and (22). Fig. 6 shows that ^1 ðtÞ; ^c1 ðtÞ and a ^2 ðtÞ; ^c2 ðtÞ of the unknown parameters converges to a1 ¼ 36; b1 ¼ 3; c1 ¼ ^2 ðtÞ; b ^1 ðtÞ; b the estimates a 20 and a2 ¼ 35; b2 ¼ 3; c2 ¼ 28 as t ! 1. Fig. 9 show that the Chen system is control to be the projective of the hyperchaotic Lü system. 5. Concluding remark In this article, reduced-order anti-synchronization problem of uncertain chaotic systems has been studied. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third-order chaotic system can be anti-synchronized with the canonical projection of a fourth-order chaotic system even though their param-
(a)
(b)
Fig. 8. Changing parameters a1 ; b1 ; c1 and a2 ; b2 ; c2 of the drive system (21) and the response system (22) with time t.
M.M. Al-sawalha, M.S.M. Noorani / Commun Nonlinear Sci Numer Simulat 15 (2010) 3022–3034
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Fig. 9. Hyperchaotic Lü system (solid line) and the controlled Chen system (dotted line).
eters are unknown. The techniques are successfully applied to two examples: hyperchaotic Lorenz system (fourth-order) and Lorenz system (third-order); Lü hyperchaotic system (fourth-order) and Chen system (third-order). Theoretical analysis and numerical simulations are shown to verify the results. Acknowledgments This work is financially supported by the Malaysian Ministry of Higher Education Grant: UKM-ST-06-FRGS0008-2008. Also the authors are grateful to the anonymous referees for their helpful comments on the earlier draft of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
Chen G, Dong X. From chaos to order. Singapore: World Scientific; 1998. Boccaletti S, Kurths J, Osipov G, Valladares DL, Zhou CS. The synchronization of chaotic systems. Phys Rep 2002;366:1–101. Nijmeijer H. A dynamical control view on synchronization. Physica D 2001;154:219–28. Boukabou A. On the control and synchronization design for autonomous chaotic systems. Nonlinear Dyn Syst Theory 2008;8(2):151–67. Yang XS. A framework for synchronization theory. Chaos Soliton Fract 2000;11:1365–8. Luo ACJ. A theory for synchronization of dynamical systems. Commun Nonlinear Sci Numer Simulat 2009;14:1901–51. Rafikov M, Balthazar J. On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commun Nonlinear Sci Numer Simulat 2008;13:1246–55. Bowong S. Adaptive synchronization between two different chaotic dynamical systems. Commun Nonlinear Sci Numer Simulat 2007;12:976–85. Chen X, Lü J. Adaptive synchronization of different chaotic systems with full unknown parameters. Phys Lett A 2007;364:123–8. Wang Z. Anti-synchronization in two non-identical hyperchaotic systems with known or unknown parameters. Commun Nonlinear Sci Numer Simulat 2009;14:1901–51. Li Ruihong, Xu Wei, Li Shuang. Anti-synchronization on autonomous and non-autonomous chaotic systems via adaptive feedback control. Chaos Soliton Fract 2007. doi:10.1016/j.chaos.2007.09.032. Wu X, Guan Z, Wu Z. Adaptive synchronization between two different hyperchaotic systems. Nonlinear Anal TMA 2008;68(5):1346–51. Sun Y, Cao J. Adaptive synchronization between two different noise-perturbed chaotic systems with fully unknown parameters. Physica A 2007;376:253–65. Yassen MT. Adaptive control and synchronization of a modified Chua’s circuit system. Appl Math Comput 2003;135:113–28. Park J. Adaptive synchronization of hyperchaotic Chen system with uncertain parameters. Chaos Soliton Fract 2005;26:959–64. Xian W, Hongmin Z. Synchronization of two hyperchaotic systems via adaptive control, Chaos Soliton Fract. doi:10.1016/j.chaos.2007.06.100. Shihua C, Lü J. Parameters identification and synchronization of chaotic systems based upon adaptive control. Phys Lett A 2002;299:353–8. Lu J, Wu X, Han X, Lü J. Adaptive feedback synchronization of a unified chaotic system. Phys Lett A 2004;329:327–33. Huang J. Chaos synchronization between two novel different hyperchaotic systems with unknown parameters. Nonlinear Anal TMA 2008;69(11):4174–81. Al-Sawalha MM, Noorani MSM. Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters. Commun Nonlinear Sci Numer Simulat 2010;15:1036–47.
3034
M.M. Al-sawalha, M.S.M. Noorani / Commun Nonlinear Sci Numer Simulat 15 (2010) 3022–3034
[21] Al-Sawalha MM, Noorani MSM. Anti-synchronization of chaotic systems with uncertain parameters via adaptive control. Phys Lett A 2009. doi:10.1016/j.physleta.2009.06.008. [22] Lòpez-Mancilla D, Cruz-Hernández C. Output synchronization of chaotic systems under nonvanishing perturbations. Chaos Soliton Fract 2008;37:1172–86. [23] Lòpez-Gutiérrez RM, Posadas-Castillo C, Lòpez-Mancilla D, Cruz-Hernández C. Communicating via robust synchronization of chaotic lasers and sliding mode control. Chaos Soliton Fract 2009. doi:10.1016/j.chaos.2008.11.019. [24] Yan JJ, Yang YS, Chiang TY, Chen CY. Robust synchronization of unified chaotic systems via sliding mode control. Chaos Soliton Fract 2007;34:947–54. [25] Lü J, Lu J. Controlling uncertain Lü system using linear feedback. Chaos Soliton Fract 2003;17:127–33. [26] Chen H. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos Soliton Fract 2005;23:1245–51. [27] Huang L, Feng R, Wang M. Synchronization of chaotic systems via nonlinear control. Phys Lett A 2004;320:271–5. [28] Zhang Q, Lu J. Chaos synchronization of a new chaotic system via nonlinear control. Chaos Soliton Fract 2008;37:175–9. [29] Park J. Chaos synchronization of a chaotic system via nonlinear control. Chaos Soliton Fract 2005;25:579–84. [30] Al-Sawalha MM, Noorani MSM. On anti-synchronization of chaotic systems via nonlinear control. Chaos Soliton Fract 2008. doi:10.1016/ j.chaos.2008.11.011. [31] Al-Sawalha MM, Noorani MSM. Anti-synchronization of two hyperchaotic systems via Nonlinear control. Commun Nonlinear Sci Numer Simulat 2009;14:3402–4311. [32] Yassen MT. On hyperchaos synchronization of a hyperchaotic Lü system. Nonlinear Anal TMA 2008;68(11):3592–600. [33] Bai E, Lonngen K. Synchronization of two Lorenz systems using active control. Chaos Soliton Fract 1997;8:51–8. [34] Li G, Zhou S. Anti-synchronization in different chaotic systems. Chaos Soliton Fract 2007;32:516–20. [35] El-Dessoky M. Synchronization and anti-synchronization of a hyperchaotic Chen system. Chaos Soliton Fract 2007. doi:10.1016/j.chaos. 2007.06.053. [36] Al-Sawalha MM, Noorani MSM. Chaos anti-synchronization between two novel different hyperchaotic systems. Chin Phys Lett 2008;25(8):2743–6. [37] Al-Sawalha MM, Noorani MSM. Active anti-synchronization between identical and distinctive hyperchaotic systems. Open Syst Inform Dyn 2008;15(4):371–82. [38] Ho MC, Hung YC, Chou CH. Phase and anti-phase synchronization of two chaotic systems by using active control. Phys Lett A 2002;296:43–8. [39] Yassen MT. Chaos synchronization between two different chaotic systems using active control. Chaos Soliton Fract 2005;23:31–140.. [40] Cao J, Wang Z, Sun Y. Synchronization in an array of linearly stochastically coupled networks with time delays. Physica A 2007;385:718–28. [41] Bowong S. Stability analysis for the synchronization of chaotic systems with different order: application to secure communications. Phys Lett A 2004;326:102–13. [42] Ho M, Hung Y, Liua Z, Jiang I. Reduced-order synchronization of chaotic systems with parameters unknown. Phys Lett A 2006;348:251–9. [43] Femat R, Perales G. Synchronization of chaotic systems with different order. Phys Rev E 2002;65:036226–33. [44] Wang C, Ge SS. Synchronization of two uncertain chaotic Systems via adaptive backstepping. Int J Bifurcat Chaos 2001;11(6):1743–51. [45] Khalil HK. Nonlinear systems, 3rd ed. Englewood Cliffs (NJ): Prentice-Hall; 2002. [46] Jia Q. Hyperchaos generated from the Lorenz chaotic system and its control. Phys Lett A 2007;366:217–22. [47] Jia Q. Projective synchronization of a new hyperchaotic Lorenz system. Phys Lett A 2007;370:40–5. [48] Lorenz EN. Deterministic nonperiodic flow. J Atmos Sci 1963;20(2):130–41. [49] Chen A, Lu J, Lü J, Yu S. Generating hyperchaotic Lü attractor via state feedback control. Physica A 2006;364:103–10. [50] Chen G, Ueta T. Yet another chaotic attractor. Int J Bifurcat Chaos 1999;9(7):1465–76. [51] Sprott JC. Chaos and time-series analysis. Oxford: Oxford University Press; 2003.
