A novel variable structure control scheme for chaotic synchronization

Transcription

1 Chaos, Solitons and Fractals 18 (2003) A novel variable structure control scheme for chaotic synchronization Chun-Chieh Wang b,c, Juhng-Perng Su a, * a Department of Electrical Engineering, National Yunlin University of Science and Technology, No. 123, Section 3, University Road, Touliu, Yunlin 640, Taiwan, ROC b Graduate School of Engineering Science and Technology (Doctoral Program), National Yunlin University of Science and Technology, No. 123, Section 3, University Road, Touliu, Yunlin 640, Taiwan, ROC c Department of Electrical Engineering C.K.I.T., No. 123, Section 3, University Road, Touliu, Yunlin 640, Taiwan, ROC Accepted 3 January 2003 Abstract This paper deals with synchronization problems of chaotic systems by applying a novel variable structure control (VSC) scheme. The synchronization of ChuaÕs circuit, the R ossler system and the Lorenz system, respectively, were used as illustrative examples. Simulation results indicated that the proposed new VSC scheme can significantly improve the synchronousness performance. These promising results justify the usefulness of the proposed controller in the application of secure communication. Ó 2003 Elsevier Science Ltd. All rights reserved. 1. Introduction Recently, researchers from different fields, such as mathematicians, physicists, as well as control engineers have touched on the issue of synchronization [1,2]. Chaotic systems, in particular, have been regarded as an important contributor to the development of secure communication systems [3 5]. The system which received the most attention among chaotic communication systems is ChuaÕs oscillator [2]. This system belongs to the general class of LurÕe systems [6]. In 1997, Nijmeijer and Mareels [7] believed the problem of chaos synchronization could be solved by nonlinear control theory when viewing the problem as a special case of observer design. Hebertt and Cesar [8] presented a nonlinear observer design for a class of systems in generalized hamiltonian form in One year after, a linear output error feedback approach was proposed to globally synchronize a class of chaotic systems by using the last state variable as the driving signal [9]. In publications regarding synchronization of chaotic systems, the controller design is often based upon the assumption that the chaotic model is well known. However, in real chaotic circuits, parameter variations do exist. These uncertainties may cause chaotic perturbations to originally regular behavior, or induce additional chaos in originally chaotic but known behavior, generating unknown chaotic motion. Lots of efforts have been dedicated to synchronization of chaotic systems which contain uncertainties [10 12]. However, they often attach one control input to each state equation, i.e. there usually exists more than one control input. Such method would significantly increase the implementation budget need. In this paper, we will present a new variable structure control (VSC) scheme to deal with synchronization of chaotic systems with uncertainties. We will define a new sliding variable as a complement to the conventional sliding variable to * Corresponding author. Tel.: x4246; fax: address: sujp@pine.yuntech.edu.tw (J.-P. Su) /03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi: /s (02)

2 276 C.-C. Wang, J.-P. Su / Chaos, Solitons and Fractals 18 (2003) form a useful error transformation by which an nth-order problem can be transformed into an equivalent first-order problem such that an efficient continuous sliding control can be devised to achieve a better performance of the system. We will prove by simulations that this control law can significantly improve the error transient responses, as well as ameliorate the steady-state tracking precision. The effectiveness of the newly developed control scheme will be demonstrated through the synchronization of ChuaÕs circuit, the R ossler system and the Lorenz system, respectively. 2. Chaotic systems Many chaotic systems, which were usually used as benchmark examples to investigate synchronizations, can be described as the following forms: ðnþ ðn 1Þ ðn 1Þ x ¼ fm t; x; _x;...; x þ g t; x; _x;...; x f þ d m ðtþ; ð1þ ðpþ ðn 1Þ f ¼ fa t; x; _x;...; x ; f; f; _...; ðp 1Þ f þ u þ d a ðtþ; ð2þ where x and f are the state variables and u is the input; d m and d a are disturbances; f m ; g : R R n! R and f a : R R n R p! R are smooth functions. Eq. (1) is called the driven subsystem and Eq. (2) the driving subsystem. The object is to design a control law such that the output of the slave system x s will approximately track a given signal of the master system x, i.e., lim t!1 eðtþ ¼0, where eðtþ ¼x s ðtþ xðtþ. For some auxiliary variable /, define the auxiliary error z as zðtþ ¼fðtÞ /ðtþ. Then, the error equations can be obtained as follows, ðnþ ðn 1Þ ðn 1Þ ðn 1Þ ðn 1Þ e ¼ fm t; x; x s ; _x; _x s ;...; x ; xs þ g t; x; x s ; _x; _x s ;...; x ; xs f þ d m ðtþ; ð3þ ðpþ z ¼ fa t; x; x s ; _x; _x s ;...; x ðn 1Þ ; xs ðn 1Þ ; z; _z;...; ðp 1Þ z þ u þ d a ðtþ / ðpþ : ð4þ We assume that f m ¼ f ^ m þ Df m, g ¼ ^g þ Dg, f a ¼ f ^ a þ Df a, where f ^ m, f ^ a are nominal parts and Df m, Df a are uncertain parts, where jdf m j 6 M, jdf a j 6 N, M, N are known functions. Set ^g 6¼ 0, jdg=^gj 6 k < 1, jd m j 6 D m and jd a j 6 D a 8t, where D m, D a are given constants or time functions Chua s circuit Take ChuaÕs circuit [9] shown in Fig. 1 as an example. This circuit is described by the following set of differential equations: dv c1 C 1 ¼ 1 dt R ðv c 2 V c1 Þ gðv c1 Þ; dv c2 C 2 ¼ 1 dt R ðv c 1 V c2 ÞþI L ; L di L dt ¼ ðv c 2 þ R 0 I L Þ; where C 1, C 2, L, R and R 0 are circuit parameters, I L is the current through the inductor L; V c1 and V c2 are the voltages across C 1 and C 2, respectively, and the piecewise linear function gðv c1 Þ describes the V I characteristics of ChuaÕs diode g, as follows: ð5þ gðv c1 Þ¼G B V c1 þ 1 2 ðg A G B ÞðjV c1 þ Ej jv c1 EjÞ with G A < 0 and G B < 0 being some appropriately chosen constants. Set z 1 :¼ I L, z 2 :¼ V c2, z 3 :¼ V c1, and E ¼ 1V. Then Eq. (5) can be reformulated in the following form: _z 1 ¼ b 0 z 1 þ b 1 z 2 ; _z 2 ¼ b 2 ðz 3 þ z 2 Þ b 3 z 1 ; _z 3 ¼ b 4 ðz 2 þ z 3 Þ b 5 z 3 b 6 ðjz 3 þ 1j jz 3 1jÞ; ð6þ ð7þ where b 0 ¼ R 0 =L, b 1 ¼ 1=L, b 2 ¼ 1=RC 2, b 3 ¼ 1=C 2, b 4 ¼ 1=RC 1, b 5 ¼ G B =C 1, b 6 ¼ðG A G B Þ=2C 1.

3 C.-C. Wang, J.-P. Su / Chaos, Solitons and Fractals 18 (2003) Fig. 1. ChuaÕs circuit and the V I characteristics of ChuaÕs diode gðv C1 Þ. Set q 1 ¼ b 0 b 2, q 2 ¼ b 0 b 2 b 1 b 3, q 3 ¼ b 1 b 2, q 4 ¼ b 4 =b 1, q 5 ¼ b 0 b 4 =b 1, q 6 ¼ ðb 4 þ b 5 Þ, q 7 ¼ b 6. Then Eq. (7) can be reformulated in the following form: z 1 ¼ q 1 _z 1 þ q 2 z 1 q 3 z 3 ; _z 3 ¼ q 4 _z 1 þ q 5 z 1 þ q 6 z 3 þ q 7 ðjz 3 þ 1j jz 3 1jÞ: ð8þ Eq. (8) is called the master system in the synchronization problem. The slave system can also be described by z 1s ¼ q 1 _z 1s þ q 2 z 1s q 3 z 3s ; _z 3s ¼ q 4 _z 1s þ q 5 z 1s þ q 6 z 3s þ q 7 ðjz 3s þ 1j jz 3s 1jÞ þ u; ð9þ where u is the controller. Set e 1 ¼ z 1s z 1, e 2 ¼ z 2s z 2, e 3 ¼ z 3s z 3. Let d 1 and d 2 be external disturbances, bounded for all time by two known constants, d 1 and d 2, respectively, i.e., jd 1 j 6 d 1 and jd 2 j 6 d. Therefore, we will restrict ourselves to the design of the sliding control of the error equations e 1 ¼ q 1 _e 1 þ q 2 e 1 q 3 e 3 þ d 1 ; ð10þ _e 3 ¼ q 4 _e 1 þ q 5 e 1 þ q 6 e 3 þ q 7 ðjz 3s þ 1j jz 3s 1j jz 3 þ 1jþjz 3 1jÞ þ u þ d 2 : ð11þ In terms of Eqs. (3) and (4), we have f m ¼ q 1 _e 1 þ q 2 e 1, g ¼ q 3, n ¼ e 3, and f a ¼ q 4 _e 1 þ q 5 e 1 þ q 6 e 3 þ q 7 ðjz 3s þ 1j jz 3s 1j jz 3 þ 1jþjz 3 1jÞ The R ossler system Consider the R ossler system _x 1 ¼ x 2 x 3 ; _x 2 ¼ x 1 þ ax 2 ; _x 3 ¼ b þ x 1 x 3 cx 3 ; ð12þ where a; b; c are constants.

4 278 C.-C. Wang, J.-P. Su / Chaos, Solitons and Fractals 18 (2003) Then Eq. (12) can be reformulated in the following form: x 2 ¼ x 2 þ a_x 2 x 3 ; _x 3 ¼ ax 2 x 3 cx 3 þ _x 2 x 3 þ b þ u; ð13þ where u is the control input to the system. Let d 1 and d 2 be external disturbances, bounded for all time by two known constants, d 1 and d 2, respectively, i.e., jd 1 j 6 d 1 and jd 2 j 6 d 2. Then, the master slave error equations can be described by the following forms: e 2 ¼ e 2 þ a_e 2 e 3 þ d 1 ; ð14þ _e 3 ¼ aðx 2s x 3s x 2 x 3 Þ ce 3 þð_x 2s x 3s _x 2 x 3 Þþb þ u þ d 2 : That is, f m ¼ e 2 þ a_e 2, g ¼ 1, n ¼ e 3, and f a ¼ aðx 2s x 3s x 2 x 3 Þ ce 3 þð_x 2s x 3s _x 2 x 3 Þþb The Lorenz system ð15þ Similarly, consider the Lorenz system which can be described by _x 1 ¼ r f x 1 þ r f x 2 ; _x 2 ¼ rx 1 x 2 x 1 x 3 ; _x 3 ¼ x 1 x 2 bx 3 : Then Eq. (16) can be reformulated in the following form: x 1 ¼ ½ ðr f þ 1Þ_x 1 þ r f ðr 1Þx 1 Šþð r f x 1 Þx 3 ; _x 3 ¼ 1 x 1 _x 1 þ x 2 1 r bx 3 þ u; f ð16þ ð17þ where u is the control input. Let d 1 and d 2 be external disturbances, bounded for all time by two known constants, d 1 and d 2, respectively, i.e., jd 1 j 6 d 1 and jd 2 j 6 d 2. Then, the master slave error equations can be described by the following forms: e 1 ¼ ½ ðr f þ 1Þ_e 1 þ r f ðr 1Þe 1 r f e 1 x 3 Š r f x 1s e 3 þ d 1 ; ð18þ _e 3 ¼ 1 ðx 1s _x 1s x 1 _x 1 Þþe 1 ðx 1s þ x 1 Þ be 3 þ u þ d 2 : ð19þ r f We know that f m ¼ ½ ðr f þ 1Þ_e 1 þ r f ðr 1Þe 1 r f e 1 x 3 Š, g ¼ r f x 1s, n ¼ e 3, and f a ¼½ð1=r f Þðx 1s _x 1s x 1 _x 1 Þþ e 1 ðx 1s þ x 1 Þ be 3 Š. Another benchmark examples including the Duffing Holmes oscillator and the Bonhoeffer van der Pol oscillator (BVP) can also be transformed into the standard form (3) and (4) [13]. 3. Injecting an a parameter into a composite sliding control scheme For any k m, a m, k a, a a > 0, define the following transformations: s m ðtþ ¼ d n 1 d þ k m þ a m n dt dt e ðtþ; ð20þ s e ðtþ ¼ d n 1 d þ k m a m n dt dt e ðtþ; ð21þ s a ðtþ ¼ d p 1 d þ k a þ a a n dt dt z ðtþ; ð22þ s z ðtþ ¼ d p 1 d þ k a a a n dt dt z ðtþ; ð23þ where n e ðtþ ¼ R t eðsþds, n z ðtþ ¼ R t zðsþds. The following important relationships can be easily established:

5 C.-C. Wang, J.-P. Su / Chaos, Solitons and Fractals 18 (2003) and _s e þ a m ðs e þ s m Þ¼_s m ; _s z þ a a ðs z þ s a Þ¼_s a ; _s m þ a m s m ¼ f ^ m þ ^w m þ Df m þ gf þ d m ; _s a þ a a s a ¼ f ^ a þ ^w a þ Df a þ u þ d a ; ð24þ ð25þ ð26þ ð27þ where ^w m ¼ Xn 1 k¼1 ^w a ¼ Xp 1 k¼1 n 1 k p 1 k k k m k k a ðn kþ e X n 1 n 1 þ 2am k k¼0 ðp kþ X p 1 p 1 z þ 2aa k k¼0 k k a k k m " ðn 1 kþ e þ a 2 m " ðp 1 kþ z þ a 2 a X p 2 X n 2 k¼0 k¼0 n 1 k p 1 k k k ðn 2 kþ m e k k ðp 2 kþ a z þ k n 1 m n e # ; ð28þ þ k p 1 a n z # / ðpþ : ð29þ To proceed with the design of sliding control, we give l m > 0 and 0 < k 1 < 1. The state-feedback control f ¼ /ðxþ was designed as follows: /ðxþ ¼G m ½^u m þ v m Š; ð30þ where G m ¼ s^g ; 1 þ s^g 2 s > 0; ð31þ ^u m ¼ ^ f m ^w m ; v m ¼ K m 1, tanh K mðs m þ s e Þ l m ð32þ : ð33þ Substituting f ¼ /ðxþ (30) into the right-hand side of (26) yields _s m þ a m s m ¼ð1 gg m Þðf ^ m þ ^w m ÞþgG m v m þ Df m þ d m : ð34þ Lemma 1. For any l m > 0 and any s 2 R, s tanhðs=l m Þ P 0, and the following inequality holds: s 0 < jsj stanh 6 cl m ; ð35þ l m where c is a constant that satisfies c ¼ e ðcþ1þ ; i.e., c ¼ 0:2785. Notation j1 gg m j 6 ð1 þ sk^g 2 Þ=ð1 þ s^g 2 Þ,,. Let, ¼ minfð1 þ sk^g 2 Þ=ð1 þ s^g 2 Þ; eg, where e < 1. Then 0 <, < 1. Notation 3. j1 gg m j 6, ) ð1 þ,þ 6 gg m 6 ð1,þ. By Lemma 1, Notations 2and 3, we have s e _s e þ s m _s m 6 a m ðs e þ s m Þ 2 þ cl m js e þ s m jðk m M D m,jf ^ m þ ^w m jþ: For g m > 0, if K m verifies K m P g m þ M þ D m þ,jf ^ m þ ^w m j; ð36þ then s e _s e þ s m _s m 6 a m ðs e þ s m Þ 2 g m js e þ s m jþcl m. For any 0 < h < 1, set ^g m :¼ð1 hþg m and U m ¼ cl m hg m : ð37þ

6 280 C.-C. Wang, J.-P. Su / Chaos, Solitons and Fractals 18 (2003) Then the following condition: s e _s e þ s m _s m 6 a m ðs e þ s m Þ 2 ~g m js e þ s m j is satisfied for js e þ s m j P U m, where ~g m > 0, ^g m P ~g m þðjgju a Þ=ð2k p 1 a Þ. The overall control law is designed as follows: u ¼ ^u a þ v a ð38þ ð39þ and ^u a ¼ ^ f a ^w a ; v a ¼ K a tanh K aðs a þ s z Þ l a ð40þ ; ð41þ where 0 < k 2 < 1. For g a > 0, 0 < h < 1. Set ^g a :¼ð1 hþg a,if K a P g a þ N þ D a ; ð42þ then it can be easily established that the control law (39) fulfills the following condition: s z _s z þ s a _s a 6 a a ðs z þ s a Þ 2 ^g a js z þ s a j; ð43þ whenever js z þ s a j P U a ¼ðcl a Þ=ðhg a Þ. This ensures that any trajectory zðtþ will reach the boundary layer, js z þ s a j 6 U a, in finite time when it starts of the boundary layer at time t ¼ 0, and after a short period of time, we have jzðtþj 6 1 ðu 2 a=k p 1 a Þ when in the boundary layer. Consequently, the inequality (38) assures that there exists T < 1 such that whenever t > T, ðiþ e ðtþ < 1 2 ð2k mþ i e; 0 6 i 6 n 1; where e ¼ U m =k n 1 m. 4. Cases study In this section, simulation results are presented for communication systems built using ChuaÕs circuit, the R ossler system and the Lorenz system Chua s circuit Consider Eqs. (8) and (9), we take the initial conditions ðz 1s ð0þ; z 2s ð0þ; z 3s ð0þþ¼ð 1; 0:3; 1:5Þ for simulations with the slave ChuaÕs circuit and ðz 1 ð0þ; z 2 ð0þ; z 3 ð0þþ ¼ ð1:5; 0:1; 1Þ with the master ChuaÕs circuit, respectively. Set ðb 0 ; b 1 ; b 2 ; b 3 ; b 4 ; b 5 ; b 6 Þ¼ð 0:0385; 15; 1; 1; 10; 6:8; 2:95Þ, then ðq 1 ; q 2 ; q 3 ; q 4 ; q 5 ; q 6 ; q 7 Þ¼ð 1:0385; 15:0385; 15; 2 ; 0:0257; 3:2; 2:95Þ. 3 According to the notations given in Section 3, we take f ^ m ¼ 1:0385_e 1 15:0385e 1, g ¼ 15, f^ a ¼ 2 _e 3 1 0:0257e 1 3:2e 3 þ 2:95ðjz 3s þ 1j jz 3s 1j jz 3 þ 1jþjz 3 1jÞ, ^g ¼ 14:99, D m ¼ D a ¼ 0, and M ¼ N ¼ 0. Take s ¼ 100 and k ¼ 0:01. By setting a a ¼ 10, a m ¼ 15, k a ¼ 1, k m ¼ 1, g a ¼ 1, g m ¼ 8:5, l a ¼ 1, l m ¼ 10, e ¼ 0:9, and h ¼ 0:5, the function /ðxþ can be computed from (30) (33), where K m ¼ g m þ M þ D m þ,jf ^ m þ ^w m j. The overall control law can, therefore, be obtained by (39) with K a ¼ g a þ N þ D a. We simulated the synchronization of ChuaÕs circuit, using SIMULINK with Runge Kutta algorithm at a fixed-step integration time of 0.01 s. This good performance can be further justified from the observation of the synchronization errors, shown in Figs. 2and 3, where the magnitude of the synchronization errors seem to fall into the bound: jeðtþj very rapidly (within 7.3 s). The results are obviously better than [9], shown in Fig The R ossler system The R ossler system can be described by (13). We take the initial conditions ðx 1 ð0þ; x 2 ð0þ; x 3 ð0þþ¼ð1; 1; 1Þ for simulations with the master system and ðx 1s ð0þ; x 2s ð0þ; x 3s ð0þþ¼ð0; 0; 0Þ with the slave system, respectively.

7 C.-C. Wang, J.-P. Su / Chaos, Solitons and Fractals 18 (2003) Fig. 2. Synchronization errors in two ChuaÕs circuits. Fig. 3. Synchronization errors in another scale Case A. Without uncertainties and disturbances In this case, we assume that all parameters of the master system are precisely known, the slave system can be constructed with those known parameters, and the dynamics of the master system are disturbance-free. The parameters of the R ossler system are a ¼ 0:4, b ¼ 2, and c ¼ 4 [8]. We take f^ m ¼ e 2 þ a_e 2, g ¼ 1, f^ a ¼ aðx 2s x 3s x 2 x 3 Þ ce 3 þð_x 2s x 3s _x 2 x 3 Þþb, ^g ¼ 0:99, and M ¼ N ¼ 0. Let s ¼ 100 and k ¼ 0:01. Take a a ¼ 6, a m ¼ 6, k a ¼ 1, k m ¼ 1, g a ¼ 1, g m ¼ 1:6, l a ¼ 1, l m ¼ 10, and h ¼ 0:5. Take K m ¼ g m þ M þ D m þ,jf ^ m þ ^w m j and K a ¼ g a þ N þ D a. The signals d m ¼ d a ¼ 0 are taken as external disturbances in the simulations. Take D m ¼ 0 ¼ D a. The overall control law can, therefore, be obtained.

8 282 C.-C. Wang, J.-P. Su / Chaos, Solitons and Fractals 18 (2003) Fig. 4. Synchronization errors in two ChuaÕs circuits [9]. Fig. 5. The trajectories of x 1 and x 1s. We simulated the R ossler system, using SIMULINK with Runge Kutta algorithm at a fixed-step integration time of 0.01 s. The state trajectories of the R ossler system along with those of the synchronizing system are depicted in Figs This good performance can be further justified from the observation of the synchronization errors, shown in Fig. 8, where the magnitude of the synchronization errors seem to fall into the bound: jeðtþj very rapidly (within 8.6 s) Case B. With uncertainties and disturbances We assume the parameter a ¼ 0:4 ift < 10, and 0.5 if t P 10. The disturbance d m ðtþ is assumed to be a white Gaussian noise, as shown in Fig. 9. The information signal is sðtþ ¼0:1 sinð20tþ, as shown in Fig. 10 [12]. We take f^ m ¼ e 2 þ a_e 2, g ¼ 1, f^ a ¼ aðx 2s x 3s x 2 x 3 Þ ce 3 þð_x 2s x 3s _x 2 x 3 Þþb, ^g ¼ 0:99, N ¼ 0 and M ¼ 0:1jx 2 j. Taking s ¼ 100. Choose k ¼ 0:01. By setting a a ¼ a m ¼ 20, k a ¼ k m ¼ g a ¼ 1, g m ¼ 1:6, l a ¼ 1, l m ¼ 10, and h ¼ 0:5. Take K m ¼ g m þ M þ D m þ,jf ^ m þ ^w m j, K a ¼ g a þ N þ D a, D m ¼ 0:0001, and D a ¼ 0. Therefore, the overall controller can be obtained. Fig. 11 shows the time response of the error between sðtþ and the recovery signal s r ðtþ. The maximal absolute error is less than when t > 5:85 s.

9 C.-C. Wang, J.-P. Su / Chaos, Solitons and Fractals 18 (2003) Fig. 6. The trajectories of x 2 and x 2s. Fig. 7. The trajectories of x 3 and x 3s The Lorenz system Consider Eqs. (17). We take the initial conditions ðx 1s ð0þ; x 2s ð0þ; x 3s ð0þþ ¼ ð0:2; 0:05; 0:4Þ for simulations with the slave Lorenz system and ðx 1 ð0þ; x 2 ð0þ; x 3 ð0þþ¼ð0; 1; 0Þ with the master Lorenz system, respectively. Set r f ¼ 10, b ¼ 8, 3 and r ¼ 28. According to the notations given in Section 3, we take f ^ m ¼ 11_e 1 þ 55 e e 1 x 3, g ¼ 10x 1s, f^ a ¼ 0:1ðx 1s _x 1s x 1 _x 1 Þþe 1 ðx 1s þ x 1 Þ 8 e 3 3, ^g ¼ 9:9, D m ¼ D a ¼ 0, and M ¼ N ¼ 0. Take s ¼ 100 and k ¼ 0:015. By setting a a ¼ 10, a m ¼ 10, k a ¼ 100, k m ¼ 10, g a ¼ 1, g m ¼ð1:5472l a jgjþ=ðk a g a Þ, l a ¼ 1, l m ¼ 1, e ¼ 0:9, and h ¼ 0:5, the function /ðxþ can be computed from (30) (33). The overall controller can, therefore, be obtained by (39).

10 284 C.-C. Wang, J.-P. Su / Chaos, Solitons and Fractals 18 (2003) Fig. 8. Synchronization errors in two R ossler systems. Fig. 9. The disturbance d m ðtþ. We simulated the synchronization of the Lorenz system, using SIMULINK with Euler algorithm at a fixed-step integration time of 0.01 s. This good performance can be further justified from the observation of the synchronization errors, shown in Figs. 12and 13, where the magnitude of the synchronization errors seem to fall into the bound: jeðtþj very rapidly (within 2.5 s).

11 C.-C. Wang, J.-P. Su / Chaos, Solitons and Fractals 18 (2003) Fig. 10. The information signal sðtþ ¼0:1 sinð20tþ. Fig. 11. The error between sðtþ and s r ðtþ. 5. Conclusion A new robust control scheme using sliding mode approach was proposed in this paper to synchronize a class of chaotic systems. A significant feature of this control scheme is the incorporation of a new sliding variable as a complement to the conventional sliding variable in order to form a more meaningful measure of errors such that an efficient control law can be derived. This suggested approach has been successfully applied to a secure communication scheme. Simulation results indicated the reaching dynamics during the reaching phase is significantly improved. To illustrate the effectiveness of the design, the synchronization of ChuaÕs circuit, the R ossler system and the Lorenz system, respectively, were used as simulated examples. Both theoretical and simulation results reveal the validity of the proposed VSC technique for synchronizing chaotic dynamics, even for uncertain chaotic systems with disturbances.

12 286 C.-C. Wang, J.-P. Su / Chaos, Solitons and Fractals 18 (2003) Fig. 12. Synchronization errors in two Lorenz system. Fig. 13. Synchronization errors in another scale. References [1] Pecora L, Carroll T. Driving systems with chaotic signals. Phys Rev A 1991;44: [2] Wu CW, Chua LO. A unified framework for synchronization and control of dynamical systems. Int J Bifurcation and Chaos 1994;4(4): [3] Kocarev L, Halle KS, Eckert K, Chua LO, Parlitz U. Experimental demonstration of secure communications via chaotic synchronization. Int J Bifurcation and Chaos 1992;2(3): [4] Parlitz U, Chua LO, Kocarev L, Halle KS, Shang A. Transmission of digital signals by chaotic synchronization. Int J Bifurcation and Chaos 1992;2(4):973 7.

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