- Email: [email protected]
A chaotic system in synchronization and secure communications
Commun Nonlinear Sci Numer Simulat 17 (2012) 1706–1713
Contents lists available at SciVerse ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
A chaotic system in synchronization and secure communications Juan L. Mata-Machuca a, Rafael Martínez-Guerra a,⇑, Ricardo Aguilar-López b, Carlos Aguilar-Ibañez c a
Departamento de Control Automático, CINVESTAV-IPN, Av. IPN 2508, 07360 DF, Mexico Departamento de Biotecnología y Bioingeniería, CINVESTAV-IPN, Av. IPN 2508, 07360 DF, Mexico c Centro de Investigacón en Computación, CIC-IPN, 07738 DF, Mexico b
a r t i c l e
i n f o
Article history: Received 11 March 2011 Accepted 17 August 2011 Available online 7 September 2011 Keywords: Synchronization Secure communications High order sliding-mode adaptative controller Chaotic parameter modulation Parameter estimation
a b s t r a c t In this paper we deal with the synchronization and parameter estimations of an uncertain Rikitake system and its application in secure communications employing chaotic parameter modulation. The strategy consists of proposing a receiver system which tends to follow asymptotically the unknown Rikitake system, refereed as transmitter system. The gains of the receiver system are adjusted continually according to a convenient high order slidingmode adaptative controller (HOSMAC), until the measurable output errors converge to zero. By using HOSMAC, synchronization between transmitter and receiver is achieved and message signals are recovered. The convergence analysis is carried out by using Barbalat’s Lemma. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The synchronization of chaotic systems has been investigated since its introduction in the paper by Pecora and Carrol in 1990 [1]. Synchronization of chaotic systems has attracted much attention due to its potential applications in secure communications, chemical reactions, biological systems and so on [2–5]. Several synchronization schemes have been proposed to tackle the problem [2,6]. In the last years, some methods to achieve synchronization have been proposed from the control theory perspective such as the observer-based approach [7,8], the so-called adaptive synchronization method [9–11] and so on [12–14]. For the synchronization problem, we consider a chaotic system, the master (or transmitter), together with the slave (or slave). The goal is to synchronize the complete response of the slave system to the master system by driving the slave with a signal derived from the master. The problem can now be easily tackled when the parameters of the master system are known. The aforementioned methods and many others are valid for chaotic systems only when the system’s parameters are known. However, to achieve synchronization between two chaotic systems is far from being straightforward, in fact there is no much work about this challenging problem because it consists of both identification of the unknown parameters and the design of a controller to achieve synchronization. Guan et al. applied an observer to identify the unknown parameter of the Lorenz system [15]. Lü et al. studied the same problem for Chen’s chaotic system with the same method [16]. The interest in parameters identification lies on its potential applications in communications, essentially when parameter modulation is used for message transmission. This is an important issue in this work since we will apply chaotic parameter modulation [17] when parameters of Rikitake system1 are used to transmit message signals. ⇑ Corresponding author. E-mail addresses: [email protected] (J.L. Mata-Machuca), [email protected] (R. Martínez-Guerra). This system resembles the reversal of polarity of the Earth’s electromagnetic field, and it is well-known that it has a chaotic behavior for some set of initial conditions and some set of parameter values. 1
1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.08.026
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The general idea for transmitting information via chaotic systems is that, an information signal is embedded in the transmitter system which produces a chaotic signal, the information signal is recovered by the receiver. Synchronization can be classified into mutual synchronization (or bidirectional coupling) [18] and master–slave synchronization (or unidirectional coupling) [1]. The chaos-based secure communications have updated their fourth generation [19]. The continuous synchronization is adopted in the first three generations while the impulsive synchronization is used in the fourth generation. Less than 94 Hz of bandwidth is needed to transmit the synchronization signal for a third-order chaotic transmitter in the fourth generation while 30-kHz bandwidth is needed to transmit the synchronization signals in the other three generations [20]. There are many applications to chaotic communication [21,22] and chaotic network synchronization [23]. The techniques of chaotic communication can be divided into three categories (a) chaos masking [24], the information signal is added directly to the transmitter; (b) chaos modulation [25], it is based on the master–slave synchronization, where the information signal is injected into the transmitter as a nonlinear filter; (c) chaos shift keying [26], the information signal is supposed to be binary, and it is mapped into the transmitter and the receiver. In these three cases, the information signal can be recovered by a receiver if the transmitter and the receiver are synchronized. In this paper is presented an adaptative asymptotic method for the synchronization, the identification of the Rikitake system with several unknown parameters and an application in secure communications via chaotic parameter modulation. By this method, we can achieve chaos synchronization, identify the unknown parameters, and recover message signals simultaneously. Roughly speaking the suggested approach consists of designing a controlled slave system by means of a high order sliding mode adaptive controller (HOSMAC), whose adaptive parameters are adjusted accordingly to a proposed adaptive algorithm. It is done in such a way that the synchronization errors between the outputs of both systems, the uncertain Rikitake and the slave, asymptotically converge to zero. The convergence analysis of the proposed scheme is carried out using the Lyapunov method in conjunction with the Barbalat’s Lemma. The remaining of this work is organized as follows. In Section 2 we introduce the problem statement. In Section 3 we develop our solution to synchronize, identify the constant unknown parameters of the Rikitake system and recover message signals. To assess the effectiveness of our method we present some numerical simulations in Section 4. Finally, we present the conclusions in Section 5. 2. Problem statement In normal chaotic communication, the transmitter and the receiver are chaotic systems. We discuss a case, the transmitter and the receiver are third order chaotic oscillators. 2.1. Transmitter In this paper, all results are based on Rikitake system [27], however, this technique can be applied to any chaotic systems such as Chua’s circuit, Chen’s circuit, and so on, satisfying Definitions 1 and 2 given in sub Section 2.2. Rikitake system is a simple mechanical model used to study the reversals of the magnetic field of the Earth, idealized by the Japanese geophysicist Rikitake [27], consists of two identical single Faraday-disk dynamos of the Bullard type coupled together. The dynamics of this system is governed by the following three dimensional system of nonlinear differential equations:
x_ 1 ¼ lx1 þ z1 y1 ; y_ 1 ¼ ly1 þ ðz1 aÞx1 ; z_ 1 ¼ 1 x1 y1 ;
ð1Þ
where the parameters l and a have some physical meaning when they are positive. For a physical meaning of the states x1, y1 and z1 we recommend to see [27]. However, the states x1 and y1 are directly related to the currents through each disc of the dynamo system, and z1 is related to the angular velocity of one of the discs. This system displays a chaotic behavior for the parameters values in a neighborhood {l = 2, a = 5} and for a large enough set of initial conditions. 2.2. Some algebraic properties In this section we present some algebraic properties that the Rikitake System satisfies. To this end we introduce the following definitions. Definition 1. Consider a smooth nonlinear system, described by a state vector X ¼ fxi gi¼n 2 Rn and by the output vector 1 i¼m m G ¼ fg i g1 2 R , of the form:
X_ ¼ f ðX; PÞ;
G ¼ hðXÞ;
ð2Þ
where h( ) is a smooth vector function and P 2 Rl is a constant parameters vector, with l < n. Let G(j) denote the jth time derivative of the vector G. We say that the vector state X is algebraically observable, if it can be uniquely expressed as
X ¼ UðG; Gð1Þ ; . . . ; GðjÞ Þ for some integer j and for some smooth function U.
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Definition 2. Under same conditions as in Definition 1. If the vector of parameters, P satisfies the following relation
X1 ðG; ::; GðjÞ Þ ¼ X2 ðY; . . . ; Y ðjÞ ÞP;
ð3Þ
where X1( ) and X2( ) are, respectively, n 1 and n n smooth matrices, then P is said to be algebraically identifiable with respect to the output vector G [28]. According to the previous definitions, it is evident that system (1) is algebraically observable with respect to the outputs g1 = x1 and g2 = y1, since the state, z1, can be rewritten, as
z1 ¼
g_ 2 lg 2 þ a; g1
ð4Þ
hence, Rikitake system is algebraically observable with respect to the selected outputs, g1 = x1 and g2 = y1. Moreover, substituting the above expression into the first differential equation of (1), we have
g_ 1 g 1 g_ 2 g 2 ¼ l g 21 þ g 22 þ ag 1 g 2 :
ð5Þ
Therefore, we conclude that system (1) vector of parameters p = (l, a) is algebraically identifiable with respect to the available outputs. That is, the non-available state z1 and the vector of parameters p can be simultaneously recovered, from the knowledge of the outputs g1 = x1 and g2 = y1. From the above definitions, it is possible to solve the synchronization problem of the uncertain Rikitake system, provided that the states x1 and y1 are always available. Moreover, it is also possible to recover the unknown parameters l and a. Thus, we are ready to establish the main control problem of this work. 2.3. Receiver Consider the uncertain Rikitake system (1), referred as the transmitter system, with the available output states x1and y1. And let us propose the following receiver controlled system:
^ x1 þ z2 y1 þ u1 ; x_ 2 ¼ l ^Þx1 þ u2; ^ y1 þ ðz2 a y_ 2 ¼ l _z2 ¼ 1 x1 y1 þ u3 :
ð6Þ
^ ¼ ðl ^Þ such that the slave system (6) follows to the unknown Riki^; a Then, the control objective is to find u = (u1, u2, u3) and p ^ converging to the actual values of (l, a). In other words, we need to find u and p ^ of system (6), such take system (6); with p ^ Þ ! ðw2 ; pÞ, as long as t ? 1.2 that ðw1 ; p 2.4. Transmission of message signals by chaotic parameter modulation In this paper we discuss the case when both parameters of system (1) are used to transmit message signals s1(t) and s2(t). We use modulation rules to modulate s1(t) and s2(t) in parameters of the transmitter in (1). The modulation rules are given by
lðtÞ ¼ l þ s1 ðtÞ; l^ ðtÞ ¼ l^ þ ^s1 ðtÞ; aðtÞ ¼ a þ s2 ðtÞ;
^ðtÞ ¼ a ^ þ ^s2 ðtÞ; a
ð7Þ ð8Þ
where ^s1 ðtÞ and ^s2 ðtÞ are the recovered message signals. Now let us introduce the following errors:
ex ¼ x1 x2 ; ey ¼ y1 y2 ; ez ¼ z1 z2 ; l~ ¼ l l^ ; a~ ¼ a a^; ~s1 ¼ s1 ^s1 ; ~s2 ¼ s2 ^s2 : and according to them, we define the following vectors:
eT ¼ ðex ; ey ; ez Þ;
~ T ¼ ðl ~Þ; ~; a p
~sT ¼ ð~s1 ; ~s2 Þ:
From Eqs. (1)–(6), and taking into account the modulation rules (7), (8) we have:
2
3 2 3 ~ x þ ez y ~s1 x u1 l e_ x 6 7 6 ~ ~Þx ~s1 y ~s2 x u2 7 e_ ¼ 4 e_ y 5 ¼ 4 l y þ ðez a 5; e_ z
ð9Þ
u3
where for simplicity, we stand for y = y1 and x = x1. As we can see, the above system can be considered as a control problem ~ must be proposed such that e asymptotically converges to zero. where the vector inputs u and p 2
Here we denote the vector state related with the master and slave system, as w1 and w2, respectively. That is, wTi ¼ ðxi ; yi ; zi Þ; for i = {1, 2}.
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3. High order sliding-mode adaptative controller (HOSMAC) In this section a HOSMAC is used at the receiver to maintain synchronization by continuously tracking the changes in the modulated parameters. Then, s1(t) and s2(t) can be recovered by using this controller. We solve the control problem (9) by means of the Lyapunov method. To this end, based on a simple quadratic Lyapunov function, we propose the needed HOSMAC and the needed estimator that assure the synchronization of both systems. Before solving the control problem we introduce the following assumptions related with the selected outputs of the transmitter system and the transmitted signals: (A1) The states x = x1 and y = y1 are available, for measurement. (A2) All the states of the transmitter system are bounded, with the generic property that the steady solution x and y, continues oscillating around zero. Comment 1. Assumption A2 is a realistic because in most case all the states of Rikitake system are bounded; for a large set of initial conditions and for a large set of positive parameters l and a. In fact Assumption A2 depends on the set of initial conditions and the values of the parameter vector q. To clarify the meaning of this property, we present a case where Assumption A2 does not hold. Selecting the parameters values as {l > 0; a > 0}, and the initial condition as w1 ð0Þ ¼ ðx1 ð0Þ ¼ 0; x1 ð0Þ ¼ 0; z1 ð0Þ ¼ zÞ; we have that x1(t) = 0, y1(t) = 0 and z1 ðtÞ ¼ t þ z. Evidently, Assumption A2 cannot be fulfilled because the states x and y remain fixed at the origin and, the state z1 is not bounded [29]. In fact, no identification method or scheme can be proposed if the transmitter system has solutions that tend either to infinity or to a constant. Consider a Lyapunov function
V¼
1 T 1 T 1 ~ p ~ þ ~sT ~s: e eþ p 2 2 2
ð10Þ
The time derivative of V along the trajectories of (9) is then given by
~_ þ l ~ey x ey u2 ez u3 ~s1 xex ~s1 yey ~s2 xey þ ~s1~s_ 1 þ ~s2~s_ 2 : ~a ~l ~_ l ~ ex x þ ex ez y ex u1 l ~ ey y þ ey ez x a V_ ¼ a
ð11Þ
~_ ; u, and ~s_ as Now, in order to make V semi-definite negative, we propose p
" ~_ ¼ p
l~_
#
~_ a 2
u1
¼
3
ex x þ ey y xey 2
;
k1 signðex Þem x
ð12Þ 3
7 6 7 6 u ¼ 4 u2 5 ¼ 4 k2 signðey Þem y 5; u3
ð13Þ
ex y þ ey x
" # e x þ ey y ~_ ~s_ ¼ s1 ¼ x ; xey ~s_ 2
ð14Þ
where k1 and k2 are strictly positive constants and m is any positive even integer. Substituting (12)–(14) into (11), we obtain
m V_ ¼ k1 jex jem x þ k2 jey jey :
ð15Þ
~; ~s1 ; ~s2 g are bounded. ~; a This implies that V_ is semi-definite negative and so V converges. Hence, the set of signals fex ; ey ; ez ; l Let us proceed to show that e converges to zero as long as t ? 1, by applying Barbalat’s Lemma [30].3 Integrating both sides of (15), we obtain
Z t 0
m k1 jex ðsÞjem x ðsÞ þ k2 jey ðsÞjey ðsÞ ds 6 Vð0Þ:
ð16Þ
Substituting the control law (13) in (9), we have that the closed-loop system can be read as
~ x þ ez y ~s1 x k1 signðex Þem e_ x ¼ l x ; ~Þx ~s1 y ~s2 x k2 signðey Þem ~ y þ ðez a e_ y ¼ l y ;
ð17Þ
e_ z ¼ ex y ey x; where the parameter dynamics are give by
l~_ ¼ ex x yey ; a~_ ¼ xey :
ð18Þ
3 Lemma (Barbalat): If the differential function f(t) has a finite limit as t ? 1, and if df/dt is uniformly continuous, then df/dt ? 0 as t ? 1. A consequence of this Lemma is that if f 2 L2 and df/dt is bounded then f ? 0 as t ? 1.
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From the equations of (17) and A2, it follows that e_ is bounded which implies that e is uniformly continuous. Using Bar€ is balat’s Lemma, it follows that vector states e ? 0 as t ? 1. Once again, differentiating (17), it is easily to show that e bounded. Thus, e_ is uniformly continuous and also e has a finite limit, as t ? 1, from Barbalat’s Lemma, we conclude that ~ converge as t ? 1 (10). ~ and a e_ ! 0 as t ? 1. Since V converges, as t ? 1, then we have that the two parameter errors l ~_ converge to zero, as t ? 1. Roughly speaking, when t is large enough l ^ are ~_ and a ^ and a Besides, from (18), it follows that l almost constant, and the differential equations of (17) implies that
^Þy ^ Þx 0 ¼ ða a 0 ¼ ðl l
ð19Þ
However once again from Assumption A2, we have that the steady states x and y remains oscillating around zero. Therefore ~ T ! 0, as t ? 1. ^. That is, p ^ and a ¼ a necessarily l ¼ l ~ have the same dynamics, defined by Eqs. (12) and (14), then the signal recovery errors Considering that ~s and p ~s1 ¼ s1 ^s1 and ~s2 ¼ s2 ^s2 asymptotically converge to zero. We summarize the previous discussion in the following proposition: Proposition 1. Under the Assumptions A1 and A2 the synchronization and the parameters estimation problem between systems (1) and (6), can be achieved for any strictly positive constants k1,k2 and for any even integer m. Furthermore, the receiver system (6) can recover the information signals s1 and s2 which are embedded in the chaotic transmitter (1) via the modulation rules (7) and (8), respectively. h
4. Numerical results Computer simulations have been carried out in order to test the effectiveness of the proposed asymptotic control strategy. The program uses the Runge–Kutta integration algorithm, with the integration step set to 0.001. The information signals s1 and s2 are chosen as sinusoidal signals with frequency of 100 Hz as in [4,25], i.e.
s1 ðtÞ ¼ 0:05 sinð200ptÞ; s2 ðtÞ ¼ 0:02 sinð200ptÞ: In the first simulation we illustrate the qualitative property described in the Assumption A2. For this end, we fixed the transmitter system parameter as p = (l = 2, a = 5); while the arbitrary initial conditions were selected as w1(0) = (x1(0) = 1, y1(0) = 1, z1(0) = 1). Fig. 1 shows the behavior of the whole state of the Rikitake system. To show the performance of the proposed control strategy we carried out a second simulation using the same set-up as above, and fixing the receiver system gains as k1 = k2 = 0.8 and m = 4; with the receiver system initialized at ^ ð0Þ ¼ 0; ^s1 ¼ 2 and ^s2 ¼ 5. In Fig. 2 we can see that the synchronization errors asymptotically converge to w2 ð0Þ ¼ 0; p zero. That is, the slave system follows almost perfectly the uncertain master system. The estimated parameters are shown in Fig. 3. As we expect, a better performance can be obtained as long as the time is increased. From this simulations we can concluded that the proposed estimator reconstructs reasonably well the parameters after elapsed 60 s. Figs. 4 and 5 show the communication process, here the waveforms of the modulated parameters are shown in subplots (a), the convergence behavior of the information recovery errors is shown in subplots (b).
Fig. 1. Qualitative behavior of the Rikitake system using modulation rules (7), (8) when is initialized at w1(0) = (1, 1, 1) and the parameters vector is fixed as p = (2, 5). Clearly, the Assumption A2 is satisfied.
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Fig. 2. Synchronization errors, when the transmitter system is initialized at w1(0) = (1, 1, 1) and its parameters vector is fixed as p = (2, 5).
Fig. 3. Parameters estimation, when the master system is initialized at w1(0) = (1, 1, 1); and the actual parameters vector is fixed as p = (2, 5).
Fig. 4. Rikitake system for chaotic communication. Numerical results for message signal s1.
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Fig. 5. Rikitake system for chaotic communication. Numerical results for message signal s2.
5. Conclusions A Lyapunov based approach for the synchronization and parameters identification of the constant unknown parameters of Rikitake system was presented. Indeed, we propose a chaotic communication approach via parameter modulation, where the receiver is controlled by a high order sliding mode adaptive controller, under the Assumption A1 that the two output states x and y are available for measurement. To accomplish this task, we first shown that the system is observable and algebraically identifiable, with respect to the available outputs. Then, we propose a receiver controlled system; where the controllers were proposed such that the vector synchronization error, the vector parameter error and the vector information recovery error, asymptotically converge to zero. The convergence proof was carried out by using the traditional Lyapunov method in combination with the Lemma of Barbalat and the Assumption A2. Finally, numerical simulations were carried out to evaluate the performance of the proposed solution.
References [1] Pecora LM, Carroll TL. Synchronization in chaotic system. Phys Rev Lett 1990;64:821–4. [2] Fradkov A. Cybernetical physics: from control of chaos to quantum control. Berlin Heidelberg: Springer Verlag; 2007. [3] Chen G, Dong X. Identification and control of chaotic systems: an artificial neural network approach. In: Proc IEEE Int Symp Circ Sys, Seattle, WA, 1995, p. 1177–1182. [4] Martı´nez-Guerra R, Yu W. Chaotic synchronization and secure communication via sliding-mode observer. Int J Bifur Chaos 2008;18:235–43. [5] Uyaroglu Y, Pehlivan I. Nonlinear Sprott94 Case A chaotic equation: synchronization and masking communication applications. Comput Elect Eng 2010;36:1093–100. [6] Martı´nez-Guerra R, Cruz J, Gonzalez R, Aguilar R. A new reduced-order observer design for the synchronization of Lorenz systems. Chaos Soliton Fract 2006;28:511–7. [7] Morgül O, Solak E. Observed based synchronization of chaotic systems. Phys Rev E 1996;54:4803–11. [8] Yu W. High-gain Observer for chaotic synchronization and secure communication. Int J Bifur Chaos 2005;18:487–500. [9] Fotsin HB, Daafouz J. Adaptive synchronization of uncertain chaotic Colpitts oscillator based on parameter identification. Phys Lett A 2005;339:304–15. [10] Pourmahmood M, Khanmohammadi S, Alizadeh G. Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller. Commun Nonlinear Sci Numer Simul 2011;16:2853–68. [11] Yu W. Synchronization of three dimensional chaotic systems via a single state feedback. Commun Nonlinear Sci Numer Simul 2011;16:2880–6. [12] Li HY, Hu YA. Robust sliding-mode backstepping design for synchronization control of cross-strict feedback hyperchaotic systems with unmatched uncertainties. Commun Nonlinear Sci Numer Simul 2011;16:3904–13. [13] Pan L, Zhou W, Zhou L, Sun K. Chaos synchronization between two different fractional-order hyperchaotic systems. Commun Nonlinear Sci Numer Simul 2011;16:2628–40. [14] Liu C, Li C, Li C. Quasi-synchronization of delayed chaotic systems with parameters mismatch and stochastic perturbation. Commun Nonlinear Sci Numer Simul 2011;16:4108–19. [15] Guan X, Peng H, Li L, Wang Y. Parameter identification and control of Lorenz chaotic system. Acta Physica Sinica 2001;50:26–9. [16] Lu J, Zhang S. Controlling Chen’s chaotic attractor using backstepping design based on parameters identification. Phys Lett A 2001;286:148–52. [17] Yang T, Chua LO. Secure communication via chaotic parameter modulation. IEEE Trans Circuits Syst I 1996;43:817–9. [18] Ushio T. Synthesis of synchronized chaotic systems based on observers. Int J Bifur Chaos 1999;9:541–6. [19] Tao Y. Chaotic secure communication systems history and new results. Telecommun Rev 1999;9:597–634. [20] Yang T, Chua LO. Impulsive stabilization for control and synchronization of chaotic systems theory and application to secure communication. IEEE Trans Circuits Syst I 1997;44:976–88. [21] Fallahi K, Leung H. A chaos secure communication scheme based on multiplication modulation. Commun Nonlinear Sci Numer Simul 2010;15:368–83. [22] Chang C. Chaotic synchronization and its research progress in secure communication. Modern Electron Technol 2010(3). [23] Chow TW, Feng JC, Ng KT. Chaotic network synchronization with application to communications. Int J Commun Syst 2001;14:217–30. [24] Kocarev L, Halle KS, Eckert K, Chua LO, Parlitz U. Experimental demonstration of secure communication via chaotic synchronization. Int J Bifur Chaos 1992;2:709–13. [25] Boutayeb M, Darouach M, Rafaralahy H. Generalized state observers for chaotic synchronization and secure communication. IEEE Trans Circuits Syst I 2002;49:345–9.
J.L. Mata-Machuca et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 1706–1713 [26] [27] [28] [29] [30]
Parlitz U, Chua LO, Kocarev L, Halle KS, Shang A. Transmission of digital signals by chaotic synchronization. Int J Bifur Chaos 1992;2:973–7. Rikitake T. Oscillations of a system of disk dynamos. Proc Cambridge Philos Soc 1958;54:89. Fliess M, Sira-Ramirez H. An algebraic framework for linear identification. ESAIM 2003;9:151–68. McMillen T. The Shape and Dynamics of the Rikitake Attractor. The Nonlinear Journal 1999;1:1–10. Áström KJ, Wittenmark B. Adaptive control. 2nd ed. Addison-Wesley; 1995.
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Contents lists available at SciVerse ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
A chaotic system in synchronization and secure communications Juan L. Mata-Machuca a, Rafael Martínez-Guerra a,⇑, Ricardo Aguilar-López b, Carlos Aguilar-Ibañez c a
Departamento de Control Automático, CINVESTAV-IPN, Av. IPN 2508, 07360 DF, Mexico Departamento de Biotecnología y Bioingeniería, CINVESTAV-IPN, Av. IPN 2508, 07360 DF, Mexico c Centro de Investigacón en Computación, CIC-IPN, 07738 DF, Mexico b
a r t i c l e
i n f o
Article history: Received 11 March 2011 Accepted 17 August 2011 Available online 7 September 2011 Keywords: Synchronization Secure communications High order sliding-mode adaptative controller Chaotic parameter modulation Parameter estimation
a b s t r a c t In this paper we deal with the synchronization and parameter estimations of an uncertain Rikitake system and its application in secure communications employing chaotic parameter modulation. The strategy consists of proposing a receiver system which tends to follow asymptotically the unknown Rikitake system, refereed as transmitter system. The gains of the receiver system are adjusted continually according to a convenient high order slidingmode adaptative controller (HOSMAC), until the measurable output errors converge to zero. By using HOSMAC, synchronization between transmitter and receiver is achieved and message signals are recovered. The convergence analysis is carried out by using Barbalat’s Lemma. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The synchronization of chaotic systems has been investigated since its introduction in the paper by Pecora and Carrol in 1990 [1]. Synchronization of chaotic systems has attracted much attention due to its potential applications in secure communications, chemical reactions, biological systems and so on [2–5]. Several synchronization schemes have been proposed to tackle the problem [2,6]. In the last years, some methods to achieve synchronization have been proposed from the control theory perspective such as the observer-based approach [7,8], the so-called adaptive synchronization method [9–11] and so on [12–14]. For the synchronization problem, we consider a chaotic system, the master (or transmitter), together with the slave (or slave). The goal is to synchronize the complete response of the slave system to the master system by driving the slave with a signal derived from the master. The problem can now be easily tackled when the parameters of the master system are known. The aforementioned methods and many others are valid for chaotic systems only when the system’s parameters are known. However, to achieve synchronization between two chaotic systems is far from being straightforward, in fact there is no much work about this challenging problem because it consists of both identification of the unknown parameters and the design of a controller to achieve synchronization. Guan et al. applied an observer to identify the unknown parameter of the Lorenz system [15]. Lü et al. studied the same problem for Chen’s chaotic system with the same method [16]. The interest in parameters identification lies on its potential applications in communications, essentially when parameter modulation is used for message transmission. This is an important issue in this work since we will apply chaotic parameter modulation [17] when parameters of Rikitake system1 are used to transmit message signals. ⇑ Corresponding author. E-mail addresses: [email protected] (J.L. Mata-Machuca), [email protected] (R. Martínez-Guerra). This system resembles the reversal of polarity of the Earth’s electromagnetic field, and it is well-known that it has a chaotic behavior for some set of initial conditions and some set of parameter values. 1
1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.08.026
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The general idea for transmitting information via chaotic systems is that, an information signal is embedded in the transmitter system which produces a chaotic signal, the information signal is recovered by the receiver. Synchronization can be classified into mutual synchronization (or bidirectional coupling) [18] and master–slave synchronization (or unidirectional coupling) [1]. The chaos-based secure communications have updated their fourth generation [19]. The continuous synchronization is adopted in the first three generations while the impulsive synchronization is used in the fourth generation. Less than 94 Hz of bandwidth is needed to transmit the synchronization signal for a third-order chaotic transmitter in the fourth generation while 30-kHz bandwidth is needed to transmit the synchronization signals in the other three generations [20]. There are many applications to chaotic communication [21,22] and chaotic network synchronization [23]. The techniques of chaotic communication can be divided into three categories (a) chaos masking [24], the information signal is added directly to the transmitter; (b) chaos modulation [25], it is based on the master–slave synchronization, where the information signal is injected into the transmitter as a nonlinear filter; (c) chaos shift keying [26], the information signal is supposed to be binary, and it is mapped into the transmitter and the receiver. In these three cases, the information signal can be recovered by a receiver if the transmitter and the receiver are synchronized. In this paper is presented an adaptative asymptotic method for the synchronization, the identification of the Rikitake system with several unknown parameters and an application in secure communications via chaotic parameter modulation. By this method, we can achieve chaos synchronization, identify the unknown parameters, and recover message signals simultaneously. Roughly speaking the suggested approach consists of designing a controlled slave system by means of a high order sliding mode adaptive controller (HOSMAC), whose adaptive parameters are adjusted accordingly to a proposed adaptive algorithm. It is done in such a way that the synchronization errors between the outputs of both systems, the uncertain Rikitake and the slave, asymptotically converge to zero. The convergence analysis of the proposed scheme is carried out using the Lyapunov method in conjunction with the Barbalat’s Lemma. The remaining of this work is organized as follows. In Section 2 we introduce the problem statement. In Section 3 we develop our solution to synchronize, identify the constant unknown parameters of the Rikitake system and recover message signals. To assess the effectiveness of our method we present some numerical simulations in Section 4. Finally, we present the conclusions in Section 5. 2. Problem statement In normal chaotic communication, the transmitter and the receiver are chaotic systems. We discuss a case, the transmitter and the receiver are third order chaotic oscillators. 2.1. Transmitter In this paper, all results are based on Rikitake system [27], however, this technique can be applied to any chaotic systems such as Chua’s circuit, Chen’s circuit, and so on, satisfying Definitions 1 and 2 given in sub Section 2.2. Rikitake system is a simple mechanical model used to study the reversals of the magnetic field of the Earth, idealized by the Japanese geophysicist Rikitake [27], consists of two identical single Faraday-disk dynamos of the Bullard type coupled together. The dynamics of this system is governed by the following three dimensional system of nonlinear differential equations:
x_ 1 ¼ lx1 þ z1 y1 ; y_ 1 ¼ ly1 þ ðz1 aÞx1 ; z_ 1 ¼ 1 x1 y1 ;
ð1Þ
where the parameters l and a have some physical meaning when they are positive. For a physical meaning of the states x1, y1 and z1 we recommend to see [27]. However, the states x1 and y1 are directly related to the currents through each disc of the dynamo system, and z1 is related to the angular velocity of one of the discs. This system displays a chaotic behavior for the parameters values in a neighborhood {l = 2, a = 5} and for a large enough set of initial conditions. 2.2. Some algebraic properties In this section we present some algebraic properties that the Rikitake System satisfies. To this end we introduce the following definitions. Definition 1. Consider a smooth nonlinear system, described by a state vector X ¼ fxi gi¼n 2 Rn and by the output vector 1 i¼m m G ¼ fg i g1 2 R , of the form:
X_ ¼ f ðX; PÞ;
G ¼ hðXÞ;
ð2Þ
where h( ) is a smooth vector function and P 2 Rl is a constant parameters vector, with l < n. Let G(j) denote the jth time derivative of the vector G. We say that the vector state X is algebraically observable, if it can be uniquely expressed as
X ¼ UðG; Gð1Þ ; . . . ; GðjÞ Þ for some integer j and for some smooth function U.
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Definition 2. Under same conditions as in Definition 1. If the vector of parameters, P satisfies the following relation
X1 ðG; ::; GðjÞ Þ ¼ X2 ðY; . . . ; Y ðjÞ ÞP;
ð3Þ
where X1( ) and X2( ) are, respectively, n 1 and n n smooth matrices, then P is said to be algebraically identifiable with respect to the output vector G [28]. According to the previous definitions, it is evident that system (1) is algebraically observable with respect to the outputs g1 = x1 and g2 = y1, since the state, z1, can be rewritten, as
z1 ¼
g_ 2 lg 2 þ a; g1
ð4Þ
hence, Rikitake system is algebraically observable with respect to the selected outputs, g1 = x1 and g2 = y1. Moreover, substituting the above expression into the first differential equation of (1), we have
g_ 1 g 1 g_ 2 g 2 ¼ l g 21 þ g 22 þ ag 1 g 2 :
ð5Þ
Therefore, we conclude that system (1) vector of parameters p = (l, a) is algebraically identifiable with respect to the available outputs. That is, the non-available state z1 and the vector of parameters p can be simultaneously recovered, from the knowledge of the outputs g1 = x1 and g2 = y1. From the above definitions, it is possible to solve the synchronization problem of the uncertain Rikitake system, provided that the states x1 and y1 are always available. Moreover, it is also possible to recover the unknown parameters l and a. Thus, we are ready to establish the main control problem of this work. 2.3. Receiver Consider the uncertain Rikitake system (1), referred as the transmitter system, with the available output states x1and y1. And let us propose the following receiver controlled system:
^ x1 þ z2 y1 þ u1 ; x_ 2 ¼ l ^Þx1 þ u2; ^ y1 þ ðz2 a y_ 2 ¼ l _z2 ¼ 1 x1 y1 þ u3 :
ð6Þ
^ ¼ ðl ^Þ such that the slave system (6) follows to the unknown Riki^; a Then, the control objective is to find u = (u1, u2, u3) and p ^ converging to the actual values of (l, a). In other words, we need to find u and p ^ of system (6), such take system (6); with p ^ Þ ! ðw2 ; pÞ, as long as t ? 1.2 that ðw1 ; p 2.4. Transmission of message signals by chaotic parameter modulation In this paper we discuss the case when both parameters of system (1) are used to transmit message signals s1(t) and s2(t). We use modulation rules to modulate s1(t) and s2(t) in parameters of the transmitter in (1). The modulation rules are given by
lðtÞ ¼ l þ s1 ðtÞ; l^ ðtÞ ¼ l^ þ ^s1 ðtÞ; aðtÞ ¼ a þ s2 ðtÞ;
^ðtÞ ¼ a ^ þ ^s2 ðtÞ; a
ð7Þ ð8Þ
where ^s1 ðtÞ and ^s2 ðtÞ are the recovered message signals. Now let us introduce the following errors:
ex ¼ x1 x2 ; ey ¼ y1 y2 ; ez ¼ z1 z2 ; l~ ¼ l l^ ; a~ ¼ a a^; ~s1 ¼ s1 ^s1 ; ~s2 ¼ s2 ^s2 : and according to them, we define the following vectors:
eT ¼ ðex ; ey ; ez Þ;
~ T ¼ ðl ~Þ; ~; a p
~sT ¼ ð~s1 ; ~s2 Þ:
From Eqs. (1)–(6), and taking into account the modulation rules (7), (8) we have:
2
3 2 3 ~ x þ ez y ~s1 x u1 l e_ x 6 7 6 ~ ~Þx ~s1 y ~s2 x u2 7 e_ ¼ 4 e_ y 5 ¼ 4 l y þ ðez a 5; e_ z
ð9Þ
u3
where for simplicity, we stand for y = y1 and x = x1. As we can see, the above system can be considered as a control problem ~ must be proposed such that e asymptotically converges to zero. where the vector inputs u and p 2
Here we denote the vector state related with the master and slave system, as w1 and w2, respectively. That is, wTi ¼ ðxi ; yi ; zi Þ; for i = {1, 2}.
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3. High order sliding-mode adaptative controller (HOSMAC) In this section a HOSMAC is used at the receiver to maintain synchronization by continuously tracking the changes in the modulated parameters. Then, s1(t) and s2(t) can be recovered by using this controller. We solve the control problem (9) by means of the Lyapunov method. To this end, based on a simple quadratic Lyapunov function, we propose the needed HOSMAC and the needed estimator that assure the synchronization of both systems. Before solving the control problem we introduce the following assumptions related with the selected outputs of the transmitter system and the transmitted signals: (A1) The states x = x1 and y = y1 are available, for measurement. (A2) All the states of the transmitter system are bounded, with the generic property that the steady solution x and y, continues oscillating around zero. Comment 1. Assumption A2 is a realistic because in most case all the states of Rikitake system are bounded; for a large set of initial conditions and for a large set of positive parameters l and a. In fact Assumption A2 depends on the set of initial conditions and the values of the parameter vector q. To clarify the meaning of this property, we present a case where Assumption A2 does not hold. Selecting the parameters values as {l > 0; a > 0}, and the initial condition as w1 ð0Þ ¼ ðx1 ð0Þ ¼ 0; x1 ð0Þ ¼ 0; z1 ð0Þ ¼ zÞ; we have that x1(t) = 0, y1(t) = 0 and z1 ðtÞ ¼ t þ z. Evidently, Assumption A2 cannot be fulfilled because the states x and y remain fixed at the origin and, the state z1 is not bounded [29]. In fact, no identification method or scheme can be proposed if the transmitter system has solutions that tend either to infinity or to a constant. Consider a Lyapunov function
V¼
1 T 1 T 1 ~ p ~ þ ~sT ~s: e eþ p 2 2 2
ð10Þ
The time derivative of V along the trajectories of (9) is then given by
~_ þ l ~ey x ey u2 ez u3 ~s1 xex ~s1 yey ~s2 xey þ ~s1~s_ 1 þ ~s2~s_ 2 : ~a ~l ~_ l ~ ex x þ ex ez y ex u1 l ~ ey y þ ey ez x a V_ ¼ a
ð11Þ
~_ ; u, and ~s_ as Now, in order to make V semi-definite negative, we propose p
" ~_ ¼ p
l~_
#
~_ a 2
u1
¼
3
ex x þ ey y xey 2
;
k1 signðex Þem x
ð12Þ 3
7 6 7 6 u ¼ 4 u2 5 ¼ 4 k2 signðey Þem y 5; u3
ð13Þ
ex y þ ey x
" # e x þ ey y ~_ ~s_ ¼ s1 ¼ x ; xey ~s_ 2
ð14Þ
where k1 and k2 are strictly positive constants and m is any positive even integer. Substituting (12)–(14) into (11), we obtain
m V_ ¼ k1 jex jem x þ k2 jey jey :
ð15Þ
~; ~s1 ; ~s2 g are bounded. ~; a This implies that V_ is semi-definite negative and so V converges. Hence, the set of signals fex ; ey ; ez ; l Let us proceed to show that e converges to zero as long as t ? 1, by applying Barbalat’s Lemma [30].3 Integrating both sides of (15), we obtain
Z t 0
m k1 jex ðsÞjem x ðsÞ þ k2 jey ðsÞjey ðsÞ ds 6 Vð0Þ:
ð16Þ
Substituting the control law (13) in (9), we have that the closed-loop system can be read as
~ x þ ez y ~s1 x k1 signðex Þem e_ x ¼ l x ; ~Þx ~s1 y ~s2 x k2 signðey Þem ~ y þ ðez a e_ y ¼ l y ;
ð17Þ
e_ z ¼ ex y ey x; where the parameter dynamics are give by
l~_ ¼ ex x yey ; a~_ ¼ xey :
ð18Þ
3 Lemma (Barbalat): If the differential function f(t) has a finite limit as t ? 1, and if df/dt is uniformly continuous, then df/dt ? 0 as t ? 1. A consequence of this Lemma is that if f 2 L2 and df/dt is bounded then f ? 0 as t ? 1.
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From the equations of (17) and A2, it follows that e_ is bounded which implies that e is uniformly continuous. Using Bar€ is balat’s Lemma, it follows that vector states e ? 0 as t ? 1. Once again, differentiating (17), it is easily to show that e bounded. Thus, e_ is uniformly continuous and also e has a finite limit, as t ? 1, from Barbalat’s Lemma, we conclude that ~ converge as t ? 1 (10). ~ and a e_ ! 0 as t ? 1. Since V converges, as t ? 1, then we have that the two parameter errors l ~_ converge to zero, as t ? 1. Roughly speaking, when t is large enough l ^ are ~_ and a ^ and a Besides, from (18), it follows that l almost constant, and the differential equations of (17) implies that
^Þy ^ Þx 0 ¼ ða a 0 ¼ ðl l
ð19Þ
However once again from Assumption A2, we have that the steady states x and y remains oscillating around zero. Therefore ~ T ! 0, as t ? 1. ^. That is, p ^ and a ¼ a necessarily l ¼ l ~ have the same dynamics, defined by Eqs. (12) and (14), then the signal recovery errors Considering that ~s and p ~s1 ¼ s1 ^s1 and ~s2 ¼ s2 ^s2 asymptotically converge to zero. We summarize the previous discussion in the following proposition: Proposition 1. Under the Assumptions A1 and A2 the synchronization and the parameters estimation problem between systems (1) and (6), can be achieved for any strictly positive constants k1,k2 and for any even integer m. Furthermore, the receiver system (6) can recover the information signals s1 and s2 which are embedded in the chaotic transmitter (1) via the modulation rules (7) and (8), respectively. h
4. Numerical results Computer simulations have been carried out in order to test the effectiveness of the proposed asymptotic control strategy. The program uses the Runge–Kutta integration algorithm, with the integration step set to 0.001. The information signals s1 and s2 are chosen as sinusoidal signals with frequency of 100 Hz as in [4,25], i.e.
s1 ðtÞ ¼ 0:05 sinð200ptÞ; s2 ðtÞ ¼ 0:02 sinð200ptÞ: In the first simulation we illustrate the qualitative property described in the Assumption A2. For this end, we fixed the transmitter system parameter as p = (l = 2, a = 5); while the arbitrary initial conditions were selected as w1(0) = (x1(0) = 1, y1(0) = 1, z1(0) = 1). Fig. 1 shows the behavior of the whole state of the Rikitake system. To show the performance of the proposed control strategy we carried out a second simulation using the same set-up as above, and fixing the receiver system gains as k1 = k2 = 0.8 and m = 4; with the receiver system initialized at ^ ð0Þ ¼ 0; ^s1 ¼ 2 and ^s2 ¼ 5. In Fig. 2 we can see that the synchronization errors asymptotically converge to w2 ð0Þ ¼ 0; p zero. That is, the slave system follows almost perfectly the uncertain master system. The estimated parameters are shown in Fig. 3. As we expect, a better performance can be obtained as long as the time is increased. From this simulations we can concluded that the proposed estimator reconstructs reasonably well the parameters after elapsed 60 s. Figs. 4 and 5 show the communication process, here the waveforms of the modulated parameters are shown in subplots (a), the convergence behavior of the information recovery errors is shown in subplots (b).
Fig. 1. Qualitative behavior of the Rikitake system using modulation rules (7), (8) when is initialized at w1(0) = (1, 1, 1) and the parameters vector is fixed as p = (2, 5). Clearly, the Assumption A2 is satisfied.
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Fig. 2. Synchronization errors, when the transmitter system is initialized at w1(0) = (1, 1, 1) and its parameters vector is fixed as p = (2, 5).
Fig. 3. Parameters estimation, when the master system is initialized at w1(0) = (1, 1, 1); and the actual parameters vector is fixed as p = (2, 5).
Fig. 4. Rikitake system for chaotic communication. Numerical results for message signal s1.
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Fig. 5. Rikitake system for chaotic communication. Numerical results for message signal s2.
5. Conclusions A Lyapunov based approach for the synchronization and parameters identification of the constant unknown parameters of Rikitake system was presented. Indeed, we propose a chaotic communication approach via parameter modulation, where the receiver is controlled by a high order sliding mode adaptive controller, under the Assumption A1 that the two output states x and y are available for measurement. To accomplish this task, we first shown that the system is observable and algebraically identifiable, with respect to the available outputs. Then, we propose a receiver controlled system; where the controllers were proposed such that the vector synchronization error, the vector parameter error and the vector information recovery error, asymptotically converge to zero. The convergence proof was carried out by using the traditional Lyapunov method in combination with the Lemma of Barbalat and the Assumption A2. Finally, numerical simulations were carried out to evaluate the performance of the proposed solution.
References [1] Pecora LM, Carroll TL. Synchronization in chaotic system. Phys Rev Lett 1990;64:821–4. [2] Fradkov A. Cybernetical physics: from control of chaos to quantum control. Berlin Heidelberg: Springer Verlag; 2007. [3] Chen G, Dong X. Identification and control of chaotic systems: an artificial neural network approach. In: Proc IEEE Int Symp Circ Sys, Seattle, WA, 1995, p. 1177–1182. [4] Martı´nez-Guerra R, Yu W. Chaotic synchronization and secure communication via sliding-mode observer. Int J Bifur Chaos 2008;18:235–43. [5] Uyaroglu Y, Pehlivan I. Nonlinear Sprott94 Case A chaotic equation: synchronization and masking communication applications. Comput Elect Eng 2010;36:1093–100. [6] Martı´nez-Guerra R, Cruz J, Gonzalez R, Aguilar R. A new reduced-order observer design for the synchronization of Lorenz systems. Chaos Soliton Fract 2006;28:511–7. [7] Morgül O, Solak E. Observed based synchronization of chaotic systems. Phys Rev E 1996;54:4803–11. [8] Yu W. High-gain Observer for chaotic synchronization and secure communication. Int J Bifur Chaos 2005;18:487–500. [9] Fotsin HB, Daafouz J. Adaptive synchronization of uncertain chaotic Colpitts oscillator based on parameter identification. Phys Lett A 2005;339:304–15. [10] Pourmahmood M, Khanmohammadi S, Alizadeh G. Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller. Commun Nonlinear Sci Numer Simul 2011;16:2853–68. [11] Yu W. Synchronization of three dimensional chaotic systems via a single state feedback. Commun Nonlinear Sci Numer Simul 2011;16:2880–6. [12] Li HY, Hu YA. Robust sliding-mode backstepping design for synchronization control of cross-strict feedback hyperchaotic systems with unmatched uncertainties. Commun Nonlinear Sci Numer Simul 2011;16:3904–13. [13] Pan L, Zhou W, Zhou L, Sun K. Chaos synchronization between two different fractional-order hyperchaotic systems. Commun Nonlinear Sci Numer Simul 2011;16:2628–40. [14] Liu C, Li C, Li C. Quasi-synchronization of delayed chaotic systems with parameters mismatch and stochastic perturbation. Commun Nonlinear Sci Numer Simul 2011;16:4108–19. [15] Guan X, Peng H, Li L, Wang Y. Parameter identification and control of Lorenz chaotic system. Acta Physica Sinica 2001;50:26–9. [16] Lu J, Zhang S. Controlling Chen’s chaotic attractor using backstepping design based on parameters identification. Phys Lett A 2001;286:148–52. [17] Yang T, Chua LO. Secure communication via chaotic parameter modulation. IEEE Trans Circuits Syst I 1996;43:817–9. [18] Ushio T. Synthesis of synchronized chaotic systems based on observers. Int J Bifur Chaos 1999;9:541–6. [19] Tao Y. Chaotic secure communication systems history and new results. Telecommun Rev 1999;9:597–634. [20] Yang T, Chua LO. Impulsive stabilization for control and synchronization of chaotic systems theory and application to secure communication. IEEE Trans Circuits Syst I 1997;44:976–88. [21] Fallahi K, Leung H. A chaos secure communication scheme based on multiplication modulation. Commun Nonlinear Sci Numer Simul 2010;15:368–83. [22] Chang C. Chaotic synchronization and its research progress in secure communication. Modern Electron Technol 2010(3). [23] Chow TW, Feng JC, Ng KT. Chaotic network synchronization with application to communications. Int J Commun Syst 2001;14:217–30. [24] Kocarev L, Halle KS, Eckert K, Chua LO, Parlitz U. Experimental demonstration of secure communication via chaotic synchronization. Int J Bifur Chaos 1992;2:709–13. [25] Boutayeb M, Darouach M, Rafaralahy H. Generalized state observers for chaotic synchronization and secure communication. IEEE Trans Circuits Syst I 2002;49:345–9.
J.L. Mata-Machuca et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 1706–1713 [26] [27] [28] [29] [30]
Parlitz U, Chua LO, Kocarev L, Halle KS, Shang A. Transmission of digital signals by chaotic synchronization. Int J Bifur Chaos 1992;2:973–7. Rikitake T. Oscillations of a system of disk dynamos. Proc Cambridge Philos Soc 1958;54:89. Fliess M, Sira-Ramirez H. An algebraic framework for linear identification. ESAIM 2003;9:151–68. McMillen T. The Shape and Dynamics of the Rikitake Attractor. The Nonlinear Journal 1999;1:1–10. Áström KJ, Wittenmark B. Adaptive control. 2nd ed. Addison-Wesley; 1995.
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