Eur. Phys. J. Special Topics 225, 2655–2667 (2016) © EDP Sciences, Springer-Verlag 2016 DOI: 10.1140/epjst/e2015-50318-7
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Regular Article
A discrete-time chaos synchronization system for electronic locking devices G. Minero-Ramales1,a , D. L´ opez-Mancilla1,b , Carlos E. Casta˜ neda1,c , G. Huerta opez1,f , R. Jaimes Re´ategui1,g , Cuellar1,d , R. Chiu Z.1,e , J. Hugo Garc´ıa L´ E. Villafa˜ na Rauda1,h , and C. Posadas-Castillo2,i 1
2
Centro Universitario de los Lagos, Unversidad de Guadalajara, Av. Enrique D´ıaz de Le´ on 1144, Col. Paseos de la Monta˜ na, Lagos de Moreno, Jalisco, M´exico Universidad Aut´ onoma de Nuevo Le´ on, Facultad de Ingenier´ıa Mec´ anica El´ectrica, Nuevo Le´ on, M´exico Received 24 November 2015 / Received in final form 20 March 2016 Published online 22 November 2016 Abstract. This paper presents a novel electronic locking key based on discrete-time chaos synchronization. Two Chen chaos generators are synchronized using the Model-Matching Approach, from non-linear control theory, in order to perform the encryption/decryption of the signal to be transmitted. A model/transmitter system is designed, generating a key of chaotic pulses in discrete-time. A plant/receiver system uses the above mentioned key to unlock the mechanism. Two alternative schemes to transmit the private chaotic key are proposed. The first one utilizes two transmission channels. One channel is used to encrypt the chaotic key and the other is used to achieve output synchronization. The second alternative uses only one transmission channel for obtaining synchronization and encryption of the chaotic key. In both cases, the private chaotic key is encrypted again with chaos to solve secure communication-related problems. The results obtained via simulations contribute to enhance the electronic locking devices.
1 Introduction The wide variety of applications for chaos communication has promoted the fast development of this field. From time immemorial humanity has had the need to a b c d e f g h i
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transmit and receive information safely. Since 1990, chaos communication systems have evolved significantly. The existence of discrete-time chaotic systems has led the generation of cipher keys that guide the application of chaos theory to secure communication [1]. Publication [2] has propelled the synchronization of chaotic systems because of their potential application to safe communications, as well as the works where two-channel communication schemes are used, such as the ones presented in [3–8]. Since then, diverse methods to perform synchronization of chaotic systems have been proposed [9–19], among others. Interest on this application is due to the intrinsic properties of chaotic signals: they possess a broad spectrum and they are pseudo-random signals, that is, they look like noise but their nature is deterministic, so they are reproducible. This fact makes it possible to hide a pseudo-random signal and to recover it without losses whenever the adequate receiver is designed. The work in [12] uses hiper-chaos to make a more complex signal. In the present article, a system for electronic locking devices based on two Chen chaos generators is proposed. It is synchronized using the model-matching approach taken from nonlinear control theory in [20] and [21]. A master/transmitter system is designed which generates a key based on a train of chaotic pulses, that is, a discrete-time chaotic signal. This key could be different every time the train of chaotic pulses is generated, according to the initial conditions. Moreover, a slave/receiver system uses the above mentioned key to unlock the electronic mechanism. To do so, the system uses the results from the synchronization of the chaotic systems (synchronization by model-matching approach). Two different schemes to broadcast private information are presented. In the first scheme two transmission channels are utilized; the first one is used to encrypt the information by means of a nonlinear mathematical function and the second one to perform the synchronization. The second scheme uses only one transmission channel. The paper is organized as follows: Sect. 2 introduces some theoretical background about the synchronization by model-matching approach. In Sect. 3 the solution to the Model-Matching problem is described. Next, the proposed method using the discretetime chaotic model and the model-matching for discrete Chen is described in Sect. 4. Examples for electronical communication locking systems based on chaos are presented in Sect. 5. Test results are provided in Sect. 6. Finally, conclusions are given at the end of the manuscript in Sect. 7.
2 Syncrhonization by model-matching approach Let us consider a plant system as the following nonlinear discrete-time system: x(k + 1) = f (x(k), u(k)) , P : y(k) = h (x(k)) ,
(1)
where x(k) ∈ Rn is the state system, u(k) ∈ R represents the input and y(k) ∈ R is the output, being f : X × U → X and h : X → Y smooth analytical functions. Let us suppose that we have a nonlinear discrete-time model system defined by: xM (k + 1) = fM (xM (k), uM (k)), M: (2) yM (k) = hM (xM (k)), where the states space xM (k) ∈ RnM , the input signal uM (k) ∈ R and the output signal yM (k) ∈ R, being fM : XM × UM → XM and hM : XM → YM smooth analytical functions. We assume that for certain numerical parameter values, u(k) = 0 and uM (k) = 0, systems (1) and (2) exhibit a chaotic behavior. We also assume that plant P (1) has an equilibrium point xo around (xo , uo ) ∈ X × U such that
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f (xo , uo ) = xo , with {u(k) = uo : k ≥ 0} as a constant input. For this input signal, there is an output signal {y(k) = h(xo ) = y o : k ≥ 0}. In the same way, system M (2) has an equilibrium point xoM around (xoM , uoM ) ∈ XM × UM . For this reason, it is possible to design a control law u(k) such that the controlled output y(k) of system P (1), achieves asymptotically the output yM (k) of system M (2), regardless the state space initial conditions of systems P and M . On the other hand, in order to solve the Model-Matching Problem (MMP), we define the output error as yE (k) = yM (k) − y(k) and we select u(k) such that yE (k) is independent of uM (k) ∀k 0, and yE (k) achieves asymptotically to zero. The first step to solve the MMP is to decouple the output from the input of an auxiliary system. This auxiliary system is defined as: xE (k + 1) = fE (xE (k), uE (k), ωE (k)) , (3) E: yE (k) = hE (xE (k)) , T
with xE (k) = [x(k)xM (k)] ∈ Rn+nM as the auxiliary state; the inputs uE (k) = u(k) and ωE (k) = uM (k) with: f (x(k), u(k) , (4) fE (xE (k), uE (k), ωE (k)) = fM (xM (k), uM (k)) and with the output:
yE (k) = h(x(k)) − hM (xM (k)).
(5)
It is important to mention that ωE (k) in system (3) is considered as a known disturbance because it represents the input u(k) of system M (2). Therefore, the above can be considered as a disturbance decoupling problem with measurable disturbances. In [1] and [10] it is shown that the nonlinear continuous-time and discrete-time MMP, respectively, have solutions decoupling from the output of system (3) by an undesired measurable disturbance (input). Now, consider systems P (1) and M (2) defined about their corresponding equilibrium points (xo , uo ) and (xoM , uoM ), with relative degrees d and dM , respectively. It is possible to find a control law u(k) such that the output y(k) of system P (1), synchronizes with the output yM (k) of system M (2) when d dM [9]. For the particular case when P and M are with the same structure such that d = dM , it is possible to find complete synchronization. Therefore, the control law can be represented as follows: u(k) = γ E (xE (k), uE (k)) , = γ E (x(k), xM (k), uM (k)) ,
(6)
= γ E (x(k), φM (xM (k), uM (k))) , where φM (xM (k), uM (k)) is the coupling signal which is sent by the model system M in order to generate the control law u(k) for solving the MMP and to obtain complete synchronization. Then, if it is possible to find a control law (6), we can assume that MMP has solution [9]. In Fig. 1, we show a block synchronization scheme using model-matching approach. The control technique explained above, has the following advantages as a synchronization method: – It is a methodical procedure, which means that it takes place in an orderly manner, following the model-matching approach control method. – It is possible to synchronize identical and nonidentical systems.
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Fig. 1. Block synchronization scheme using model-matching approach.
– It uses unidirectional coupling, thus allowing the coupling signal to require less transmitter channels. This is due to the fact that the model does not need any prior knowledge of the plant. – It represents a robust control technique, as it is explained in [22].
3 Solution to the model-matching problem Consider the auxiliary system E (3) in new coordinates: (ς(xE (k)), xM (k)) = φ(xE (k)) = φ(x(k), xM (k)) ,
(7)
where ς(xE (k)) = ς(xE (k)), ..., ςd+1 (xE (k))T , and with:
i−1 i−1 ςi (xE (k)) = hEi ◦ fEi (xE (k)) = ξi (x(k)) − hM i ◦ fM 0,
∀i = 1, 2, ..., d + 1, where h·i ◦ f·i , according to the definition of relative degree, are independent functions and can be selected as new coordinates (see [9] and references therein). Then, in a closed loop, using the control law u(k) = γ E (xE (k), uE (k)) (6), system E (3) takes the form: ςi (k + 1) = ςi+1 (k), i = 1, ..., d, ςd+1 (k + 1) = v(k) = −α0 ς1 (k) − . . . − αd ςd+1 (k), xM (k + 1) = fM (xM (k), uM (k)),
(8)
yE (k) = ς1 (k), where v ∈ R represents an external control. Note that the output of system P (1) is different from the output of system M (2) (now transformed into new coordinates (8)) by the signal yE (k) according to the next linear equation: yE (k + 1 + d) + αd yE (k + d) + . . . + α1 yE (k + 1) + α0 yE (k) = 0.
(9)
In Eqs. (8) and (9), α0 , . . . , αd are constants. Therefore, it is possible to find two subsystems in closed loop form (8) as follows: having a subsystem that is described by xM (k + 1) = fM (xM (k), uM (k)) which represents the dynamics of system M (2), and a subsystem that is described by ς(k + 1) = Aς(k), with: ⎤ ⎡ 0 1 0 ··· 0 0 1 ··· 0 ⎥ ⎢ 0 ⎢ . . .. . . . ⎥. .. A=⎢ . .. ⎥ . ⎥ ⎢ .. ⎣ 0 0 0 ··· 1 ⎦ −α0 −α1 −α2 · · · −αd
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The system M (2) is stable and a control law v(k) is chosen such that the eigenvalues of matrix A have real negative part. Then, the closed loop system is exponentially stable and the synchronization condition is achieved [1].
4 Synchronization of two discrete-time Chen maps 4.1 Discrete-time chaotic Chen model and Plant In this paper we use a discrete-time chaotic Chen described by the following set of equations [23]: x1 (k + 1) = 1 − a(x21 (k) + x22 (k)), (10) x2 (k + 1) = −2abx1 (k)x2 (k). System (10) has chaotic dynamics with parameters a = 1.95 and b = 1, as it is shown in [23]. The equations of system (10) can be rewritten as in (1) by adding the control input u(k) and defining the output y(k) = x2 (k) such that the resulting relative degree is d = 1. Therefore, let us consider the following dynamical system with the form (1) and with relative degree d = 1: ⎧ x (k + 1) = 1 − a(x21 (k) + x22 (k)) + u(k), ⎪ ⎨ 1 P : x2 (k + 1) = −2abx1 (k)x2 (k), ⎪ ⎩ y(k) = x2 (k).
(11)
For this reason, the following dynamical system with the form (2) and with the same structure as (1) is proposed: ⎧ x (k + 1) = 1 − (ax2M 1 (k) + x2M 2 (k)) + uM (k), ⎪ ⎨ M1 M : xM 2 (k + 1) = −2abxM 1 (k)xM 2 (k), ⎪ ⎩ yM (k) = xM 2 (k).
(12)
Note that system (12) has relative degree dM = d = 1. Furthermore, the MMP has solution and it is possible to achieve synchronization between the output of plant system (11) and the output of model system (12). Thus, the synchronization error is given by: (13) yE (k) = yM (k) − y(k) = xM 2 (k) − x2 (k).
4.2 Model-matching for discrete Chen Using the model-matching methodology for discrete-time systems [1], it is possible to find a control law u(k) such that the outputs of systems P (11) and M (12) synchronize. The control law u(k) is selected as: β(k) 1 − a x2M 1 (k) + x2M 2 (k) + uM (k) + v(k) u(k) = β(k) 2 2 −1 + a x1 (k) + x2 (k) ,
(14)
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Fig. 2. Communication scheme with two transmission channels.
where
β(k) = 4a2 b2 xM 1 (k)xM 2 (k), v(k) = v1 (k) − v2 (k), v1 (k) = (2α1 abx1 (k) − α0 ) x2 (k),
(15)
v2 (k) = (2α1 abxM 1 (k) − α0 ) xM 2 (k). Considering Eq. (7), the coupling signal is given by: φ (xM (k), uM (k)) = β(k)[1 − a x2M 1 (k) + x2M 2 (k) + uM (k)] − v2 (k)
(16)
such that, the linearized part of the auxiliary system with form (3) in new coordinates becomes: ς1 (k + 1) = ς2 (k), ς2 (k + 1) = −0.2ς1 (k) − 0.4ς2 (k),
(17)
yE (k) = ς1 (k). With parameters α0 = 0.2 and α1 = 0.4 system (17) is exponentially stable. Notice that, the output yE (k) tends to zero, which means that the output of systems P (11) and M (12) synchronize.
5 Electronical communication locking system based on chaos In this section, two locking communication systems based on a discrete-time Chen system are proposed, where the above theory is applied. 5.1 Two-transmission channels The two-transmission channels communication system is based on output synchronization between two chaotic oscillators by means of the model-matching approach. Figure 2 presents the communication scheme with two transmission channels. Let us consider the coupling signal of the form (16) and the block diagram presented in Fig. 2 in which φM (xM (k), uM (k)) represents the input to the controller block, and u (x(k), xM (k), uM (k)) (or simply u(k)) is the input to slave (receiver) system P (11). Authors propose two transmission channels: the first channel is for sending the coupling signal φM (xM (k), uM (k)) in order to achieve output synchronization without any chaotic key; the second channel is used for transmitting the chaotic key as follows: the model (transmitter) system M (12) with two outputs that correspond to discrete-time Chen states xM 1 (k) and xM 2 (k), respectively. In this
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Fig. 3. Communication scheme with one transmission channel.
case, xM 1 (k) is sent to a normalizer in order to generate the chaotic code key which is summed with state xM 2 (k) to create the encrypted code key. In order to recover the code key, the state x2 (k) of system P (11) is subtracted from the encrypted code key.
5.2 One-transmission channel The communication scheme with one transmission channel is presented in Fig. 3. The procedure consists on using only one channel for achieving output synchronization as well as the encryption of the information. As can be seen in Fig. 3, the encrypted code is inserted in the model (transmitter) system M (12) through the input signal uM (k) multiplied by a gain β. The transmitter output signal is a non-linear function u1 (xM (k), uM (k)), while signal u(k) is considered as the output of plant (receiver) system P (11) which, once the synchronization between the outputs of model (12) and plant (11) is achieved, the chaotic code key is recovered (u1 (k) → u(k)=chaotic key).
6 Results T
The simulation results are made with: initial conditions x(0) = [0.5 − 0.5] for system T P (11), and xM (0) = [0.1 − 0.1] for system M (12); α0 = 0.2 and α1 = 0.04. It is important to remark that the key could be different every time the sequence of chaotic pulses is generated, according to the initial conditions; thus, any initial conditions can be proposed.
6.1 Syncrhonization results. By using the synchronization scheme presented in Fig. 1, it is possible to make numerical simulations in order to show the proposed synchronization methodology. Figure 4 shows the chaotic dynamics for system P (4 a) for state x1 (k) and 4 b) for state x2 (k)). Synchronization results between master and slave systems are presented, where the states of system P reach the states of system M : in Fig. 5(a) it is shown how x1 (k) reaches xM 1 (k) and Fig. 5(b) displays how y(k) = x2 (k) reaches yM (k) = xM 2 (k). Synchronization error is presented as follows: e1 (k) = xM 1 (k) − x1 (k) is portrait in Fig. 6(a), as well as e2 (k) = xM 2 (k) − x2 (k) is displayed in Fig. 6(b). It can be seen that after approximately five steps for k, it is possible to achieve complete synchronization. Regarding the obtained results, it is shown that with the
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x1(k)
1 0 −1 0
10
20
10
20
b)
30
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50
30
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x2(k)
1 0 −1 0
k
Fig. 4. Chaotic dynamic for system P : (a) state x1 (k) and (b) state x2 (k).
x1(k), xM1(k)
a) 1
x (k)
0
xM1(k)
1
−1 0
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x2(k), xM2(k)
b) 1
x (k)
0
x
2
M2
−1 0
10
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k
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(k)
50
Fig. 5. Synchronization results: (a) x1 (k) reaches xM 1 (k), (b) x2 (k) reaches xM 2 (k). a)
0.5
1
e (k)
1
0 −0.5 0
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b)
0
2
e (k)
0.5
−0.5 −1 0
10
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k
Fig. 6. Synchronization error: (a) e1 (k) = xM 1 (k) − x1 (k), (b) e2 (k) = xM 2 (k) − x2 (k).
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0.1
Code key
0.08 0.06 0.04 0.02 0 0
10
20
k
30
40
50
Fig. 7. Code key for two-transmission channels.
Encrypted code key
1
0.5
0
−0.5
−1 0
5
10
15
20
25 k
30
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Fig. 8. Encrypted code key for two-transmission channels.
discrete-time Chen system, it is possible to present complete synchronization. The proposed synchronization method has shown flexibility in order to be applied in electronic locking using chaos synchronization due to the fact that it is possible to avoid the violation of the electronic locking system. 6.2 Two-transmission channels results In this case the input signal uM (k) = 0 was applied to the master (transmitter) system M (12) of Fig. 2 and using the same parameters as described above. The code key is presented in Fig. 7 as well as the encrypted code key is displayed in Fig. 8. In Fig. 9, the coupling signal φM (xM (k), uM (k)) is shown, the recovered code key is presented in Fig. 10 and the control signal u(k) is portrayed in Fig. 11. In order to show the locking in communications for the present paper, the Fourier spectrum of the generated binary code (see Fig. 12) is compared to its corresponding chaotic key (see Fig. 13). As it can be seen on Fig. 12, it presents two amplitudes (0 and 1), thus, it is easy to recover the binary code sent. Meanwhile, Fig. 13 presents different amplitudes. Therefore, the chaotic key cannot be recovered using conventional methods.
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0.5
M
φ (x (k),u (k))
1
M
M
0
−0.5
−1 0
10
20
k
30
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50
Fig. 9. Coupling signal φM (xM (k), uM (k)).
Recovered code key
0.5
0
−0.5
−1 0
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k
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Fig. 10. Recovered code key for two-transmission channels.
0.6 0.5
u(k)
0.4 0.3 0.2 0.1 0 −0.1 −0.2 0
10
20
k
30
40
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Fig. 11. Control signal u(k) for two-transmission channels.
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0.1
Normalized units
0.08 0.06 0.04 0.02 0 0
10
20
k
30
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Fig. 12. Fourier spectrum of the binary code for two-transmission channels.
Normalized units
1
0.5
0
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−1 0
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k
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Fig. 13. Fourier spectrum of the chaotic code key for two-transmission channels.
Encrypted chaotic key
1
0.5
0
−0.5
−1 0
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Fig. 14. Encrypted chaotic key for one-transmission channel.
6.3 One-transmission channel results For simplicity, in this subsection we show the next results: Fig. 14 displays the encrypted chaotic key. In Fig. 15 the recovered code key is shown, and Fig. 16 presents the coupling signal uM (k). The locking communication results are similar to those in the above subsection.
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Recovered code key
1
0.5
0
−0.5
−1 0
10
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k
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Fig. 15. Recovered code key for one-transmission channel.
0.6 0.5 0.4
u(k)
0.3 0.2 0.1 0 −0.1 −0.2 0
10
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k
30
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Fig. 16. Coupling signal u(k) for one-transmission channel.
7 Conclusions In this paper a novel electronic locking key based on discrete-time chaos synchronization is presented. The communication schemes presented in this work have advantages and disadvantages. For example, the need to utilize two transmission channels implies a greater cost. Nevertheless this two channel system possesses the capability to obtain a fast and efficient output synchronization and a high speed to encrypt the chaotic code depending on the used gains in u(k). The one single transmission channel system is cheaper because of the use of only one channel to perform both synchronization and encryption of the code. In both schemes the chaotic code is robust in comparison to the traditional scheme. In addition, the chaotic code obtained throughout these systems contains multiple frequencies that make it immune to the conventional filtering techniques. The authors thanks to Consejo Nacional de Ciencia y Tecnolog´ıa (M´exico) under research grant 166654 and to retention program no. 120489.
References 1. A.Y. Aguilar-Bustos, C. Cruz-Hernandez, Proc. of the 2003 American Control Conference (2003), vol. 3, p. 2335 2. L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64, 821 (1990)
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2667
3. J.H. Garc´ıa, R. Jaimes, A.N. Pisarchik, A. Murgu´ıa-Hernandez, C. Medina-Guti´errez, R. Valdivia-Hernandez, E. Villafana-Rauda, J. Phys.: Conf. Ser. 23, 276 (2005) 4. A.N. Pisarchik, F.R. Ruiz-Oliveras, Patent with publication info: MX2009001860 (2009) 5. A.N. Pisarchik, F.R. Ruiz-Oliveras, IEEE J. Quantum Electron. 46, 279 (2010) 6. R.V. Hernandez, R. Jaimes, J.H. Garc´ıa, A.N. Pisarchik, Patent with publication info: MX2007005976 (2014) 7. M. Jim´enez-Rodr´ıguez, R. Jaimes, A.N. Pisarchik, Discontinuity Nonlinearity Complexity 1, 57 (2012) 8. A.N. Pisarchik, R. Jaimes, J.R. Sevilla-Escoboza, F.R. Ruiz-Oliveras, J.H. Garc´ıaL´ opez., J. Franklin Inst. 349, 3194 (2012) 9. C. Cruz-Hernandez, H. Nijmeijer, Synchronization through extended Kalman filtering, in Neuw Trends in Nonlinear Observer Design, edited by H. Nijmeijer and T. I. Fossen, Lecture Notes in Control and Information Sciences 244 (Springer-Verlag, 1999), pp. 469–490 10. D. L´ opez Mancilla, C. Cruz-Hernandez, Nonlinear Dyn. Syst. Teory 5, 141 (2005) 11. D. L´ opez Mancilla, C. Cruz-Hernandez, Output synchronization of chaotic oscillators and private communication, 16th IFAC World Congress Prague, Czech Republic (2005) 12. H. Sira-Ramirez, C. Cruz-Hernandez, Int. J. Bifurc. Chaos 11, 1381 (2001) 13. M.P. Kennedy, M.J. Ogorzalek (Eds.) Chaos Synchronization and Control: Theory and Applications, vol. 44 of IEEE Trans. Circuit Syst. I (1997) 14. G. Chen, M.J. Ogorzalek (Eds.) Control and Synchronization of Chaos, vol. 10 of Int. J. Bifurc. Chaos (World Scientific, 2000) 15. M.J. Ogorzalek, IEEE Trans. Circuits Syst. I 40 (1993) 16. C.W. Wu, O. Chua, Int. J. Bifurc. Chaos 3, 1619 (1993) 17. Y. Wang, K. Sun, S. He, H. Wang, Eur. Phys. J. Special Topics 223, 1591 (2014) 18. A. Ray, A. Roychowdhury, Eur. Phys. J. Special Topics 223, 1509 (2014) 19. S. Banerjee, J. Kurths, Eur. Phys. J. Special Topics 223, 1441 (2014) 20. M.D. Di Benedetto, J.W. Grizzle, IEEE Trans. Automatic Control 39, 1539 (1994) 21. A. Isidori, Nonlinear control systems, 3rd edn. (Springer-Verlag, Berlin, 1995) 22. D. L´ opez Mancilla, C. Cruz-Hernandez, Chaos, Solitons Fractals 34, 1172 (2008) 23. L.Q. Chen, Phys. Lett. A 281, 327 (2001)
