A new adaptive observer-based synchronization scheme for private communication

Physics Letters A 355 (2006) 193–201 www.elsevier.com/locate/pla

A new adaptive observer-based synchronization scheme for private communication Samuel Bowong a,∗ , F.M. Moukam Kakmeni b , Hilaire Fotsin c a Laboratory of Applied Mathematics, Department of Mathematics and Computer Science, Faculty of Science, University of Douala,

P.O. Box 24157 Douala, Cameroon b Laboratory of Research in Advanced Material and Nonlinear Science, Department of Physics, Faculty of Sciences, University of Buea,

P.O. Box 63 Buea, Cameroon c Laboratoire d’électronique, Département de Physique, Faculté des sciences, Université de Dschang, B.P. 67 Dschang, Cameroon

Received 1 June 2005; received in revised form 12 February 2006; accepted 14 February 2006 Available online 28 February 2006 Communicated by C.R. Doering

Abstract The problem of secure communication via parameter modulation in a class of uncertain chaotic systems is considered. For a given uncertain master chaotic system, a robust adaptive observer-based response can be constructed to synchronize the drive system. The information signal is used to modulate one parameter of a given chaotic system. The resulting chaotic signal is later demodulated and the information signal is recovered using an adaptive demodulator. The convergence of the demodulator is established. Theoretical analysis and numerical simulation on Chua’s circuit show the effectiveness and efficiency of the proposed scheme. © 2006 Elsevier B.V. All rights reserved. Keywords: Chaotic systems; Adaptive synchronization; Robust adaptive observers; Secure communication

1. Introduction In the last decades, chaos synchronization has attracted a lot of interests to study [1–5]. One important reason is the successful application to private communication [6,7]. In their seminal paper, Pecora and Carroll [1] addressed the synchronization of chaotic systems using a drive-response conception. The idea is to use the output of the driving system to control the response system so that they oscillate in a synchronized manner. System decomposition [1], iteration method [8], observer method [9–13], feedback method [14,15], etc. have been proposed to realize different synchronization phenomena. One important reason for chaos synchronization is the successful application of chaos to secure communication. So far, many ideas and methods have been proposed to tackle the problem of chaotic secure communication including chaotic masking [6,16], chaotic shift keying [17] and chaos modulation [6,13– 16]. In chaotic masking, the message to be transmitted is added to a much stronger chaotic signal in order to hide the information, the overall signal is then transmitted to the receiver. Under certain conditions the message may be recovered at the receiver, see [5–14]. In chaos shift keying, the transmitted signal is obtained by switching between N chaotic generators according to the information level of an N -ary message (usually binary messages are used with two chaotic generators). In chaotic modulation, the message modifies the state or the parameters of the chaotic generator through an invertible procedure, thus the generated chaotic signal inherently contains the information on the transmitted message. As in chaotic masking scheme, the message may be recovered in the receiver under certain conditions [18–20]. Even though these approaches have been successful demonstrated in simulations, theoretically, performance of the communication schemes were usually quantified by assuming an identical chaos * Corresponding author. Tel.: +237 996 41 64; fax: +237 231 02 90.

E-mail addresses: [email protected] (S. Bowong), [email protected] (F.M. Moukam Kakmeni), [email protected] (H. Fotsin). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.02.035

194

S. Bowong et al. / Physics Letters A 355 (2006) 193–201

synchronization. This may impose some limitations to the applicability of these techniques. Of particular interest is the problem of synchronizing two or more systems when the designer of the receiver does not know not only the initial states but also some or all parameters and external disturbances. This is a more complicated problem referred to as adaptive synchronization [16,19,20]. Its solution is important in communications when the parameter modulation is used for message transmission (see, e.g. [16,19] and references therein), where a solution based on adaptive observers in an idealized setting with neglected noise and parameter mismatch was proposed. This effect should be taken into account when we want to evaluate the performance of a practical chaos communication scheme. As a consequence, secure communication via parameter modulation in chaotic systems in the presence of unknown parameters and external disturbances is an important issue. In the present Letter, we show that it is possible using synchronization techniques of two chaotic systems to design a secure communication system based on modern non-linear control theory. In the chaotic drive system with the hidden message, not only the Lipschitz constants on function matrices but also the bounds on uncertainties are unknown. In this case, a robust adaptive observer based response system is designed to synchronize the given uncertain chaotic system. If certain conditions are satisfied, two adaptation laws are chosen to repress external disturbances and to estimate unknown constants and uncertain parameters vector, respectively. Then, we take an information signal and use it to modulate one parameter of a chaotic system. The resulting transmitted signal consists of the information hidden in the signal from the chaotic system. The recovery of the information at the receiver is achieved with an appropriate demodulator which is described in this Letter. Lyapunov stability theory and Barbalat lemma ensure the global synchronization between the transmitter and receiver even if the drive system Lipschitz constants on function matrices and bounds on uncertainties are unknown. The results are illustrated by the numerical example of Chua’s circuit. A fairly good agreement is obtained between the analytical and numerical results. Throughout this Letter, it is noted that λ(W ) denotes an eigenvalue of W , λmax (W ) and λmin (W ) represent respectively the max[λi (W )] and the min[λi (W )], i = 1, . . . , n. |w| represents the absolute value of w and W  represents the Euclidian norm when W is a vector or the induced norm when W is a matrix. In is the identity matrix of dimension n. 2. Problem statement We consider the following form of chaotic systems:  x˙ = A(μ)x + Bf (x, μ) + Bg(x, μ)s + d(t), y = Cx,

(1)

where x ∈ Rn is the state vector, μ ∈ Rm is the parameter vector, y ∈ Rp is the output vector, s ∈ R is an unknown parameter which can be considered as a signal to be reconstructed in the receiver, f : Rn × Rm → Rq and g : Rn × Rm → Rq are nonlinear vectorvalued functions and d : R → Rn is the external disturbance vector. A(μ) is a matrix that may include parametric perturbations, B and C are constant known matrices with appropriate dimension. Generally speaking, only partial states of system (1) can be measured. Therefore, without loss of generality, the matrix C can be denoted by C = [Ip , 0]. Further, in the secure communication scheme, the gain B could be arbitrarily. In the Letter, the gain B is chosen as B = [Ip , 0] ∈ Rn×q , which implies that matrices B and C have the same rank. Note that many chaotic systems are already in this form, e.g., Lur’e systems, Rössler system, Lorenz system, almost all forced chaotic oscillators, etc. In order to recover the message s, we must make the following assumptions: Assumption 1. The matrix A(μ) and the non-linear functions f (x, μ) and g(x, μ) satisfy the following Lipschitz conditions:   A(μ) − A(μ) ˆ ∀μ, μˆ ∈ Rm , ˆ   ka μ − μ,   f (x, μ) − f (x, ˆ ∀x, xˆ ∈ Rn , ∀μ ∈ Rm , ˆ μ)  kf x − x,   f (x, μ) − f (x, μ) ˆ ∀x ∈ Rn , ∀μ, μˆ ∈ Rm , ˆ   kμ μ − μ,   g(x, μ) − g(x, ˆ ∀x, xˆ ∈ Rn , ∀μ ∈ Rm , ˆ μ)  kg x − x,   g(x, μ) − g(x, μ) ˆ ∀x ∈ Rn , ∀μ, μˆ ∈ Rm , ˆ   ks μ − μ,

(2) (3) (4) (5) (6)

where ka , kf , kμ , kg and ks are appropriate positive constants. Assumption 2. The uncertain parameter μ, the external disturbance d(t) and the message s(t) are norm bounded by three unknown positive constants μm , dm and sm , respectively. Assumption 3. The pair (A, B) is controllable while the pair (C, A) is observable. Further, there exists a constant vector L ∈ Rn×1 to make the transfer function H (s1 ) = C(s1 In − (A − LC))−1 B be strictly positive real. The following must be pointed out. (i) The Lipschitz properties are satisfied locally if A(μ), f (x, μ) and g(x, μ) are differentiable with respect to μ. Let U ⊂ Rn be a region which contains the chaotic attractor of (1) and let M ⊂ Rm be a region containing

S. Bowong et al. / Physics Letters A 355 (2006) 193–201

195

Fig. 1. Block diagram of a chaotic communication system. The region between the transmitter and the receiver is a hostile environment.

the relevant parameter values for which (1) exhibits chaotic behavior. The following analysis will remain valid if (1) is satisfied locally for x ∈ U and μ ∈ M. Note also that the Lipschitz constants ka , kf , kμ , kg and ks are often required to be known for the control design purpose. However, it is often difficult to obtain the precise values of ka , kf , kμ , kg and ks in some practical systems, hence the Lipschitz constants are often selected to be larger, which will induce the control gain to be higher, and the obtained results would be conservative. (ii) From Assumption 3 and Kalman–Yakubovich–Popov lemma [21], there exist two positive definite matrices P = P T and Q = QT such that the following algebraic equations hold:    T A(μ) − LC P + P A(μ) − LC = −Q, (7) and B T P = C.

(8)

Note that the equality (8) implies that the span of rows of B T P belongs to the span of rows of C. 3. Chaotic secure communication The secure communication system involves the development of a signal that contains the information that is to remain undetectable by others within a carrier signal. We can ensure the security of this information by inserting it into a chaotic signal that is transmitted to a prescribed receiver who would be able to detect and recover the information from the chaotic signal. In the present application, we propose a secure communication system as shown in Fig. 1. The technique takes the information and modulates one parameter of a non-linear signal that is generated by a chaotic signal generator. The resulting signal is transmitted through the hostile environment to a receiver. The receiver consists of a chaotic signal generator of the type that was used in the chaotic transmitter and additive terms. This will permit the demodulation of the received signal and the recovery of the information. As the signal is transmitted through the hostile environment, it is secure since it requires that an interloper possesses an identical chaotic signal generator and additive terms in order to intercept the information. In this Letter, we try to apply this idea to the secure communication problem. Let the signal received by the receiver be y(t). Within the receiver, an additional signal generator creates a signal which is similar to that created in the transmitter (1). If y(t) is the signal transmitted to the receiver, a robust adaptive observer for system (1) is constructed as follows: 1 ˆ ˆ − C x) ˆ + βB(y − C x), ˆ x˙ˆ = A(μ) ˆ xˆ + Bf (x, ˆ μ) ˆ + Bg(x, ˆ μ)ˆ ˆ s (t) + B α(y (9) 2 ˆ and sˆ (t) are estimates of x(t) and s(t), respectively, αˆ is an estimated feedback gain where μˆ ∈ Rm is the parameter vector, x(t), which is updated according to the following adaptation algorithm: α˙ˆ = γ y − C x ˆ 2,

(10)

sˆ is the demodulator: s˙ˆ (t) = ηg T (x, ˆ μ)(y ˆ − C x), ˆ

(11)

with γ and η two positive constants and βˆ 

β , 2(y − C x) ˆ T (y − C x) ˆ

for some β > 0. We can summarize our result on the following theorem.

(12)

196

S. Bowong et al. / Physics Letters A 355 (2006) 193–201

Theorem 1. If condition (12) holds, then the receiver (9) associated with the estimated feedback gain (10) and the demodulator (11) can globally synchronize the transmitter (1). Proof. Let e = x − xˆ be the synchronization error. By adding and subtracting likewise terms, the synchronization error dynamics is described by       e˙ = A(μ) − LC e + LCe + A(μ) − A(μ) ˆ xˆ + B f (x, μ) − f (x, ˆ μ)     + B f (x, ˆ μ) − f (x, ˆ μ) ˆ + d(t) + Bg(x, ˆ μ) ˆ s(t) − sˆ (t)     + B g(x, μ) − g(x, ˆ μ) s(t) + B g(x, ˆ μ) − g(x, ˆ μ) ˆ s(t) 1 T ˆ − αBB P e − βBCe. ˆ 2

(13)

Consider the following Lyapunov function candidate: V (e, α, ˆ sˆ ) = eT P e +

1 1 (α − α) ˆ 2 + (s − sˆ )2 , 2γ η

(14)

where α is defined as in Eq. (24). Let x ˆ  xˆm be satisfied for some xˆm > 0. Moreover, let us define μ = μ − μ ˆ and ˆ sˆ ) with respect to time is σ = [(BB T )−1 ]T B. So the time-derivative of V (e, α,    T V˙ (e, α, ˆ sˆ ) = eT A(μ) − LC P + P A(μ) − LC e + eT P LCe  −1      A(μ) − A(μ) ˆ xˆ + 2eT P B f (x, μ) − f (x, ˆ μ) + 2eT P BB T BB T  −1    ˆ μ) − f (x, ˆ μ) ˆ + 2eT P BB T BB T d(t) + 2eT P B f (x,     ˆ μ) s(t) + 2eT P B g(x, ˆ μ) − g(x, ˆ μ) ˆ s(t) + 2eT P B g(x, μ) − g(x,   ˆ μ) ˆ s(t) − sˆ (t) − αe ˆ T P BB T P e + 2eT P Bg(x, 1 2 (α − α) ˆ α˙ˆ − (s − sˆ )2 s˙ˆ γ η     −eT Qe + 2B T P eLT P e + 2ka σ μxˆm BP e       + 2B T P ekf e + 2sm B T P ekg e + 2sm ks μB T P e       ˆ μ) ˆ s(t) − sˆ (t) + 2kμ μB T P e + 2σ dm B T P e + 2eT P Bg(x, 2  1 2 ˆ T C T Ce − (α − α) − αˆ B T P e − 2βe ˆ α˙ˆ − (s − sˆ )2 s˙ˆ . γ η ˆ T P BCe − − 2βe

(15)

By Assumption 1, we get the following inequalities: kf2    2 2B T P ekf e  B T P e + ε1 e2 , ε1 2 k2   2  sm g T B P e + ε2 e2 , 2sm B T P ekg e  ε2  T  T  1  T 2  2 1  2   2B P eL P e  B P e + δ LT P e  B T P e + δλmax P LLT P e2 , δ δ  T  (kμ μ)2  T 2 B P e + β1 , 2kμ μB P e  β1   (σ ka xˆm μ)2  T 2 B P e + β2 , 2ka xˆm μB T P e  β2   (ks sm μ)2  T 2 B P e + β3 , 2ks sm μB T P e  β3

(16) (17) (18) (19) (20) (21)

and   (σ dm )2  T 2 B P e + β4 , 2dm B T P e  β4

(22)

S. Bowong et al. / Physics Letters A 355 (2006) 193–201

where ε1 , ε2 , δ, β1 , β2 , β3 and β4 are seven suitable positive constants. Then, we get   T 2 α˙ˆ      T T   ˙ ˆ B Pe − V (e, α, ˆ sˆ )  −e Q − ε + δλmax P LL P In e + (α − α) γ  ˙sˆ ˆ T C T Ce + 2(s − sˆ ) eT P Bg(x, + β − 2βe ˆ μ) ˆ − , η

197

(23)

where α=

kf2

+

2 k2 sm g

+

ε1 ε2 β = β1 + β2 + β3 .

1 (kμ μ)2 (σ ka xˆm μ)2 (ks sm μ)2 (σ dm )2 + + + , + δ β1 β2 β3 β4 (24)

Thus, if condition (12) holds, by applying the adaptation law (10) and the demodulator (11), one obtains V˙ (e, α, ˆ sˆ )  −λmin (S)e2 ,

(25)

where S = Q − (ε + δλmax (P LLT P ))In . Note that the free parameters ε and δ can be selected to be small enough such that ˆ sˆ ) is semi negative definite. Whence system (13) is Lyapunov stable. Which implies that e ∈ L∞ . λmin (S) > 0 so that V˙ (e, α, Integration (25), one has   V (e, α, ˆ sˆ )  V e(0), α(0), ˆ sˆ (0) − λmin (S)

t eT e dτ. 0

Then, from Eq. (13), we have e˙ ∈ L∞ . By Barbalat’s Lemma [22], we have e(t) → 0 as t → ∞. This implies that the receiver (9) associated with the adaptation law (10) and the demodulator (11) can globally synchronize the transmitter (1), i.e., x(t) ˆ → x(t) as t → ∞ and this achieves the proof. 2 ˆ Remark 1. Since the error y − C xˆ tends to zero, the magnitude of βB(y − C x) ˆ in the receiver (9) would increase unboundedly and become infeasible in computation. In practice, we can replace βˆ in the robust adaptive observer (9) with  β ˜ βˆ  2(y−C x) , if y − C x ˆ  β, ˆ T (y−C x) ˆ (26) ˜ βˆ = 0, if y − C x ˆ  β, where β˜ is a sufficiently small positive constant. Therefore, the state error would be contained within a neighborhood of the origin. Applying the persistency of excitation property [21], one easily has the following result. Corollary 1. If there exist two positive constants T and θ such that along the synchronized system trajectory x(t) it holds for all t  t0  0 that t+T

 T   g x(τ ), μ B T Bg x(τ ), μ dτ  θ In > 0.

(27)

t

Then, s(t) → sˆ (t) as t → ∞. Remark 2. If the parameter s(t) ∈ R is used as an information signal, Corollary 1 gives a possibility to recover it. It is worth noting that the persistency of excitation property may actually hold, thanks to the well-known properties of the chaos such as it topological transitivity. Thus, the unknown information signal s(t) can be recovered simultaneously in the receiver from sˆ (t). 4. Numerical studies The aim of this section is to demonstrate that the information signal can be “hidden” within the chaotic signal as it propagates in the hostile environment. The Chua’s circuit has been choosen to show the effectiveness and efficiency of the proposed secure communication scheme. The transmitter model in the dimensionless form is as follows:  x˙1 = δ1 [x2 − f1 (x1 ) + g1 (x1 )s], (28) x˙2 = x1 − x2 + x3 , x˙3 = −δ2 x2 ,

198

S. Bowong et al. / Physics Letters A 355 (2006) 193–201

Fig. 2. (a) The transmitted information, (b) x2 versus x1 and (c) the square wave information s(t) = 0.01(1 + sign(sin(0.05t))).

where f1 (x1 ) is a piecewise linear function given as:  2 3 f1 (x1 ) = x1 − (29) |x1 + 1| − |x1 − 1| , 7 14 g1 (x1 ) = |x1 + 1| − |x1 − 1| and s = s(t) ∈ R. This system is known to exhibit double scroll characteristics for δ1 = 9 and δ2 = 14.286, see e.g. [23] for more references as well as an electronic circuit implementation of this system. Assume that the transmitted signal is y = x1 , i.e., C = (1, 0, 0). Thus, this system is in the form given by (1) with

1 0 δ1 0 B= 0 , f (x, μ) = δ1 f1 (x1 ) and g(x, μ) = δ1 g1 (x1 ). A(μ) = 1 −1 1 , 0 0 −δ2 0 Clearly, the pair (A, C) is observable, thereby permitting the choice of the gain matrix L = (2, 2.6984, 3.2858)T to make the transfer function   −1 s 2 + s1 + 14.286 H (s1 ) = C s1 I3 − A(μ) − LC B = 31 s1 + 3s12 + 3s1 + 1 be strictly positive real. As derived earlier, the receiver is modelled as follows: ⎧ ˆ 1 − xˆ1 ), ˆ 1 − xˆ1 ) + β(x ⎨ x˙ˆ 1 = δˆ1 [xˆ2 − f (xˆ1 ) + g(x1 )ˆs ] + 12 α(x ˙xˆ 2 = xˆ1 − xˆ2 + xˆ3 , ⎩ x˙ˆ 3 = −δˆ2 xˆ2 ,

(30)

where the adaptation law on αˆ is chosen to be α˙ˆ = γ (x1 − xˆ1 )2

(31)

and sˆ is the demodulator s˙ˆ = ηδˆ1 (x1 − xˆ1 )g1 (x1 ),

(32)

S. Bowong et al. / Physics Letters A 355 (2006) 193–201

199

Fig. 3. Behavior of the communication of two coupled Chua’s circuits for μ = 0. (a) The information signal s(t) (solid line) and the demodulated signal sˆ (t) from the receiver signal (dashed line) and (b) the message error s(t) − sˆ (t).

Fig. 4. Behavior of the communication of two coupled Chua’s circuits for μ = 0.168. (a) The information signal s(t) (solid line) and the demodulated signal sˆ (t) from the receiver signal (dashed line) and (b) the message error s(t) − sˆ (t).

with γ and η two positive constants and βˆ satisfies the following condition: 

β βˆ  2(x − 2, 1 xˆ1 ) βˆ = 0,

˜ if |e1 |  β, ˜ if |e1 |  β,

(33)

with β˜ a sufficiently small positive constant. We simulated (28)–(29) with the parameters indicated above and the initial conditions (x1 (0), x2 (0), x3 (0)) = (1, 0, 0). The ˆ = 0, response system was simulated with the following initial conditions and parameters: (xˆ1 (0), xˆ2 (0), xˆ3 (0)) = (2, 0, 0), α(0) γ = β = 1 and β˜ = 0.01. We chose the message as the square wave information s(t) = 0.01(1 + sign(sin(0.05t)). Fig. 2(a) shows the transmitted information x1 , and Fig. 2(b) presents x2 versus x1 . As can be seen, the solutions are chaotic. The square wave information s(t) is depicted in Fig. 2(c). For the simulations, we considered the following cases: Case 1. We simulated (30) with δ1 = δˆ1 and δ2 = δˆ2 . The real message and the recovered message are shown in Fig. 3(a). In this figure, the hidden message s(t) and the recovered message sˆ (t) are denoted by the solid and dashed lines, respectively. It follows that the hidden message has been recovered with good accuracy. In order to add evidence of the effectiveness and efficiency of the proposed secure communication scheme, we have plotted the message error s(t) − sˆ (t) as shown in Fig. 3(b). As can be seen, the message error is quite small, and that the real message is reconstructed without any error. Case 2. Wesimulated (30) with δˆ1 = 8.91 and δˆ2 = 14.143. Note that in this case, δ1 = δ1 − δˆ1  = 0.09, δ2 = δ2 − δˆ2  = 0.143 and μ = (δ1 )2 + (δ2 )2 = 0.168. Fig. 4(a) presents the real message s(t) (solid line) together with the recovered message sˆ (t) (dashed line) while Fig. 4(b) shows the message error s(t) − sˆ (t). Note also that the demodulator recovers the unknown information signal well. As can be seen, the message error is quite small and very close to zero.

200

S. Bowong et al. / Physics Letters A 355 (2006) 193–201

Fig. 5. Behavior of the communication of two coupled Chua’s circuits for μ = 0.337. (a) The information signal s(t) (solid line) and the demodulated signal sˆ (t) from the receiver signal (dashed line), (b) the message error s(t) − sˆ (t) for γ = 1 and (c) the message error s(t) − sˆ (t) for γ = 5.

Case 3.We simulated (30) with δˆ1 = 8.82 and δˆ2 = 14. In this case, δ1 = δ1 − δˆ1  = 0.18, δ2 = δ2 − δˆ2  = 0.286 and μ = (δ1 )2 + (δ2 )2 = 0.337. We first show γ = 1 and the real message (solid line) together with the recovered message (dashed line) and the message error are respectively shown in Figs. 5(a) and 5(b). Although the message error is higher as compared to case 2 (due to increased parameter mismatch and noise), the message error is also quite small and very close to zero. To show the effect of increased gain, we also choose γ = 5. Fig. 5(c) shows the message error s(t) − sˆ (t). As can be seen, by increasing the gain, we may reduce the error in the message recovery. 5. Conclusion In this Letter, we have introduced a technique of creating a secure communication system that is based on the modulation of one parameter of an equation that admits chaotic solutions. The propagating chaotic signal that contains the information cannot be easily distinguished from a chaotic signal that contains no information. In addition, the information signal can be recovered from this propagating chaotic signal using techniques from modern control theory. Numerical example of the Chua circuit has been presented to show the effectiveness and efficiency of the proposed secure communication scheme. We also presented some simulation results indicating the robustness of the proposed scheme. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821. T.L. Carroll, L.M. Pecora, IEEE Trans. Circuits Systems I 38 (1991) 453. S. Bocaletti, J. Kurths, G. Osipov, D. Valladares, C. Zhou, Phys. Rep. 366 (2002) 1. G. Chen, Control and synchronization of chaotic systems (a bibliography), ftp.egr.uh.edu/pub/TeX/chaos.tex, loginname: anonymous, password: your e-mail address, 1997. M. Lakshamanam, K. Murali, Chaos in Nonlinear Oscillators, Controlling and Synchronization, World Scientific, Singapore, 1996. K.M. Cuomo, A.V. Oppenheim, Phys. Rev. Lett. 71 (1993) 65. M. Hasler, Philos. Trans. R. Soc. London 1701 (1995) 115. M.Y. Chen, Z.Z. Han, Y. Shan, Int. J. Bifur. Chaos 14 (2004) 347. H. Nijmeijer, I.M.Y. Mareels, IEEE Trans. Circuits Systems I 44 (1997) 882. O. Morgül, E. Solak, Int. J. Bifur. Chaos 7 (1997) 1303. G. Grassi, S. Mascolo, IEEE Trans. Circuits Systems I 44 (1997) 1011. M. Feki, Phys. Lett. A 309 (2003) 53.

S. Bowong et al. / Physics Letters A 355 (2006) 193–201

[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

S. Bowong, F.M. Moukam Kakmeni, R. Koina, Int. J. Bifur. Chaos 14 (2004) 2477. R. Femat, J. Alvarez-Ramirez, G. Fernandez-Anaya, Physica D 139 (2002) 231. S. Bowong, Phys. Lett. A 326 (2004) 102. T.L. Liao, S.H. Tsai, Chaos Solitons Fractals 11 (2000) 1387. G. Kolumbán, M.P. Kennedy, L.O. Chua, IEEE Trans. Circuits Systems I 44 (1997) 927. E.-W. Bai, K.E. Lonngren, A. Ucar, Chaos Solitons Fractals 23 (2005) 1071. B. Andrievsky, Math. Comput. 58 (2002) 285. M. Feki, Chaos Solitons Fractals 18 (2003) 141. R. Marino, P. Tomei, Nonlinear Control Design—Geometric, Adaptive, Robust, Prentice–Hall, Englewood Cliffs, NJ, 1995. H.K. Khalil, Nonlinear Systems, third ed., Springer-Verlag, United Kingdom, 1995. M.E. Yalçn, J.A. Suykens, J. Vanderwalle, IEEE Trans. Circuits Systems I 47 (2000) 425.

201