Synchronization of simplest two-component Hartley's ...

Nonlinear Dyn DOI 10.1007/s11071-013-1024-3

O R I G I N A L PA P E R

Synchronization of simplest two-component Hartley’s chaotic circuits: influence of channel Robert Tchitnga · Patrick Louodop · Hilaire Fotsin · Paul Woafo · Anaclet Fomethe

Received: 17 November 2012 / Accepted: 30 July 2013 © Springer Science+Business Media Dordrecht 2013

Abstract Firstly, the synchronization problem of the simplest two-component Hartley chaotic systems is considered. A simple and effective controller is used to achieve synchronization between the drive and response systems. The proposed controller is built around a linear and a nonlinear parts with each contributing to the achievement of the synchronization process. The stability of the drive–response systems framework is proved through the Lyapunov stability theory. Secondly, the impact of channel on the signal coming from the drive system to synchronize the re-

sponse system is taken into consideration. In this second part, the conditions to obtain synchronization between both master and slave systems are investigated. For the purpose of illustration, PSpice simulations are given as complement of the numerical analysis. Keywords Hartley’s oscillator · Simplest chaotic circuit · Synchronization · Nonlinear controller · Delay

1 Introduction R. Tchitnga (B) · P. Louodop Research Group on Experimental and Applied Physics for Sustainable Development (EAPhySuD), Department of Physics, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon e-mail: [email protected] R. Tchitnga · P. Louodop · H. Fotsin Laboratory of Electronics and Signals Processing, Department of Physics, Faculty of Science, University of Dschang, P.O. Box 412, Dschang, Cameroon R. Tchitnga · P. Woafo Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, Faculty of Science, University of Yaounde, I, P.O. Box 812, Yaounde, Cameroon A. Fomethe Laboratoire de Mécanique et de Modélisation des Systèmes, Faculté des Sciences, Université de Dschang, BP. 67, Dschang, Cameroun

Since the pioneering work of Pecora and Caroll [1], the dynamics, the control and the synchronization of chaotic systems have became major topics thanks to their multiple applications in sciences and technology [2–4]. Hence, a wide variety of approaches have been developed and presented in the literature [5–11]. However, for applications in engineering, the investigation of small and simple chaotic systems has been widely done in the last decade [12–15], but still remains actual. In this sense, there are some new chaotic systems recently proposed in the literature such as the simple jerk chaotic systems [16, 17], or circuits with a maximum of four electronic components [18, 19] or even three as the simplest chaotic circuit proposed by Muthuswamy and Chua [20]. More recently, a twocomponent chaotic Hartley oscillator was proposed and its dynamics investigated in Ref. [21]. It belongs to the low voltage and high frequency oscillating circuit,

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in the range of GHz such as that proposed by Li et al. [22]. The chaotic behavior of this recent chaotic circuit is based on the exploitation of the nonlinear properties of a Junction Field Effect Transistor (JFET) as single locally active electronic component such as the memristor in Ref. [20]. According to cryptographic requirements for chaotic secure communications [23], investigating the impacts of the channel on chaos-based secure communication strategy helps to present the new cryptosystem in more rigorous way. This conclusion could be added to various or even all synchronization strategies because chaos-based secure communication is obtained only if the transmitter and the receiver are synchronized [24–31]. In Ref. [32] Andrievsky and Fradkov worked on the information transmission by adaptive synchronization with chaotic carrier and noisy channel. They considered the presence of additive noise in the channel and basing themselves on the adaptive observations, they designed receivers to achieve signal transmission. More recently, Shen Cheng et al. [33] worked on the adaptive synchronization of chaotic Colpitts circuits against parameter mismatches and channel distortions. Also they used additive noise and demonstrated that two chaotic Colpitts circuits can properly synchronized with the use of adaptive controllers. Furthermore, the impact of disturbances and delay are intensively studied in biological systems [34–38]. In the framework of this paper our aim is to achieve synchronization of the systems given in Ref. [21] so that it could be applied to a secure communication scheme in spite of its simplicity. Here, we consider the transmission through a channel and we take into account the noise or perturbation introduced by the channel and the delay due to the time taken by the signal coming from the master to synchronize the slave. Both channel characteristics are considered individually. Some numerical simulations are given to observe their impacts on the synchronization and therefore, in this case, to understand what could happen if such strategy is applied for secure communication. The controller used to drive the slave system to follow the master has two major peculiarities: on one hand, it is a two-part controller made of a linear part, and a nonlinear one with an exponential-based function which can be shaped using semiconductors diodes. On the other hand, it is built using the delayed or the perturbed signal coming from the master. Under such a hypothesis,

neither the master system nor the slave system are subjected to perturbation or delay but just the signal coming from the master is late or perturbed when reaching to the slave. Both parts of this controller have complementary role on the achievement of the synchronization between the drive and the response systems. To the best of our knowledge, the literature makes no mention of such considerations. Generally, the authors consider that delays are between different parts of the same system or between different systems (drive system(s) and response system(s)) [39, 40]. This work is organized as follows: In Sect. 2, the chaotic circuit is lightly introduced and its model given. Section 3 is devoted to the synchronization of two oscillators in the case of an ideal channel. The problem is formulated in the first subsection and some assumptions are given; then the stability of the scheme is proved using Lyapunov stability theory. We show that with the given two-part controller the drive– response systems are practically synchronized. In the second subsection, numerical results are presented. In Sect. 4, the problem is tackled under consideration of a real channel. After the problem statement and solution in the first subsection, the results of simulations are given in the next. The last section is devoted to the conclusion.

2 Brief presentation of the system The considered circuit diagram and its small signal and high frequency equivalent circuit are shown on Fig. 1(a) and Fig. 1(b). According to Ref. [21], the internal parasitic capacitors of the JFET occurring at high frequency can be used to complete the two sides of the tapped coil L1 and L2 to form the necessary resonant circuit of the feedback loop. The state equations of the circuit under such considerations is given by ⎧ dVGS 1 ⎪ ⎪ = (−i1 + i2 − iD − Id ), ⎪ ⎪ dt C GS ⎪ ⎪ ⎪ ⎪ ⎪ dVDG 1 ⎪ ⎪ ⎪ ⎨ dt = C (−i2 + Id ), DG

⎪ di1 1 ⎪ ⎪ VGS , = ⎪ ⎪ ⎪ dt L1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 di ⎪ ⎩ 2= (−VGS + VDG + E). dt L2

(1)

Synchronization of simplest two-component Hartley’s chaotic circuits: influence of channel Fig. 1 (a) Circuit diagram and (b) small signals high frequency equivalent circuit

The current through the diode is given by iD = VGS VT

Is (e − 1), with Is its reverse saturation current, meanwhile the current source between the drain and the source of the JFET delivers the piecewise current Id below (see Ref. [41]): ⎧ 0 ⎪ ⎪ ⎪ ⎪ if VGS ≤ VGSoff , ⎪ ⎪ ⎨ gmo (VGS − VGSoff )2 Id = if VDG ≤ VGSoff , ⎪ ⎪ ⎪ ⎪ ⎪ (V − VGD )(VGS + VDG − 2VGSoff ) g ⎪ ⎩ mo DS if VDG ≥ VGSoff .

(2)

VGSoff ≤ 0 is the cutoff value of the voltage between the gate and the source electrodes of the JFET and, gmo is the current gain of this transistor. Normalizing voltages, currents, and time with respect to VT , I0 and τ = ω0 t, respectively, where ω0 is the resonant radian frequency of the system, and introducing the dimensionless states variables x1 = VGS /VT , x2 = VDG /VT , x3 = i1 /IS and x4 = i2 /IS , the model can now be described by the following set of dimensionless coupled nonlinear differential equations: ⎧ x˙1 = a1 (x4 − x3 ) − a2 exp(x1 ) − 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − a3 g(x1 , x2 ), ⎪ ⎪ ⎨ x˙2 = −α a1 x4 − a3 g(x1 , x2 ) , ⎪ ⎪ ⎪ ⎪ ⎪ x˙3 = b1 x1 , ⎪ ⎪ ⎪ ⎩ x˙4 = b2 e − x1 + x2 ) ,

gm0 VT I0 Is VT CGS ω0 , a2 = VT CGS ω0 , a3 = CGS ω0 , b1 GS b2 = L2VIT0 ω0 , α = CCGD and e = VET .

where a1 = VT L1 I0 ω0 ,

(3)

=

The dimensionless function g(x1 , x2 ) in Eq. (4) recalls the piecewise nonlinear function (2), which rep-

resents the drain-source current: g(x1 , x2 ) ⎧ ⎨0 = (x1 − xm )2 ⎩ (x1 − xm )(x1 + x2 − 2xm )

if x1 ≤ xm , if x2 ≤ xm , if x2 ≥ xm .

(4)

For the selected values of parameters a1 = 10.7066381, a2 = 0.0000359, a3 = 0.0117372, b1 = 0.0010204, b2 = 0.00625, α = 1.1152239, xm = −56.36 and e = 112, and with the initial conditions (x1 (0), x2 (0), x3 (0), x4 (0)) = (0.00, 0.00, 0.05, 0.03), the chaotic attractors obtained are displayed on Fig. 2(a) x2 versus x1 and Fig. 2(c) x4 versus x2 for Matlab simulation as well as on Fig. 2(b) VGD (V ) versus VGS (V ) and Fig. 2(d) i2 (A) versus VGD (V ) for Pspice simulation. For more details about the dynamics of this autonomous two-component chaotic oscillator, one should refer to Ref. [21].

3 Ideal channel 3.1 Synchronization of simplest Hartley’s oscillators The problem in this subsection is to propose a strategy to synchronize two units of the given Hartley oscillators defined by the set of Eq. (3). To achieve this goal, let us consider that the master system is described by Eq. (3) while the slave system, unidirectionally coupled to the master is defined by the set of Eq. (5) below: ⎧ y˙1 = a1 (y4 − y3 ) − a2 exp(y1 ) − 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − a3 g(y1 , y2 ) − u(τ ), ⎪ ⎪ ⎨ (5) y˙2 = −α a1 y4 − a3 g(y1 , y2 ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪y˙3 = b1 y1 , ⎪ ⎪ ⎩ y˙4 = b2 e − y1 + y2 ) .

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Fig. 2 Chaotic attractors (a) x2 versus x1 and (c) x4 versus x2 for Matlab simulation, (b) VGD (V ) versus VGS (V ) and (d) i2 (A) versus VGD (V ) for Pspice simulation

Synchronization of simplest two-component Hartley’s chaotic circuits: influence of channel

In the above coupled nonlinear differential equations, u(τ ) is the proposed two parts controller defined by the following relation: u(τ ) = β exp(x1 ) − 1 − exp(y1 ) − 1 (6) + k(y1 − x1 ), where k is a positive constant gain factor and β the controller parameter. Lets us make precise that β is a fixed or an adaptive parameter defined by the designer according to the aims to be achieved. In the present case, it is chosen as β = a2 . According to Eq. (6) and if the error between both master and slave systems is defined by the relation ei = yi − xi , i = 1, 2, 3, 4, the errors state dynamics is derived as follows: ⎧ e˙1 = a1 (e4 − e3 ) − a3 G − ke1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ e˙2 = −α(a1 e4 − a3 G), (7) ⎪ e˙3 = b1 e1 , ⎪ ⎪ ⎪ ⎪ ⎩ e˙4 = b2 −e1 + e2 ) , where G = g(y1 , y2 ) − g(x1 , x2 ). At this stage, we need to prove that the dynamics of the errors system Eq. (5) practically converges to a limit value, in the sense that, when τ → ∞, |e(τ )| → ε, which is a very small positive constant. This can be qualitatively analyzed with the Lyapunov stability theory as follows: Consider a candidate Lyapunov function in the following form: V=

1 2 e22 a1 e32 a1 e42 . e1 + + + 2 α b1 b2

(8)

The time derivative of this function with respect to system Eq. (7) satisfies V˙ = a3 G(e2 − e1 ) − ke12 .

(9)

As we deal with chaotic systems, they can be considered to be bounded. This means that there exist two sufficiently small and positive constants r and Q which verify the following relations: |G| ≤ Q|e1 + e2 |, |ei | ≤ 2r,

i = 1, 2, 3, 4.

Remark 1 Since in the present work, the master and slave systems are coupled unidirectionally, there is no doubt on the boundedness of the master system. Only the boundedness of the slave system under the control

action could be questionable. Therefore, the design of the controller is critical. We assume that the trajectory of the controlled slave system is always contained in a bounded and closed domain for the purpose of designing the controller, which to our own opinion can actually be confirmed by the numerical simulations [42]. Hence, Eq. (9) becomes V˙ ≤ a3 Qe22 + (a3 Q − k)e12 .

(10)

Let us take k ≥ a3 Q; the above equation can be rewritten as V˙ ≤ a3 Qe22 − λe12 ,

where

λ = |k − a3 Q|, V˙ ≤ a3 Qen2 − λe12 , where

en (τ ) = e12 (τ ) + e22 (τ ) + e32 (τ ) + e42 (τ ), V˙ ≤ 64a3 Qr 2 − λe12 as en (τ ) ≤ e1 (τ ) + e2 (τ ) + e3 (τ ) + e4 (τ )

(11)

≤ 8r, V˙ ≤ p − λe12

where p = 64a3 Qr 2 .

Then, from Eq. (11), it follows that if e1 (τ ) > p , λ then V˙ < 0 [39]. Hence V decreases, which implies that |en (τ )| decreases as well. It then follows from the standard invariance arguments that asymptotically the error satisfies the following bound [39, 43]: en (τ ) ≤ C, where C≥

p . λ

From Eq. (11), one can see that the asymptotic error depends linearly on the free parameter p. Hence, if this parameter is small, the resulting error will be small as well. The dependence of the error on |en (τ )| deserves special attention. Note that |en (τ )| ≤ 8r. Hence, if p decreases, r decreases too and |en (τ )| decreases as well. This argument shows that with the proposed

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Fig. 3 Time histories of drive system (solid lines) and slave system (dashed lines)

synchronization scheme, p should be made as small as possible. Therefore, the state error would be contained within a neighborhood of the origin. Note that if |en (τ )| = 0, i.e., xi = yi , i = 1, 2, 3, 4 and r = 0, the synchronization error system is globally asymptotically stable.

3.2 Results of simulations The numerical results of this case, with k = 0.3 and the initial conditions (x1 (0), x2 (0), x3 (0), x4 (0)) = (0.00, 0.00, 0.05, 0.03), (y1 (0), y2 (0), y3 (0), y4 (0)) = (0.00, 0.20, −0.05, −0.03) are recorded on Figs. 3 and 4. The time histories of the drive system (solid lines) and response system (dashed lines) are presented on Fig. 3. One can notice that both master– slave systems achieve synchronization. On Figs. 4(a) to 4(d), the time histories of errors states are presented. According to these graphs, it can be observed that the synchronization is reached as the time histories of these errors are converging with time towards a really small limit value. Completing this development about the graphs on Fig. 3, we can conclude on the synchronization of the drive–response systems according to the stabilization of the two-part proposed controller.

4 Real channel 4.1 Problem statement The behavior of the synchronization process of two simplest Hartley oscillators coupled unidirectionally is going to be investigated, when the signal coming from the master system is subjected to delay due to the time taken to pass through the channel or due to its parameters. To expressed it mathematically, the above proposed two-part controller Eq. (6) is going to be redefined by the following relation: u(τ ) = a2 exp x1 (τ − θ ) − 1 − exp(y1 ) − 1 (12) + k y1 − x1 (τ − θ ) , where θ represents the time-lag taken by the master signal to reach the slave system. From here, the errors state dynamics takes the following form: ⎧ e˙1 = a1 (e4 − e3 ) − a2 ϕ − a3 G − kξ, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ e˙2 = −α(a1 e4 − a3 G), (13) ⎪ e˙3 = b1 e1 , ⎪ ⎪ ⎪ ⎪ ⎩ e˙4 = b2 (−e1 + e2 ), where ξ = y1 − x1 (τ − θ ) and ϕ = ((exp(x1 (τ − θ )) − 1) − (exp(x1 (τ )) − 1)).


Fig. 4 Time histories of errors states between master–slave systems

Remark 2 The statement in Remark 1 is also valid in the present case. Hence, x1 (τ − θ ) is also bounded. To ensure the boundedness of the slave system, let us assume that the trajectory of the controlled slave system is always contained in a bounded and close domain, which can actually be confirmed by the numerical simulations. Let us add and subtract the quantity (exp(y1 ) − 1) in the expression of the above function ϕ. The function in question becomes ϕ = −[exp(y1 )−exp(x1 (τ −θ ))]+[exp(y1 )−exp(x1 (τ ))]. According to our assumption, ϕ is supposed to be bounded by two positive constants Q1 and Q2 such that |ϕ| ≤ Q1 |ξ | + Q2 |e1 |.

(14)

Let us prove that the dynamics of the errors system Eq. (13) practically converges to a limit value. To achieve this goal, we redefine the Lyapunov candidate function as in Eq. (15)

0 1 2 e22 a1 e32 a1 e42 +k + e12 (τ + s) ds. V = e1 + + 2 α b1 b2 −θ (15) The time derivative of this function with respect to system Eq. (13) satisfies V˙ = a3 G(e2 − e1 ) − kξ e1 − a2 ϕe1 + ke12 − kξ 2 . (16)

Otherwise, the following inequality can be considered: V˙ ≤ a3 |G||e2 − e1 | + k|ξ ||e1 | − a2 |ϕ||e1 | + ke12 − kξ 2 . Taken into account Eq. (16) and the relation 2ab ≤ a 2 + b2 , the above relation becomes 3k 2 ˙ V ≤ a3 Q(e2 + e1 ) + + a2 Q1 + a2 Q2 e12 2 a2 Q1 − k 2 ξ . + (17) 2 Remembering that en (τ ) > ei (τ ); i = 1, 2, 3, 4 and considering k ≥ a2 Q1 , we have 3k + a2 Q1 + a2 Q2 en2 − λ0 ξ 2 , V˙ ≤ 4a3 Q + 2 |k − a2 Q1 | , where λ0 = 2 3k ˙ V ≤ 4a3 Q + + a2 Q1 + a2 Q2 en2 − λ0 ξ 2 2

where en (τ ) = e12 (τ ) + e22 (τ ) + e32 (τ ) + e42 (τ ), 3k ˙ V ≤ 64 4a3 Q + + a2 Q1 + a2 Q2 r 2 − λ0 ξ 2 2 (18)

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Fig. 5 Time histories of errors norm between master–slave systems

as en (τ ) ≤ e1 (τ ) + e2 (τ ) + e3 (τ ) + e4 (τ ) x ≤ 8r, V˙ ≤ p0 − λ0 ξ 2

3k + a2 Q1 + a2 Q2 r 2 . with p0 = 64 4a3 Q + 2

Then, from Eq. (18), it follows that if ξ(t) > p0 , λ0 then V˙ < 0 [39]. Hence V decreases, which implies that |en (τ )| decreases as well. It then follows from the standard invariance arguments that asymptotically the error satisfies the following bound [39, 43]: en (τ ) ≤ C0 , where p0 C0 ≥ . λ0 From Eq. (18), one can see that the asymptotic error depends linearly on the free parameter p0 . Note that |en (τ )| ≤ 8r. Hence, if p0 decreases, r and |en (τ )| as well decrease. This argument shows that with the proposed synchronization scheme, p0 should be made as small as possible. Therefore, the state error would be contained within a neighborhood of the origin. Note that if |en (τ )| = 0, i.e., xi (τ − θ ) xi (τ ), i = 1, 2, 3, 4

that is θ is sufficiently small and r = 0, the synchronization error system is globally asymptotically stable. Remark 3 Let us consider that the channel introduces a noise on the transmitted signal x1 (τ ). That noise affects directly the controller in such a way that u(τ ) = a2 exp xn (τ ) − 1 − exp(y1 ) − 1 + k y1 − xn (τ ) ,

(19)

where xn (τ ) represents the noisily signal x1 coming from the master system. From this, the errors state dynamics takes the following form: ⎧ e˙1 = a1 (e4 − e3 ) − a2 φ − a3 G − kη, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ e˙2 = −α(a1 e4 − a3 G), ⎪ e˙3 = b1 e1 , ⎪ ⎪ ⎪ ⎪ ⎩ e˙4 = b2 (−e1 + e2 ),

(20)

where η = y1 − xn (τ ) and φ = ((exp(xn (τ )) − 1) − (exp(x1 (τ )) − 1)). The errors state dynamics, here in Eq. (20), has the same expression as (13). But the output signal xn (τ ) coming from the channel to control the response system is not always bounded. Let us assume that we are under conditions (small perturbations, noise with low amplitudes or with high signal– noise rate) allowing to observe the confining of the slave system. One can see that, under this hypothesis, we obtain the practical synchronization as above.


Fig. 6 Time histories of errors norm with xn (τ ) = x1 (τ ) + Nwgn(1, 1, 1)

Fig. 7 Time histories of errors norm with xn (τ ) = awgn(x1 (τ ), r , measured )

4.2 Results of simulations The numerical results considering the presence of delay are obtained with k = 1 and the same parameter values and initial conditions either for the transmitter or for the receiver. On the graphs of Fig. 5, one can easily observe that the synchronization is reached and destroyed when the delay is increasing. In Fig. 5(a) we

show the time history of the error norm en (τ ) = e12 (τ ) + e22 (τ ) + e32 (τ ) + e42 (τ ) when the numerical time-lag θ = 0. For Figs. 5(b) and 5(c) the time-lags are, respectively, θ = 0.25 and θ = 0.3125. If we increase this time-lag to θ = 0.314, both transmitterreceiver systems diverge. Without the delay and considering the introduction of noise, we investigate two cases. We first suppose that the perturbations output signal coming from the channel has the form xn (τ ) = x1 (τ ) + N wgn(1, 1, 1), where wgn(1, 1, 1) is a Matlab white Gaussian noise generator, and N is the amplitude of noise. Secondly, we suppose that the signal coming from the master x1 (τ ) is intrinsically modified by the channel and thus the channel output signal form is xn (τ ) = awgn(x1 (τ ), r , measured ), where r represent the signal–noise rate. Figure 6 presents the time history of en (τ ) with numerical time τ and noise amplitude N . It appears from this 3D graph that the synchronization is wrecked when the amplitude of noise increases. Figure 7 shows the behavior of the errors norm en (τ ) with numerical time τ and the signal–noise ratio r. It appears that when r is high the synchronization is better. Let us investigate the robustness of the proposed scheme in front of noise and mismatches. In what follows, we suppose that the mismatches are inserted in the value of system parameter e. Here, Fig. 8 presents the behavior of the error norm en (τ ) considering that there is no noise introduced by the channel. It appears

Fig. 8 Behavior of en (τ ) according to when k = 12

Fig. 9 Behavior of en (τ ) according to k when = 0.01

from this curve that the amplitude of en (τ ) decreases towards zero more rapidly when the mismatches are small. Figure. 9 shows the relative behavior of en (τ ) amplitude according to the control parameter k and for

= 0.01. One can see that the synchronization can be easily obtained when k is high. Taking into account only the presence of noise introduced by the channel, it is obvious in Fig. 10 that the synchronization is better when the amplitude N of noise is small and according to Fig. 11, the high value of signal–noise ratio helps to deal with a small synchronization error en (τ ).

5 Conclusion In this paper, we have studied a strategy for synchronizing two simplest two-component Hartley’s chaotic

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Fig. 10 Behavior of en (τ ) according to N when = 0 and k = 10

Fig. 11 Behavior of en (τ ) according to r when = 0 and k = 10

circuits. The proposed method can be easily realized in practice through electrical components. To investigate our procedure, we based ourselves on the boundedness of the chaotic oscillators. The controller is built with the help of the nonlinear device used in the given circuits. The systems are practically synchronized and the stability conditions of the schemes are defined. We have considered the case for which the signal coming from the master to synchronize the slave pass through a channel. In such a case we have taken into account and separately the impacts of delay and the introduction of noise or perturbations. The results by numerical simulations show the effectiveness of the applied method. Our future objective is to realize this synchronization procedure experimentally and to apply it in chaos secure communication schemes.

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