Synchronization of PWL function-based 2D and 3D multi-scroll chaotic ...

Nonlinear Dyn (2012) 70:1633–1643 DOI 10.1007/s11071-012-0562-4

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

Synchronization of PWL function-based 2D and 3D multi-scroll chaotic systems Jesus M. Muñoz-Pacheco · Ernesto Zambrano-Serrano · Olga Félix-Beltrán · Luz C. Gómez-Pavón · Arnulfo Luis-Ramos

Received: 8 June 2012 / Accepted: 30 July 2012 / Published online: 21 August 2012 © Springer Science+Business Media B.V. 2012

Abstract We study the synchronization of a piecewise linear function-based chaotic system. That system generates multiple scrolls in multiple directions (two- and three-directions) on phase space. In this scenario, the design of a controller based on Generalized Hamiltonian forms is possible. As function of control signals, we propose a master–slave synchronization scheme using 2n − 1 combinations to drive a nonlinear state observer. Associated with this, the piecewise linear functions of the slave are directly controlled by the state-variables of the master system. We computed the synchronization error for each combinations. Besides, the circuit synthesis based on operational amplifiers validates our synchronization scheme by means of SPICE simulations. We observed that the synchronization error at circuit level depends on the number of the control signals used. Our numerical and SPICE simulation results are in agreement showing the usefulness of the proposed approach. Keywords Chaotic system · Synchronization · Multi-directional attractor · Hamiltonian forms · Operational amplifier · Synthesis

J.M. Muñoz-Pacheco () · E. Zambrano-Serrano · O. Félix-Beltrán · L.C. Gómez-Pavón · A. Luis-Ramos Facultad de Ciencias de la Electrónica, Benemérita Universidad Autónoma de Puebla, Apdo. Postal 542, C.P. 72570 Puebla, Pue., Mexico e-mail: [email protected]

1 Introduction Chaotic oscillators have their origin in nonlinear dynamic systems [1]. Those generate chaotic signals with a wide frequency spectrum in most of the cases. As a consequence, the chaotic behavior is well characterized by its extremely sensitivity to initial conditions [2] as well as to perturbations in chaotic oscillator parameters [3]. Theoretically, chaotic systems produce infinite numbers of chaotic signals, unlike pseudorandom signals, which are limited in number and are periodic. Those properties and the broadband nature of chaotic signals are of particular interest in secure communications by applying some synchronization approach [4–7]. In this context, the encoding of messages using a chaotic signal is widely studied in theoretical and experimental approaches [8–11]. A special class of chaotic systems is based on piecewise linear (PWL) functions in order to generate two-directional (2D) and three-directional (3D) multiscroll chaotic systems (MSCS) [2, 12]. This multidirectionality is obtained by adding extra nonlinear functions to the chaotic system. As a result, the dynamic of the whole attractor is more complex than in one-directional (1D) MSCS due to the number of equilibrium points is related with the saturation levels of PWL functions [2]. Besides, the effort to estimate chaotic dynamics according the system parameters is also increased. However, the synchronization conditions for 2D and 3D MSCS need to be investigated

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in order to exploit those characteristics in future applications [16]. This could imply strong cryptographic properties to increase the security in confidential information transmission [13–15]. To the best of our knowledge, chaos synchronization has been principally researched on 1D MSCS [3– 7, 9, 10] in contrast to the approaches about synchronization on 2D and 3D MSCS, and their synthesis with electronic devices [13–15]. Based on the aforementioned discussion, we report the synchronization of a PWL function-based 2D and 3D MSCS. Advantages of our approach are listed as follows. First, the approach lies on Hamiltonian forms avoiding to compute the Lyapunov exponents contrary to other work reported in literature based on a vector Lyapunov function and hyperbolic functions [14]. Second, we introduce a synchronization scheme with 2n − 1 possible combinations to control the nonlinear state observer. Therefore, this approach can be considered as a enhanced version of the paper introduced in [13], where only one state-variable as control signal was used. From practical point of view, we consider an Operational Amplifier (OpAmp) model with second-order effects to realize the circuit synthesis of the synchronization schemes. It is worthy to note that an electronic circuit has not been synthesized in few papers dealing with 2D and 3D MSCS synchronization [13, 14]. Furthermore, we design a Hamiltonian-based controller to investigate how many signals can be used to reach the synchronization. Consequently, we compute the relationship between the number of control signals and the synchronization error, using one, two and three state-variables of the master system, or a combination of them. Finally, taking into account slew-rate, saturation, offset and finite bandwidth of the OpAmp, SPICE simulations show a good agreement between theoretical results and circuit synthesis.

2 Hamiltonian-based controller design for 2D and 3D MSCS In this section, we introduce a controller with a gain matrix in order to drive the trajectories of the observer system. To obtain the control, PWL function-based 2D

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and 3D MSCS are represented by (1a) and (1b), respectively, as follows: d2 x˙1 = x2 − f (x2 , k, h, p, q, α), b x˙2 = x3 , (1a) x˙3 = −ax1 − bx2 − cx3 + d1 f (x1 , k, h, p, q, α) + d2 f (x2 , k, h, p, q, α), and d2 f (x2 , k, h, p, q, α), b d3 x˙2 = x3 − f (x3 , k, h, p, q, α), b x˙3 = −ax1 − bx2 − cx3 + d1 f (x1 , k, h, p, q, α) (1b) x˙1 = x2 −

+ d2 f (x2 , k, h, p, q, α) + d3 f (x3 , k, h, p, q, α), being a, b, c, d1 , d2 , d3 positive constants in the range from 0 to 1, k is the slope of the saturated function series and multiplier factor in the saturated plateaus, α is a scaling factor for the breakpoints between slopes and plateaus, h is the shift of slope center, and p, q are positive integers. Let us consider the nonlinear functions f (x1 ), f (x2 ) and f (x3 ) are modeled by using PWL approximations as f (xj ; k, h, p, q, α) ⎧ (2q + 1)k, ⎪ ⎪ ⎪ k (x − ih) + 2ik, ⎪ j ⎪ ⎪ ⎨α = (2i + 1)k, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −(2p + 1)k,

if xj > qh + α if |xj − ih| ≤ α (2) −p ≤ i ≤ q if α < xj − ih < h − α −p ≤ i ≤ q − 1 if xj < −ph − α,

where j represents the state-variable associated with its nonlinear function and i is an integer number between −p and q. Figure 1 shows a generalized phase diagram for those saturated function series. Details about the frequency scaling and amplitude-level scaling of the PWL functions, and electronic circuit implementation of these chaotic systems have been reported in [3, 12]. In those references, a 4 × 4 × 4scroll chaotic attractor was obtained by simulating (1b) with a = b = c = d1 = d2 = d3 = 0.7, p = q = 1, k = 1, h = 2 and α = 0.1, as is shown in Fig. 2. Note that there are 4 scrolls in each direction of the phase space. Associated with this, we adopted the generalized Hamiltonian forms approach [5] given by ∂H ∂H + S(x) , x ∈ Rn, (3) x˙ = J (x) ∂x ∂x

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being y the output vector, C a constant matrix, and F (y) a nonlinear function related with the PWL functions of MSCS in (1). Then, if the synchronization error is defined as limt→∞ [y(t) − η(t)] = 0, the nonlinear state observer associated with (4) is given in next form: ∂H ξ˙ = J (y) ∂H ∂ξ + S(y) ∂ξ + F (y) + K(y − η),

ξ ∈ Rn,

Fig. 1 PWL approximation f (xj )

Fig. 2 3D 4-scroll chaotic attractor

in order to synchronize the 2D and 3D MSCS in (1a) and (1b), where x is the state vector, H describes an energy function which is globally positive-definite in R n , and J (x) and S(x) are squared matrices. According to Hamilton theory, the gradient vector ∂H ∂x is assumed that exists everywhere [5]. Furthermore, the energy structure of the system is represented by J (x) and S(x), these must satisfy J (x) + J T (x) = 0 and S(x) = S T (x). In this paper, a quadratic energy function given by H (x) = 12 [ax12 + bx22 + x32 ] is used. In particular, we are interested in a special class of generalized Hamiltonian systems with destabilizing vector fields and linear output given by ∂H ∂H + S(y) + F (y), x˙ = J (y) ∂x ∂x ∂H , y ∈ Rm, y=C ∂x

x ∈ Rn, (4)

η = C ∂H ∂ξ ,

η ∈ Rm,

(5)

with ξ as the estimated state vector of x, η as the estimated output in terms of ξ , and K a constant matrix known as the observer gain. Thereby, the Hamiltonian system in (4) and its nonlinear state observer in (5) can be represented as a master–slave system, respectively. In this way, we recast the 2D MSCS in (1a) by taking into account (4) and (5) as follows [15]: ⎡ ⎤ ⎡ 1⎤ 1 0 x˙1 2b 2 ∂H ⎣ x˙2 ⎦ = ⎣ − 1 0 1⎦ 2b ∂x x˙3 − 12 −1 0 ⎡ 1 1⎤ 0 2b − 2 ∂H 1 + ⎣ 2b 0 0 ⎦ ∂x − 12 0 −c ⎡ ⎤ − db2 f (x2 ) ⎦, +⎣ (6) 0 d1 f (x1 ) + d2 f (x2 ) ⎡ ⎤ ⎡ 1⎤ 1 0 ξ˙1 2b 2 ⎣ ξ˙2 ⎦ = ⎣ − 1 ⎦ ∂H 0 1 2b ∂ξ ξ˙3 − 12 −1 0 ⎡ 1 1⎤ 0 2b − 2 ∂H 1 + ⎣ 2b 0 0 ⎦ ∂ξ − 12 0 −c ⎡ ⎤ − db2 f (x2 ) ⎦ +⎣ 0 d1 f (x1 ) + d2 f (x2 ) ⎡ ⎤ k1 k4 + ⎣ k2 k5 ⎦ (y − η). (7) k3 k6 Here ∂H ∂x = ax1 + bx2 + x3 and C is the form of

d1 d2  0 a b C= , a bd22 0

∂H ∂ξ

= aξ1 + bξ2 + ξ3 . (8)

which includes the parameters related to nonlinear functions f (x1 ) and f (x2 ) with the purpose of controlling both nonlinear functions in the observer. In a

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Fig. 3 Block diagram of the synchronization schemes for 2D and 3D cases

similar way, the master–slave system for the 3D MSCS case is defined as follows [15]: ⎡ ⎤ ⎡ 1 1⎤ 0 x˙1 2b 2 ⎦ ∂H ⎣ x˙2 ⎦ = ⎣ − 1 0 1 2b ∂x x˙3 − 12 −1 0 ⎡ 1 1⎤ 0 2b − 2 ∂H 1 + ⎣ 2b 0 0 ⎦ ∂x − 12 0 −c ⎡ ⎤ − db2 f (x2 ) ⎦ , (9) +⎣ − db3 f (x3 ) d1 f (x1 ) + d2 f (x2 ) + d3 f (x3 ) ⎡ ⎤ ⎡ 1⎤ 1 0 ξ˙1 2b 2 ∂H ⎣ ξ˙2 ⎦ = ⎣ − 1 0 1⎦ 2b ∂ξ ξ˙3 − 12 −1 0 ⎡ 1 1⎤ 0 2b − 2 ∂H 1 + ⎣ 2b 0 0 ⎦ ∂ξ − 12 0 −c ⎡ ⎤ − db2 f (x2 ) ⎦ +⎣ − d3 f (x3 ) b

d1 f (x1 ) + d2 f (x2 ) + d3 f (x3 ) ⎤ ⎡ k1 k4 k7 + ⎣ k2 k5 k8 ⎦ (y − η). k3 k6 k9

available to be sent, as control signal, to the second one (slave). Note that the observability needs to be demonstrated by each row in matrix C. In addition, the state x of the master can be asymptotically estimated by the state ξ of the observer when the matrix in (12) is negative definite (sufficient and necessary condition) [5, 15], 2S − (KC + C T K T ) < 0.

(12)

As this approach is based on Hamiltonian forms, we obtained the synchronization in a structured way, we did not need compute the Lyapunov exponents to design the controller. Moreover, the initial conditions can be chosen without being close to the same equilibrium point at the attractor, and we can extend this method to other PWL function-based 2D and 3D MSCS [2]. Therefore, this approach is expected to be less complex than paper published in [14], which is based on selecting a vector Lyapunov function and hyperbolic functions.

3 Synchronization schemes for 2D and 3D MSCS (10)

Now C includes the parameters related to f (x1 ), f (x2 ), and f (x3 ); then, ⎤ ⎡ d1 d2 d3 a b (11) C = ⎣ 0 bd22 0 ⎦ . d3 0 0 b In this manner, the synchronization of 2D and 3D MSCS is obtained when the matrices S and C are observable (sufficient condition). This means that at least one of the state-variables of the master system must be

The aim of our work is study the synchronization of identical orders MSCS generating scrolls in 2D and 3D. According to previous section, we propose a chaotic synchronization scheme presented in Fig. 3. This scheme is based on a different number of statevariables as driving signals (control signal). Therefore, 2n − 1 combinations arise as possibilities to control the slave system. 3.1 Synchronization of 2D MSCS In order to be familiar with the synchronization process in 2D MSCS, two different synchronization

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Fig. 4 Numerical simulation results for the synchronization schemes of 2D MSCS

schemes are evaluated by computing its synchronization error. For 2D MSCS case, the state-variables x1 and x2 from the master system are chosen as driving signals. They are associated with the destabilization vector in the Hamiltonian forms given in (6), and projected to directly control the PWL functions f (x1 ) and f (x2 ) of the observer in (7). Hence, three combina-

tions are achieved. The first approach considers only one state-variable. This is accomplished by switching on/off either switch S1 or S2 in Fig. 3. Switch S3 is always turned off for 2D synchronization. Therefore, the synchronization is fulfill using either x1 [case I(a)] or x2 [case I(b)] as driving signal when (S1 = on, S2 = off ), and (S1 = off , S2 = on), respectively. The other

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Fig. 5 Synchronization in the phase space for 2D MSCS

approach to synchronize 2D MSCS is obtained when both switches (S1 , S2 ) are turned on. Consequently, the slave system is simultaneously controlled by both master system signals (x1 , x2 ) [case II(a)]. By proposing a set of gains that satisfies the relation in (12), we computed the error of those three cases as is shown in Fig. 4. The set of gains in (7) are: (k1 , k2 , k3 )T = (1, 2, 6) and (k4 , k5 , k6 )T = (0, 0, 0) when x1 drives; (k1 , k2 , k3 )T = (0, 0, 0) and (k4 , k5 , k6 )T = (1, 2, 6) when x2 drives; and, (k1 , k2 , k3 )T = (1, 2, 6) and (k4 , k5 , k6 )T = (1, 2, 6) when (x1 , x2 ) drive [15]. Note that each column vectors in matrix (7) is linked with a state-variable. Then, from k1 to k3 and from k4 to k6 correspond to x1 and x2 , respectively. In Fig. 4(a) we plot a comparison of the three possible cases for 2D synchronization considering the error between x1 − ξ1 . Similarly, Figs. 4(b), 4(c) compares the error of x2 − ξ2 and x3 − ξ3 . As is shown, the synchronization error approaches to zero when x1 or (x1 , x2 ) are driving. The difference between two schemes relies on the magnitude of the error overshooting. That is expected due to the destabilization vector in (7) depends on two PWL functions [f (x1 ) and f (x2 )], which are controlled by the two statevariables from master in case II(a). In addition, the synchronization error converges to a constant value when x2 is driving. As a consequence, the signals synchronize but they have an offset. By solving the term K(y − n) in (7), it can observe that only the error of x2 − ξ2 is sent as a feedback signal towards the slave system for case I(b). Contrary to that, the errors of x1 − ξ1 and x2 − ξ2 are sent for the other cases. Therefore, a 2D MSCS is synchronized by applying both approaches. In Fig. 5, we present the synchronization in the phase space for each state-variables using (x1 , x2 ) as driving signals. A straight line with its limits defined

Table 1 Values of the gain matrix for 3D MSCS Case

(k1 , k2 , k3 )T

(k4 , k5 , k6 )T

(k7 , k8 , k9 )T

I(a)

(1, 2, 6)

(0, 0, 0)

(0, 0, 0)

I(b)

(0, 0, 0)

(1, 2, 6)

(0, 0, 0)

I(c)

(0, 0, 0)

(0, 0, 0)

(1, 2, 6)

II(a)

(1, 2, 6)

(1, 2, 6)

(0, 0, 0)

II(b)

(1, 2, 6)

(0, 0, 0)

(1, 2, 6)

II(c)

(0, 0, 0)

(1, 2, 6)

(1, 2, 6)

III(a)

(1, 2, 6)

(1, 2, 6)

(1, 2, 6)

by the amplitude-level of the chaotic signals represents synchronized signals. 3.2 Synchronization of 3D MSCS In a similar way, we attain the synchronization schemes for 3D MSCS by switching (S1 , S2 , S3 ) in Fig. 3. As a result, we divide the approach in three cases related to the number of the state-variables used as control signals. Case I: One state-variable. (a) x1 driving (S1 = on, S2 = off , S3 = off ) (b) x2 driving (S1 = off , S2 = on, S3 = off ) (c) x3 driving (S1 = off , S2 = off , S3 = on) Case II: Two state-variables. (a) (x1 , x2 ) driving (S1 = on, S2 = on, S3 = off ) (b) (x1 , x3 ) driving (S1 = on, S2 = off , S3 = on) (c) (x2 , x3 ) driving (S1 = off , S2 = on, S3 = on) Case III: Three state-variables. (a) (x1 , x2 , x3 ) driving (S1 = on, S2 = on, S3 = on) Indeed, we computed the error for the seven possible combinations as shown in Fig. 6. Those simulations was carried out by choosing a set of values for the matrix K in (10) as given in Table 1 [15]. For

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Fig. 6 Numerical simulation results for the synchronization schemes of 3D MSCS

cases I(a), II(a), II(b), and III(a), the synchronization error is approaching to zero as shown in Fig. 6(a). Analogous to 2D MSCS synchronization, cases I(b), I(c), and II(c) have an offset denoted by constant error. Again, this offset depends on the feedback signals sent towards the slave system as can be proved by solving K(y − n) in (10). An important remark observed from this studied is the fact that the state-

variable x1 is extremely related to the synchronization error; i.e., when this variable is used as driving signal in association with other state-variables, the error is reduced to zero. This behavior can be in congruence with the PWL function f (x1 ) is the dominant nonlinear function in the 3D MSCS [3]. Finally, we analyzed how many signals can be used to reach the synchronization as well as the relationship be-

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Fig. 7 OpAmp-based synthesis for the synchronization schemes of 2D MSCS

tween the number of control signals and the synchronization error in contrast to the paper introduced in [13].

4 OpAmp based-circuit synthesis In this section, the master–slave system defined by (6) and (7) is synthesized with OpAmp’s in order to validate the approach introduced in previous section. In Fig. 7, we show the proposed electronic circuit. As well known, the second-order effects that degrade the behavior of the circuits are introduced by the OpAmp’s [17]. Besides, the tolerances of the

passive elements (typically about 10 % to 20 %) or matching effects at integrated circuit level are also related with a reduced performance of the circuits [18]. Consequently, deviations from ideal response obtained by mathematical computations are produced by those non-idealities. From practical point of view, this is a critical issue. Therefore, we use an electrical model for the OpAmp which includes second-order effects, to simulate the synchronization schemes in circuit simulator SPICE [19]. The circuit synthesis for 2D MSCS in (1a) is carried out taking into account the work reported in [3, 12]. Consequently, the 2D-4-scroll master and slave chaotic systems are sketched by a dotted

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Fig. 8 SPICE simulations of the synchronization error represented in phase diagrams for 2D MSCS. (a) (x1 ) drives and x1 − ξ1 and x2 − ξ2 errors as feedback signals; (b) (x2 ) drives

and x2 − ξ2 error as feedback signal; (c) (x1 , x2 ) drive and x1 − ξ1 and x2 − ξ2 errors as feedback signals

and long-dash box shown in Fig. 7. Regarding to the synchronization circuit (dash–dot–dot box), we design differential amplifier configurations with the purpose of establishing the feedback trajectories obtained by solving K(y − n) in (7). In this manner, each OpAmp gain stages is associated with an individual observer gain of matrix K. From the circuit, it is observed that the synchronization error between (x1 , ξ1 ) and

(x2 , ξ2 ) is applied as feedback signal to the slave system through the voltage-to-current converter resistors Ra. As expected, the nonlinear functions of the slave are directly controlled by the state-variables from the master. Note that the synchronization schemes for 2D case can be emulated by these circuit obeying the switch combinations in Fig. 3. For instance, the synchroniza-

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tion scheme when x1 drives is obtained if the states of switches are S1 = on and S2 = off . By choosing Rg1 = Rg4 = 10 kΩ, Rg2 = Rg5 = 20 kΩ, Rg3 = Rg6 = 60 kΩ, Ra = 10 Ω, the observer gains are adjusted to (k1 , k2 , k3 )T = (1, 2, 6) and (k4 , k5 , k6 )T = (1, 2, 6). The SPICE simulations for x1 , x2 and (x1 , x2 ) as driving signals are shown in Figs. 8(a), 8(b), 8(c), respectively. We observe a design trade-off between the synchronization error and the number of feedback signals. This means that deviations from ideal behavior shown in Fig. 5 are more evident for Fig. 8(b) due to we only sent one feedback signal. Contrary to that, the response in Fig. 8(c) is more close to ideal behavior. However, the form factor (active and passive devices count) increases as a function of the number of state-variables. By comparing the SPICE circuit simulations with those of the mathematical model, the synchronization error is acceptable considering that a real OpAmp model is included. In this way, we achieve the synchronization of PWL function-based 2D MSCS at electronic circuit level.

5 Conclusions We showed the synchronization of 2D- and 3D-4scroll chaotic attractors is possible by using the generalized Hamiltonian forms. The synchronization was achieved by controlling the PWL functions in the slave chaos generator with the state-variables from the master. Simulations results of the mathematical model reveal that the proposed synchronization scheme is suitable for using one, two, or three state-variables as driving signals. Associated with this, we introduced a OpAmp-based electronic circuit synthesized from the synchronization scheme. A design trade-off at circuit level between the synchronizing error and the number of feedback signals in 2D and 3D MSCS was established. SPICE simulations confirms the usefulness of the proposed approach in future practical applications. This approach has the potential to be used as a basic topology in full-duplex (send and receive data simultaneously) encrypted communications system. Acknowledgements The first author thanks National Council for Science and Technology (CONACyT/MÉXICO) for the support granted through project “Retención” number 174330.

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19. Technical documentation for TL081 OpAmp. Texas Instruments. http://www.ti.com/product/tl081. Accessed 03 June 2012