Inverse Lag Synchronization in Mutually Coupled ... - wseas.us

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Inverse Lag Synchronization in Mutually Coupled Nonlinear Circuits CHRISTOS K. VOLOS Department of Mathematics and Engineering Sciences Hellenic Army Academy Athens, GR16673 GREECE [email protected] IOANNIS M. KYPRIANIDIS, STAVROS G. STAVRINIDES, IOANNIS N. STOUBOULOS AND ANTONIOS N. ANAGNOSTOPOULOS Department of Physics Aristotle University of Thessaloniki Thessaloniki, GR54124 GREECE [email protected], [email protected], [email protected], [email protected] Abstract: - In this paper, the case of inverse lag synchronization phenomenon of two mutually coupled chaotic oscillators, was studied. This type of synchronization is observed for the first time in continuous dynamical systems. The system consists of two identical double scroll chaotic circuits coupled via a linear resistor. For a specific region of the coupling value, the system appears inverse lag synchronization. The experimental observation of this synchronization phenomenon is in full agreement with the numerical simulations. Key-Words: - Inverse lag synchronization, mutual coupling, double scroll circuits where x = (x1, x2, …, xn)T ∈ Rn is the vector of drive states and y = (y1, y2, …, yn)T ∈ Rn is the vector of response state, with positive N as time k → ∞. In this paper inverse lag synchronization, in the case of mutually coupled nonlinear continuous dynamical systems, is studied for the first time, both numerically and experimentally. These systems were two identical autonomous double-scroll circuits. The present paper is organized as follows: in Section 2, the proposed system and the analog equivalent circuit are presented; in Section 3, numerical and experimental results are presented, supporting conclusion that the observed type of synchronization is inverse lag synchronization; and finally, conclusion remarks appear in Section 4.

1 Introduction The notion of chaotic synchronization was introduced by Pecora and Carroll [1] in 1990. A wide range of research activity, in a variety of complex physical, chemical and biological systems has been stimulated, ever since [2-12]. In particular, the topic of synchronization of coupled chaotic electronic circuits has been studied intensively. Many interesting applications based on chaotic synchronization have came out of this research, such as broadband telecommunication systems [13], secure communications and cryptography [14-18]. Various types of synchronization have been reported, especially the last decade, namely complete synchronization [9, 10, 19-23], phase synchronization [24], generalized synchronization [25], anti-synchronization and anti-phase synchronization [26, 27], lag synchronization [28], anticipating synchronization [29] and projective synchronization [30]. Recently, a new type of synchronization in unidirectionally coupled dynamical maps, named inverse lag synchronization, has been reported [31]. In this case, the response states become synchronized to the driving states, in the form: y(k) = –x(k – N) ≡ –xN

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2 The Coupled Circuits A system of two identical nonlinear double scroll circuits, which are linearly coupled, via a resistor RC, implementing a mutual or bidirectional coupling, is shown in Fig. 1. The state equations (2) describe the coupling system. State variables x1,2, y1,2, and z1,2, represent the voltages at the outputs of the operational amplifiers numbered as “1”, “2” and “3”, respectively (Fig.1). The first three equations of system (2) describe the first of the two coupled

(1)

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identical double-scroll circuits, while the other three describe the second one.  x& 1 = y1 &  y1 = z1 + ξ ⋅ (y 2 − y1 ) z& 1 = −α ⋅ ( x1 + y1 + z1 ) + c ⋅ f (x1 )   x& 2 = y 2  y& 2 = z 2 + ξ ⋅ (y1 − y 2 )  z& 2 = −α ⋅ ( x 2 + y 2 + z 2 ) + c ⋅ f (x 2 )

where k = R 2

and it is realized by the circuit R3 inside the dotted frame in Fig. 1. This implementation demonstrates an i-v characteristic with two saturation plateaus at ±1 and an intermediate linear part with slope 1 . k R Coupling coefficient is ξ = and it is present RC at the equations of both circuits, since the coupling between them is bidirectional. The operational amplifiers “6” implement the functions –f(x1,2) respectively. The values of the circuit elements were: R = 20kΩ, R1 = 1kΩ, R2 = 14.3kΩ, R3 = 20.4kΩ, RX = 12.5kΩ, and C = 1nF. All the operational amplifiers were of the type LF411. The voltages of the positive and negative power supplies were set ±15V. Consequently, α = 0.5, c = 0.8 and k = 0.7. For these values, the Lyapunov exponents were calculated to be LE1 = 0.13271, LE2 = 0, and LE3 = -0.8541. Since, according to the numerical analysis, one of the Lyapunov exponents is positive, it is ensured that both the coupled circuits operate in a chaotic mode and exhibit a chaotic double-scroll attractor.

(2)

In the above state equations, α, and c are the circuit parameters and are defined as follows: α = ( R ⋅ C) , −1

c = (R x ⋅ C)

−1

(3)

3 Dynamic Behavior of the System The two mutually coupled chaotic double-scroll circuits described by the equation system (2), exhibit a variety of dynamical behaviors, including various types of synchronization and regions of desynchronization, depending on the coupling factor ξ and initial conditions. Experimental study, as well as numerical simulation of the system state equations (2), by employing a fourth order Runge-Kutta algorithm, illustrate the former dynamic behavior, as it is shown in the bifurcation diagram of x2 – x1 versus the coupling factor ξ, presented in Fig. 2a. This diagram was numerically produced by increasing the coupling factor ξ value, beginning from ξ = 0 (its corresponds to uncoupled systems) to ξ = 2 with step ∆ξ = 0.004, while the initial conditions in each iteration remain the same. The set of initial conditions for the two coupled systems were set to (x1, y1, z1) = (-0.5, -0.6, 0.1) and (x2, y2, z2) = (0.8, 0.3, -0.05). More analytically, a number of various regions, where the system displays chaotic, periodic or quasi-periodic behavior, appeared in the bifurcation diagram, for ξ > 1.045 (Fig. 2b). In this, work we focused in the quasi-periodic region that appears for 1.553 ≤ ξ < 1.758, since in

Fig. 1: The schematic of the double scroll circuits, mutually coupled via a linear resistor RC. The functions f(x1,2) used in system’s Equation (2) are saturation functions, defined by the following expression:  1, 1 f (x1,2 ) =  x1,2 , k −1,

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if x1,2 > k if − k ≤ x1,2 ≤ k

(4)

if x1,2 < −k

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this region the coupling system demonstrates the phenomenon of inverse lag synchronization. This means, that the signal x2 of the second circuit is opposite to the signal x1 of the first circuit with time lag τmin: x 2 (t) = − x1 (t − τ min ), τ min > 0

(5)

(a)

(a)

(b)

(b) Fig. 2: The bifurcation diagrams of x2 – x1 vs. ξ, for α = 0.5, c = 0.8 and k = 0.7 with initial values, (a) (x1, y1, z1) = (–0.5, –0.6, 0.1) and (x2, y2, z2) = (0.8, 0.3, –0.05) and (c) blow-up of region ξє[1, 1.9] for the case (a).

(c) Fig. 3: Coexisting dynamics of the system of Fig.1 for α = 0.5, c = 0.8, k = 0.7 and ξ = 1.57. (a) Simulation and (b) experimental phase portrait in y versus x plane of the first circuit and (c) simulation phase portrait of the second circuit.

Consequently, in this region the two coupled circuits have symmetrical quasi-periodic attractors as shown in Figs. 3a, and 3b, while the system also presents a quasi-periodic behavior, as shown in Figs. 4a and 4b.

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The quasi-periodic behavior of the coupling system is confirmed by the calculation of the Lyapunov exponents, LE1 = 0, LE2 = 0, LE3 = -0.3086, LE4 = -0.74650, LE5 = -0.92721 and

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LE6 = -4.26775. The existence of two zero Lyapunov exponents is a proof that the system has a quasi-periodic behavior. The time series of x1, x2 and –x2, presented in Figs. 5a and 5b confirm the inverse lag synchronization behavior of the system, since signals x1 and x2 are opposites with time lag. Thus, x1(t) + x2(t + τmin) ≈ 0 (Fig. 5c).

(a)

(a)

(b)

(b) Fig. 4: (a) Simulation and (b) experimental phase plots in x2 versus x1 plane, for α = 0.5, c = 0.8, k = 0.7 and ξ = 1.57.

In order to quantify lag synchronization, we used the similarity function, defined with respect to state variables, x, of the chaotic oscillators,  x 2 ' ( t + τ ) - x1 ( t ) 

S(τ) =

2

 ( x ( t ) )2 × ( x ' ( t ) ) 2  2  1 

1

(c) Fig 5: Time series of, (a) x1 (gray line) and x2 (dot line), (b) x1 (gray line) and –x2 (dot line) and (c) x1(t) + x2(t + τmin), for α = 0.5, c = 0.8, k = 0.7 and ξ = 1.57. Inverse lag synchronization is observed.

(6) 2

where, x2΄(t) = –x2(t).

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References: [1] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., Vol. 64, 1990, pp. 821-824. [2] A. C. J. Luo, A theory for synchronization of dynamical systems, Commun. Nonlinear Sci. Numer. Simul., Vol. 14, 2009, pp. 1901-1951. [3] A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A universal concept in nonlinear sciences, Cambridge University Press, 2003. [4] E. Moselkide, Y. Maistrenko and D. Postnov, Chaotic synchronization: applications to living systems, World Scientific, 2002. [5] S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares and C. S. Zhou, The synchronization of chaotic systems, Phys. Rep., Vol. 366, 2002, pp. 1-101. [6] N. Axmacher, F. Mormann, G. Fernández, C. E. Elger and J. Fell, Memory formation by neuronal synchronization, Brain Research Reviews, Vol. 52, 2006, pp. 170-182. [7] J. Wang, Y. Q. Che, S. S. Zhou and B. Deng, Unidirectional synchronization of HodgkinHuxley neurons exposed to ELF electric field, Chaos Solitons & Fractals, Vol. 39, 2009, pp. 1335-1345 [8] Q. Y. Wang, Q. S. Lu, G. R. Chen and D. H. Guo, Chaos synchronization of coupled neurons with gap junctions, Phys. Lett. A, Vol. 356, 2006, pp. 17-25. [9] I. M. Kyprianidis and I. N. Stouboulos, Synchronization of two resistively coupled nonautonomous and hyperchaotic oscillators, Chaos Solitons & Fractals, Vol. 17, 2003, pp. 317-325. [10] I. M. Kyprianidis and I. N. Stouboulos, Chaotic synchronization of three coupled oscillators with ring connection, Chaos Solitons & Fractals, Vol.17, 2003, pp. 327-336. [11] A. Caneco, C. Grácio and L. Rocha, Symbolic Dynamics and Chaotic Synchronization in Coupled Duffing Oscillators, J. Nonlinear Mathematical Physics, Vol. 15, 2008, pp. 102111. [12] I. M. Kyprianidis, Ch. K. Volos, S. G. Stavrinides, I. N. Stouboulos and A. N. Anagnostopoulos, Master-Slave double-scroll circuit incomplete synchronization, J. Engineering Science and Technology Review, Vol. 3, 2010, pp. 41-45. [13] S. G. Stavrinides, A. N. Anagnostopoulos, A. N. Miliou, A. Valaristos, L. Magafas, K. Kosmatopoulos and S. Papaioannou, Digital chaotic synchronized communication system,

The time lag value τmin = 1.1722ms, that came out of the calculation of the similarity function (Fig. 6), does not remain the same for the rest values of the coupling factor ξ, for which the system appears inverse lag synchronization. Let Smin be the minimum value of S(τ) and let τmin be the amount of lag when Smin is achieved. Lag synchronization between the two circuits is characterized by the conditions Smin = 0 and τmin ≠ 0.

Fig 6: The similarity function (S) versus time (t), for α = 0.5, c = 0.8, k = 0.7 and ξ = 1.57. S(τmin) = 0 means lag with time shift of τmin = 1.1722ms.

4 Conclusion In the present paper a new type of synchronization was observed. This kind of synchronization is called inverse lag synchronization because the state variable of the first system and the opposite of the state variable of the second system are synchronized with a time lag τ, with respect to each other. This phenomenon was observed for the first time in coupled continuous dynamical systems. The coupling system consists of two identical double-scroll circuits which operate in a chaotic mode for the chosen set of parameters and initial conditions. For a certain range of the coupling factor the inverse lag synchronization, which was first detected numerically, was further confirmed by the experimental results. It is very important to notice that the inverse lag synchronization was found when the oscillators had quasi-periodic attractors. Finally, it would be interesting to investigate the whether the inverse lag synchronization is observed in other coupled dynamical systems, based on nonlinear circuits and to explore (or even prove) the relation between quasi-periodic behavior and inverse lag synchronization.

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