Unidirectional Synchronization of Jerk Circuit. and it s Uses in Secure Communication System

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1 Advanced Studies in Theoretical Physics Vol. 9, 05, no., HIKARI Ltd, Unidirectional Synchronization of Jerk Circuit and it s Uses in Secure Communication System Aceng Sambas,, Mustafa Mamat *, Mada Sanjaya W.S.,, Zabidin Salleh and Fatma Susilawati Mohamad 5,5 Faculty of Informatics and Computing Universiti Sultan Zainal Abidin, Kuala Terengganu, Malaysia Department of Physics, Universitas Islam Negeri Sunan Gunung Djati Bandung, Indonesia Bolabot Techno Robotic Institute Sanjaya Star Group Corp, Bandung, Indonesia Pusat Pengajian Informatik dan Matematik Gunaan Universiti Malaysia Terengganu, K. Terengganu, Malaysia Copyright 05 Aceng Sambas et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Information is masked by chaotic signals at the transmitter, and then sent to the receiver by the public channel. Finally the encrypted signals are decrypted at the receiver. In this scheme, the key issue is that the two identical chaos generators in the transmitter end and the receiver end need to be synchronized. In this paper, in order to show some interesting phenomena of three dimensional autonomous ordinary differential equations, the chaotic behavior as a function of a variable control parameter, has been studied. The initial study in this paper is to analyze the phase portraits, the Lyapunov exponents, the Poincaré maps and the bifurcation diagrams. Moreover, some appropriate comparisons are made to contrast some of the existing results. Finally, the effectiveness of the unidirectional coupling scheme between two identical Jerk circuits in a secure

2 56 Aceng Sambas et al. communication system is presented in details. Finally, the simulation and the experimental results are shown to demonstrate that the proposed method is correct and feasible Keywords: Jerk circuit, bifurcation diagram, secure communication system. Introduction Chaos is a term used to describe behavior that is seemingly random, but has an underlying mathematical order to it. Chaos is very common in nature, but is often mistaken for random behavior. Chaos can occur only in nonlinear systems and is characterized by a breakdown of predictability known as sensitive dependence on initial conditions which is the most important distinguishing feature of chaos. The Characteristics of chaotic systems mentioned above, can be used many practical application, such as in communication system [-], bits generators [], robotics [], biology [5], psychology [6], ecology [7-] and economy [9]. The paper by Pecora and Carroll [0] considers how identical or almost identical chaotic systems can be synchronized by a chaotic reference signal so that the two systems follow the same chaotic orbit. R. He and P.G. Vaidya [] showed how this synchronization can be understood in many representative cases by the existence of a global Lyapunov function of the difference signals. It is another milestone in chaos research. Subsequently, many synchronization methods were presented such as state feedback synchronization [ ], impulsive synchronization [5 ]. Chaos theory was applied to many engineering areas. Specially, secure communication based chaos rapidly developed [9 0]. In 99, Cuomo and Oppenheim presented the first scheme of a communications device made by two identical Lorenz oscillators []. In 997, Kolumban, Kennedy and Chua realized digital communications based on chaos synchronization []. After that, symmetric cryptography based on chaos grew up fast. In 00, Kocarev and Tasev presented a public-key cryptography based on Chebyshev chaotic maps []. The paper is organized as follows. In section, the details of the proposed autonomous Jerk circuit s simulation using MATLAB 00 and MultiSIM 0.0, are presented. In Section, the unidirectional coupling method is applied in order to synchronize two identical autonomous Jerk circuits. The chaotic masking communication scheme by using the above mentioned synchronization technique is presented in Section. Finally, in Section 5, the concluding remarks are given.. Jerk Circuit Sprott found the functional form of three-dimensional dynamical systems which exhibit chaos. Jerk equation has a simple nonlinear function, which can be implemented with an autonomous electronic circuit []. Furthermore, Alpana Pandey modifies the system of equations Jerk into a system of simple quadratic

3 Unidirectional synchronization of Jerk circuit 57 equations. In this work, the Jerk circuit, which was firstly presented by Alpana Pandey in 0 [5-6], is used. This is a three-dimensional autonomous nonlinear system that is described by the following system of ordinary differential equations: x y y z z x y az bx () The new system has one quadratic term and two positive real constants a and b. The parameters and initial conditions of the Jerk system () are chosen as: a = 0.5, b = 0.5 and (x0, y0, z0) = (0.00, 0.0, 0.), so that the system shows the expected chaotic behavior. 0.6 Phase Space Jerk Circuit signal y signal x (a) 0.6 Phase Space Jerk Circuit signal z signal y (b)

4 5 Aceng Sambas et al. 0.6 Phase Space Jerk Circuit signal z signal x Figure : Numerical simulation results using MATLAB 00, for a = 0.5, b = 0.5, (a) x-y plane, (b) y-z plane, (c) x-z plane.. Numerical Simulations In this section, we present numerical simulation to illustrate the dynamical behavior of Jerk circuit from system (). For numerical simulation of chaotic system defined by a set of differential equation such as Jerk circuit, different integration techniques can be used in simulation tools. In the MATLAB 00 numerical simulation, ODE5 solver yielding a fourth-order Runge-Kutta integration solution has been used. Figure.(a)-(c) show the projections of the phase space orbit on to the x y plane, the y z plane and the x z plane, respectively. As it is shown, for the chosen set of parameters and initial conditions, the Jerk system presents chaotic attractors of Rössler type. Also, it is known from the nonlinear theory, that the spectrum of Lyapunov exponents provides additional useful information about system s behavior. In a three dimensional system, like this, there has been three Lyapunov exponents (,, ). In more details, for a D continuous dissipative system the values of the Lyapunov exponents are useful for distinguishing among the various types of orbits. So, the possible spectra of attractors, of this class of dynamical systems, can be classified in four groups, based on Lyapunov exponents [7-9]. For a fixed point, 0,,, 0, For a limit point 0 0 For a two-torus 0 0 For a strange attractor 0 0 So, in Figure (a) and (b) the dynamics of the proposed system s Lyapunov exponents for the variation of the parameter a , is shown. For 0.9 (c)

5 Unidirectional synchronization of Jerk circuit 59 c 0.57 a strange attractor is displayed as the system has one positive Lyapunov exponent, while for values of 0.57 c 0.65 is a transition to limit point behavior as the system has two negative Lyapunov exponents. (a) (b) Figure : Nonlinear dynamics of system () for specific values set b=0.5. (a) Lyapunov exponents versus the parameter control a (b) Lyapunov exponents versus the parameter control a , with MATLAB 00. Analog Circuit Simulations A simple electronic is designed that can be used to study chaotic phenomena. The circuit employs simple electronic elements, such as resistors, capacitors, multiplier and operational amplifiers. In Figure 5, the voltages of C, C, C are

6 550 Aceng Sambas et al. used as x, y and z, respectively. The nonlinear term of system () are implemented with the analog multiplier. The corresponding circuit equation can be described as: x C R y C R z C R y z 7 x C R y C R 9 z 0C R 0 x () We choose R = R = R = R = R5 = R6 = R7 = R = 00 kω, R9 = 00 kω, R0 = 0 kω. C = C = C C = nf. The circuit has three integrators in a feedback loop and a multiplier, Op-amp TL0CD and IC AD6 is used as a multiplier. The supplies of all active devices ± 9 V. With MultiSIM 0.0, we obtain the experiment observations of system () as Figure 6. As compared with Figures (a)-(c) good qualitative agreement between the numerical simulation and the MultiSIM 0.0 results of the Jerk circuit is confirmed. The parameter variable a of system () is changed by adjusting the resistor R9, and obeys the following relations: a () C R 9 Y X A V/V 0 V R7 R R9 00kΩ R0 0kΩ C nf U5A TL0CD Zsignal R6 R5 UA TL0CD R C nf UA TL0CD Ysignal R R UA TL0CD R C nf UA VCC - XSignal TL0CD VCC Figure 5: Schematic of the proposed Jerk circuit using MultiSIM 0.0.

7 Unidirectional synchronization of Jerk circuit 55 (a) (b) (c) Figure 6: Various projections of the chaotic attractor using MultiSIM in (a) x-y plane, (b) x-z plane and (c) y-z plane.. Unidirectional Chaotic Synchronization of Coupled Jerk Circuits Generally, there are various methods of coupling between coupled nonlinear systems available in the literature. However, two are the most interesting. In the first method due to Pecora and Caroll, a stable subsystem of a chaotic system could be synchronized with a separate chaotic system under certain suitable conditions [0]. In unidirectional synchronization, the evolution of the first system (the drive) is unaltered by the coupling, the second system (the response) is then constrained to copy the dynamics of the first. So, in Figure 7 the simulation results of this scheme which confirms the case of chaotic synchronization are shown. Finally, in Figure the implementation of the unidirectional synchronization scheme of coupled Jerk circuits, with MultiSIM 0.0, is displayed. (a) (b) Figure 7: Phase portrait of x vs. x, with MATLAB 00. (a) 0 mω. (b) 00 mω.

8 55 Aceng Sambas et al. A Y X V/V 0 V R7 R R9 00kΩ R0 0kΩ C nf U5A Zsignal TL0CD R6 R5 UA TL0CD R C nf UA Ysignal TL0CD R R UA R TL0CD C nf VCC - XSignal UA TL0CD VCC VCC - UA Y X A R7 R R9 00kΩ R0 0kΩ V/V 0 V VCC C6 nf U0A Zsignal TL0CD R6 R5 U9A TL0CD R C5 nf UA Ysignal TL0CD R R U7A VCC - R TL0CD C nf XSignal U6A TL0CD R 0mΩ TL0CD VCC Figure : Schematic of the unidirectional coupled Jerk circuits. Synchronization numerically appears for a coupling strength R 0 mω as shown in Figure 7(a). For different initial condition, if the resistance coupling strength R > 0 mω, the synchronization cannot occur as shown in Figure7 (b). The synchronization occurs when R 0 mω with errors ex x x 0 which implies the complete synchronization for this resistance coupling strength as shown in Figure 7(a).. Application to Secure Communication Systems In chaos-based secure communication scheme, chaos synchronization is the critical issue, because two identical chaos generators, in the transmitter and the receiver end, need to be synchronized. Information signal is added to the chaotic signal at transmitter and at receiver the masking signal is regenerated and subtracted from the receiver signal. For synchronization of transmitter and receiver, unidirectional synchronization method of identical coupled Jerk circuits, is used. The sinusoidal wave signal of amplitude V and frequency khz is added to the generated chaotic signal x and the S(t) = x + i(t) is fed into the receiver. The chaotic signal x is regenerated allowing a single subtraction to retrieve the transmitted signal, [x + i(t)] - xr= i(t), if x = xr. Figure 9 shows the MultiSIM 0.0 simulation results for the proposed masking signal communication scheme. Figure 0 shows the circuit schematic for implementing the Jerk s circuit chaotic masking communication system.

9 Unidirectional synchronization of Jerk circuit 55 (a) (b) (c) Figure 9: MultiSIM 0.0 outputs of Jerk circuit masking communication systems, for A = V and f = KHz: (a) Information signal, (b) Chaotic masking transmitted signal, (c) Retrieved signal. A Y X V/V 0 V R7 R R9 00kΩ R0 0kΩ C nf VCC U5A Zsignal R6 TL0CD R5 UA TL0CD R C nf UA Ysignal TL0CD R R UA R TL0CD C nf XSignal UA VCC - TL0CD VCC - V Vrms khz 0 R kω R5 kω U OPAMP_T_VIRTUAL R R7 kω U OPAMP_T_VIRTUAL R9 5kΩ kω kω U OPAMP_T_VIRTUAL R kω R0 Y X A R7 R R9 00kΩ R0 0kΩ V/V 0 V VCC C6 nf U0A Zsignal R6 TL0CD R5 U9A TL0CD R C5 nf UA Ysignal TL0CD R R U7A VCC - R TL0CD C nf XSignal U6A TL0CD UA R 0mΩ TL0CD VCC U6 OPAMP_T_VIRTUAL R kω R6 kω R kω R7 kω U5 OPAMP_T_VIRTUAL Figure 0: Jerk s circuit masking communication system.

10 55 Aceng Sambas et al. 5. Conclusion In this paper, the chaotic synchronization in coupled identical Jerk circuits has been investigated by implementing unidirectional synchronization technique. We have demonstrated with simulations that chaotic circuits can be synchronized and used in a secure communication scheme. Chaos synchronization and chaos masking were realized using MultiSIM 0.0 program. Furthermore, some comparisons are made with some existing results. Finally, the simulation results demonstrate the effectiveness of the proposed scheme. References [] A. Sambas., M. Sanjaya W.S and Halimatussadiyah, Unidirectional Chaotic Synchronization of Rossler Circuit and Its Application for Secure Communication, WSEAS Transactions on System, 9 (0), no., [] A. Sambas., M. Sanjaya W.S., M. Mamat and Halimatussadiyah, Design and Analysis Bidirectional Chaotic Synchronization of Rossler Circuit and Its Application for Secure Communication, Applied Mathematical Sciences, 7 (0), no., -. [] Ch. K. Volos, I. M. Kyprianidis and I. N. Stouboulos, Motion Control of Robots Using a Chaotic Truly Random Bits Generator, Journal of Engineering Science and Technology Review, 5 (0), no., 6 -. [] Ch. K. Volos, N. Doukas, I. M. Kyprianidis, I. N. Stouboulus, T. G. Kostis, (0), Chaotic Autonomous Mobile Robot for Military Missions, Recent Advances in Telecommunications and Circuit Design. WSEAS 7th International Conference on Communications Rhodes Island, Greece July 6-9, , ISBN: [5] J. L. Hindmarsh, and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Philosophical Transaction of the Royal Society B: Biological Sciences, (9), [6] J. C. Sprott, Dynamical Models of Love, Nonlinear Dyn. Psych. Life Sci., (00), 0 -. [7] M. Sanjaya W. S, I. Mohd, M. Mamat and Z. Salleh, Mathematical Model of Three Species Food Chain Interaction with Mixed Functional Response, International Journal of Modern Physics: Conference Series, 9 (0),

11 Unidirectional synchronization of Jerk circuit 555 [] M. Sanjaya W. S, M. Mamat, Z. Salleh, and I. Mohd, Bidirectional Chaotic Synchronization of Hindmarsh-Rose Neuron Model, Applied Mathematical Sciences, 5 (0), no. 5, [9] Ch. K. Volos, I. M. Kyprianidis, and I. N. Stouboulos, Synchronization Phenomena in Coupled Nonlinear Systems Applied in Economic Cycles, WSEAS Trans. Systems, (0), no., [0] L. M. Pecora and T. L. Carroll, Synchronization in Chaotic Systems, Physical Review Letters, 6 (990), [] R. He and P. G. Vaidya, Analysis and Synthesis of Synchronous Periodic andchaotic Systems, Physical Review A, 6 (99), no., [] F. Liu, Y. Ren, X. Shan, and Z. Qiu, A linear feedback synchronization theorem for a class of chaotic systems, Chaos, Solitons and Fractals, (00), [] Y. Wang, Z.-H. Guan, and H. O. Wang, Feedback and adaptive control for the synchronization of Chen system via a single variable, Physics Letters A, (00), no., [] Y. Hong, H. Qin, and G. Chen, Adaptive synchronization of chaotic systems via state or output feedback control, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, (00), [5] T. Yang, L.-B. Yang, and C.-M. Yang, Impulsive synchronization of lorenz systems, Physics Letters A, 6 (997), no. 6, [6] J. Sun, Y. Zhang, and Q. Wu, Impulsive control for the stabilization and synchronization of Lorenz systems, Physics Letters A, 9 (00), no., [7] W. Xie, C. Wen, and Z. Li, Impulsive control for the stabilization and synchronization of Lorenz systems, Physics Letters A, 75 (000), no., [] A. I. Panas, T. Yang, and L. O. Chua, Experimental results of impulsive synchronization between two Chua s circuits, International Journal of Bifurcation and Chaos, (99), no.,

12 556 Aceng Sambas et al. [9] A. Sambas, Mada Sanjaya W. S, Mustafa Mamat, N. V Karadimas and O. Tacha (0), Numerical Simulations in Jerk Circuit and It s Application in a Secure Communication System. Recent Advances in Telecommunications and Circuit Design. WSEAS 7th International Conference on Communications Rhodes Island, Greece July 6-9, , ISBN: [0] A. Sambas, Mada Sanjaya W. S, Mustafa Mamat and O. Tacha. 0, Design and Numerical Simulation of Unidirectional Chaotic Synchronization and Its Application in Secure Communication System. Recent Advances in Nonlinear Circuits: Theory and Applications. Journal of Engineering Science and Technology Review. 6(), ISSN: [] K. M. Cuomo and A. V. Oppenheim, Circuit implementation of synchronized chaos with applications to communications, Phys. Rev. Lett., 7 (99), no., [] G. Kolumban, M. Kennedy, and L. Chua, The role of synchronization in digital communications using chaos. i. fundamentals of digital communications. Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, (997), no. 0, [] L. Kocarev and Z. Tasev, Public-key encryption based on chebyshev maps, Circuits and System. Proceedings of the 00 International Symposium on, (00), -. [] J. C. Sprott, Simple Chaotic Systems and Circuits, Am. J. Phys., 6 (000), [5] A. Pandey, R.K. Baghel and R. P. Singh, An Autonomous Chaotic Circuit for Wideband Secure Communication. International Journal of Engineering, Business and Enterprise Applications, (0), no., - 7. [6] A. Pandey, R.K. Baghel and R. P. Singh, Analysis and Circuit Realization of a New Autonomous Chaotic System, International Journal of Electronics and Communication Engineering, 5 (0), no., [7] C. Li and J. C. Sprott, Finding coexisting attractors using amplitude control, Nonlinear Dynamics, 7 (0), no., [] A. Wolf,. Quantity Chaos with Lyapunov Exponents, Chaos Princeton University Press, (96),

13 Unidirectional synchronization of Jerk circuit 557 [9] F. Han, Multi-Scroll Chaos Generation Via Linear Systems and Hysteresis Function Series, PhD thesis, Royal Melbourne Institute of Technology University, Australia: 00. Received: March 0, 05; Published: August 7, 05

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