IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS Numerical Simulation Bidirectional Chaotic Snchronization of Spiegel-Moore Circuit and Its Application for Secure Communication To cite this article: W S M Sanjaa et al 07 IOP Conf. Ser.: Mater. Sci. Eng. 80 0066 Related content - Bi-directional Secure Communication Based on Discrete Chaotic Snchronization Long Min, Qiu Shui-Sheng and Peng Fei - Complete snchronization of doubledelaed Rössler sstems with uncertain parameters Sang Jin-Yu, Yang Ji and Yue Li-Juan - Snchronizing modified van der Pol Duffing oscillators G J Fodjouong, H B Fotsin and P Woafo View the article online for updates and enhancements. This content was downloaded from IP address 7.44.96.8 on 05/0/08 at 8:6
st Annual Applied Science and Engineering Conference IOP Conf. Series: Materials Science and Engineering 80 (07) 0066 doi:0.088/757-899x/80//0066 International Conference on Recent Trends in Phsics 06 (ICRTP06) Journal of Phsics: Conference Series 755 (06) 000 doi:0.088/74-6596/755//000 Numerical Simulation Bidirectional Chaotic Snchronization of Spiegel-Moore Circuit and Its Application for Secure Communication W S M Sanjaa,4*, D Anggraeni,4, R Dena,, N Ismail Department of Phsics, Universitas Islam Negeri Sunan Gunung Djati, Bandung, Indonesia Department of Phsics Education, Universitas Islam Negeri Sunan Gunung Djati, Bandung, Indonesia Department of Electrical Engineering, Universitas Islam Negeri Sunan Gunung Djati, Bandung, Indonesia 4 Bolabot Techno Robotic Institute, CV. Sanjaa Star Group, Bandung, Indonesia *madasws@gmail.com Abstract. Spiegel-Moore is a dnamical chaotic sstem which shows irregular variabilit in the luminosit of stars. In this paper present the performed the design and numerical simulation of the snchronization Spiegel-Moore circuit and applied to securit sstem for communication. The initial stud in this paper is to analze the eigenvalue structures, various attractors, Bifurcation diagram, and Lapunov exponent analsis. We have studied the dnamic behavior of the sstem in the case of the bidirectional coupling via a linear resistor. Both experimental and simulation results have shown that chaotic snchronization is possible. Finall, the effectiveness of the bidirectional coupling scheme between two identical Spiegel- Moore circuits in a secure communication sstem is presented in details. Integration of theoretical electronic circuit, the numerical simulation b using MATLAB, as well as the implementation of circuit simulations b using Multisim has been performed in this stud.. Introduction Chaos explain the behavior of certain dnamical nonlinear sstems, i.e., sstems which state variables evolve with time, exhibiting complex dnamics that are highl sensitive on initial conditions. Sensitivit to initial conditions of chaotic sstems is familiarl known as the butterfl effect. Small changes in an initial state will make a larger difference in the behavior of the sstem at future states (e.g., []-[]). Chaos behavior have been discovered in phsical [], ecolog [4], neuroscience [5], chemical reaction [6], pscholog [7], and economics [8]. In man implementation of engineering and computer science such as robotic sstem [9], text encrption [0], image encrption [], image encrption [], speech encrption [] and other. One of most important engineering implementation Content from this work ma be used under the terms of the Creative Commons Attribution.0 licence. An further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence b Ltd
st Annual Applied Science and Engineering Conference IOP Conf. Series: Materials Science and Engineering 80 (07) 0066 doi:0.088/757-899x/80//0066 is secure communication because of the properties of random behaviors and sensitivit to initial conditions of chaotic sstems (e.g., [4]-[7]). Christiaan Huijgens (665) the Dutch scientist noted the snchronizing behavior of pendulum clocks. Man scientists have been investigate the snchronization of several dnamical sstems. When Pecora and Carroll published their observations of snchronization in unidirectionall coupled chaotic sstems, snchronization of chaotic oscillators in particular became popular [8]. Man researchers simulated the chaos can be snchronized and applied to secure communication schemes (e.g., [4]- [0]). Generall, this research focus on the development of chaos and non-linear dnamical sstem behavior in chaotic electrical oscillator. We investigate and analze some basic properties to stud the non-linear dnamics and chaotic behavior, such as eigenvalues structure, phase plane, Lapunov exponent, and diagram bifurcation analsis, while the analsis of the snchronization in the case of bidirectional coupling between two identical generated chaotic sstems. Moreover, some appropriate comparisons are made to contrast some of the existing results. And presented the effectiveness of the bidirectional coupling between two identical Spiegel-Moore [] chaotic circuits in a secure communication sstem. The paper is organized as follows. In section, the details of the proposed autonomous Spiegel- Moore circuit s simulation using MATLAB. In section, build an analog circuit using Multisim. In Section 4, the bidirectional coupling method is applied in order to snchronize two identical autonomous Spiegel-Moore chaotic circuits. The chaotic masking communication scheme b using the above mentioned snchronization technique is presented in Section 5. Finall, in Section 6, the concluding remarks are given.. Mathematical Model of Spiegel-Moore Circuit Moore-Spiegel (966) found a model the irregular variabilit in the luminosit of stars []. This is a three-dimensional autonomous nonlinear sstem that is described b the following sstem of ordinar differential equations: x z () z z a x bx The sstem has one cubic non-linerities term and two positive real constants a and b. The parameters and initial conditions of the Moore-Spiegel sstem () are chosen as: a = 9, b = 5 and (x 0; 0; z 0) = (, 7, 4), so that the sstem shows the expected chaotic behavior... Equilibrium Point Analsis Spiegel-Moore sstem has one equilibrium points E 0 (0, 0, 0). The dnamical behavior of equilibrium points can be studied b computing the eigenvalues of the Jacobian matrix J of sstem () where: 0 0 J ( x,, z) 0 0 () x b a x For equilibrium points E 0 (0, 0, 0) and a = 9, b = 5, the eigenvalues are obtained b solving the det J 0 which is: characteristic equation, 9 5 0 () Yielding eigenvalues of λ =.65554, λ = -.7507679 8.66054040 i, λ = 0.6648555 + 8.66054040 i. The above eigenvalues show that the sstem has unstable spiral behavior. In this case, the phenomenon of chaos is presented.
st Annual Applied Science and Engineering Conference IOP Conf. Series: Materials Science and Engineering 80 (07) 0066 doi:0.088/757-899x/80//0066.. Numerical Simulation In this section, software MATLAB used for numerical simulations. To solve the sstem of differential equations () used the fourth-order Runge-Kutta method. Figure -(c) show the projections of the phase space orbit on to the x plane, the z plane and the xz plane, respectivel. As it is shown, for the chosen set of parameters and initial conditions, the Spiegel-Moore sstem presents chaotic attractors. (c) Figure. Numerical simulation results using MATLAB, for a = 9, b = 5, in x- plane, -z plane, (c) x-z plane... Lapunov Exponent Analsis Three Lapunov exponents (λ, λ, λ ) it is also known from the theor of nonlinear dnamics that for a three dimensional sstem (). In more details, for a D continuous dissipative sstem the values of the Lapunov exponents are useful for distinguishing among the various tpes of orbits. So, the possible spectra of attractors, of this class of dnamical sstems, can be classified in four groups, based on Lapunov exponents (e.g., [6], []). (λ, λ, λ ) (,, ): a fixed point (λ, λ, λ ) (0,, ): a limit point (λ, λ, λ ) (0, 0, ): a -torus (λ, λ, λ ) (+, 0, ): a strange attractor Therefore, the last configuration just possible third-order chaotic sstem. In this case, a positive Lapunov exponent reflects a direction of stretching and folding and determines chaos in the sstem. So, in figure and the dnamics of the proposed sstem s Lapunov exponents for the variation of the parameter a 6 0 and b 9. For 8 a 0 and 4. b 5.7 a strange attractor is displaed as the sstem has one positive Lapunov exponent, while for values of 6 a < 8 and 4. > b > 5.7 is a transition to limit point behavior as the sstem has two negative Lapunov exponents.
st Annual Applied Science and Engineering Conference IOP Conf. Series: Materials Science and Engineering 80 (07) 0066 doi:0.088/757-899x/80//0066 Figure. Nonlinear dnamics of sstem () Lapunov exponents versus the parameter control a at b = 5, Lapunov exponents versus the parameter control b at a = 9..4. Bifurcation Diagram Analsis Bifurcation indicate a situation in which the solutions of a nonlinear sstem of differential equations alter their character with a change of a parameter on which the solutions depend [6]. Bifurcation theor studies these changes (e.g. dependence of their stabilit on the parameter, appearance and disappearance of the stationar points, etc.). Figure. Nonlinear dnamics of sstem () Bifurcation diagram of z vs the control parameter a, Bifurcation diagram of z vs the control parameter b. Spiegel-Moore circuit of figure and, was written to result the bifurcation diagrams b MATLAB program. In this diagram a possible bifurcation diagram for sstem (), in the range of 6 a 0. For the chosen value of 8 a < 0 and 4. b 5.7 the sstem displas the expected chaotic behavior. Also, for 6 a < 8 and 4. > b >5.7, a reverse period doubling route is presented.. Analog Circuit Simulation using Multisim A simple electronic circuit was designed using Multisim software, and that can be used to stud chaotic phenomena. The circuit emplos simple electronic elements, such as resistors, capacitors, multiplier and operational amplifiers. Figure 4, the voltages of C, C, C are used as x, and z, respectivel. The nonlinear term of sstem () are implemented with the analog multiplier. The corresponding circuit equation can be described as: 4
st Annual Applied Science and Engineering Conference IOP Conf. Series: Materials Science and Engineering 80 (07) 0066 doi:0.088/757-899x/80//0066 x z 6 7 z z 8 00 9 x We assume R = R = R = R 4 = R 5 = R 6 = R 7 = 00 kω, R 8=. kω, R 9 = kω, R 0 = 0 kω. C = C = C = nf. The circuit has three integrators (b using Op-amp TL08CD) in a feedback loop and a multiplier (IC AD6). The supplies of all active devices are ± 5 V. With Multisim, we obtain the simulation results of sstem () as shown in figure 5. Compared with figure, a good qualitative agreement between the numerical simulation and the Multisim results of the Spiegel-Moore circuit is confirmed. The parameter variable a of sstem () is changed b adjusting the resistor R 8, and obes the following relation: a (5) 8 0 x (4) Figure 4. Schematic of the proposed Spiegel-Moore circuit b using Multisim. Figure 5. Various projections of the chaotic attractor using Multisim, for a = 9, b = 5 x- plane x-z plane (c) -z plane. 5
st Annual Applied Science and Engineering Conference IOP Conf. Series: Materials Science and Engineering 80 (07) 0066 doi:0.088/757-899x/80//0066 4. Bidirectional Chaotic Snchronization 4. Mathematical Model of Bidirectional Coupling The case of bidirectional coupling two sstems interact and coupled with each other creating a mutual snchronization. Following bidirectional coupling configuration (e.g., [4]-[0]), described: x z x z z z z z g g c a c a x x The coupling coefficient g c is present in the equations of both sstems, since the coupling between them is mutual. Numerical simulations of sstem (6) using the 4th-order Runge-Kutta method, are used to describe the dnamics of chaotic snchronization of bidirectionall coupled Spiegel-Moore circuits. bx bx (6) (c) (d) Figure 6. Phase portrait of x vs x and error x -x in the case of bidirectionall coupled Spiegel-Moore circuits, for g c =.47 (full snchronization) and (c) g c = 0.5 (full desnchronization), for a = 9, a = 8.985, b = 5. In bidirectional coupling, the coupled sstems are connected in such a wa that the mutuall influence each other s behavior. Snchronization numericall appears for a coupling factor g c.47 as shown in figure 6 -, with error e x x x 0, which implies the complete snchronization. 4. Analog Circuit Simulation using Multisim Snchronization of chaotic motions among the coupled dnamical sstems is an important generalization for the phenomenon of snchronization of linear sstem, which is useful and indispensable in communications. Simulation results show that two sstems snchronize well. Figure 7 shows the circuit schematic for implementing the bidirectional snchronization of coupled Spiegel- 6
st Annual Applied Science and Engineering Conference IOP Conf. Series: Materials Science and Engineering 80 (07) 0066 doi:0.088/757-899x/80//0066 Moore sstems. Chaotic snchronization appears for a coupling strength R 680 mω, as shown in figure 8. For different initial conditions or resistance coupling strength R > 680 mω, the snchronization cannot occur as shown in figure 8. Figure 7. Bidirectional chaotic snchronization Spiegel-Moore circuit b using Multisim Figure 8. Snchronization phase portrait of x vs x, for R = 680 mω and R = Ω, with Multisim. 5. Applications in Secure Communication Sstem Caused the fact that output signal can recover input signal, it can be implement secure communication for a chaotic sstem. The attendance of the chaotic signal between the transmitter and receiver has proposed the use of chaos in secure communication sstems. The sstem design depends as we described on the self snchronization propert of the Spiegel-Moore circuits. Transmitter and receiver sstems are identical except for their control value a as equation (5), in which the transmitter sstem R 8 is. kohm and the receiver sstem R 8 is. kohm as shown in figure 7 and 9. In this masking scheme, a low-level message signal is include to the snchronizing driving chaotic signal in order to regenerate a clean driving signal at the receiver. So, the message has been perfectl recovered b using the signal masking approach through snchronization in the Spiegel-Moore circuits. Sinusoidal wave is added to the generated chaotic x signal, and the S(t) = x + i(t) is feed into the receiver. The chaotic x signal is regenerated allowing a single subtraction to retrieve the transmitted signal, [x+i(t)]-x r = i (t), If x = x r. The simulation results shows that Spiegel-Moore chaotic circuit is an 7
st Annual Applied Science and Engineering Conference IOP Conf. Series: Materials Science and Engineering 80 (07) 0066 doi:0.088/757-899x/80//0066 excellent for chaotic masking communication when the frequenc information is at intervals of 0.8 khz khz. Otherwise, when the frequenc information is more than khz or less than 0.8 khz, the chaotic masking communication is not occur. Figure 9. Spiegel-Moore circuit masking communication sstem for sinusoidal wave (c) Figure 0. Multisim outputs of Spiegel-Moore circuit masking communication sstem with input V and KHz: Information signal, Chaotic masking transmitted signal, (c) Retrieved signal. 6. Conclusion In this paper, Spiegel-Moore chaotic circuit sstem including chaotic motions, b means of Lapunov exponent spectrum, diagram bifurcation analsis has been studied. Moreover, it is implemented via a designed circuit with Multisim showing ver good agreement with the numerical simulation result. The chaotic snchronization of two identical Spiegel-Moore circuits sstem has been investigated b implementing bidirectional method technique. Chaotic snchronization, realization circuit and chaos masking were realized b using MATLAB and Multisim programs. Finall, the comparison between MATLAB and Multisim simulation results demonstrate the effectiveness of the proposed secure communication scheme. 7. Acknowledgment The authors would like gratefull acknowledgment the financial support from LPMM UIN Sunan Gunung Djati Bandung, Indonesia. 8
st Annual Applied Science and Engineering Conference IOP Conf. Series: Materials Science and Engineering 80 (07) 0066 doi:0.088/757-899x/80//0066 References [] Zhang H 00 Chaos Snchronization and Its Application to Secure Communication (Canada: PhD thesis, Universit of Waterloo) [] Lorenz E N 96 Deterministic nonperiodic flow J.of the Atmospheric Sciences 0 0-4 [] Shinbrot T, Grebogi C, Wisdom J and Yorke J A 99 Chaos in a double pendulum American Journal of Phsics 60 49-499 [4] Sanjaa W S M, Mohd I, Mamat M and Salleh Z 0 Mathematical Model of Three Species Food Chain Interaction with Mixed Functional Response International Journal of Modern Phsics: Conference Series 9 440 [5] Sanjaa W S M, Mamat M, Salleh Z and Mohd I 0 Bidirectional Chaotic Snchronization of Hindmarsh-Rose Neuron Model Applied Mathematical Sciences 5 54 685-695 [6] Rossler O E 976 An equation for continuous chaos Phsics Letters A. 57 97-98 [7] Sprott J C 004 Dnamical Models of Love Nonlinear Dn. Psch. Life Sci. 8 0-4 [8] Volos Ch K, Kprianidis I M and Stouboulos I N 0 Snchronization Phenomena in Coupled Nonlinear Sstems Applied in Economic Ccles WSEAS Trans. Sstems 68-690 [9] Volos Ch K, Kprianidis I M, and Stouboulos I N 0 Experimental Investigation on Coverage Performance of a Chaotic Autonomous Mobile Robot Robotics and Autonomous Sstems 6 4- [0] Volos Ch K, Kprianidis I M and Stouboulos I N 0 Text Encrption Scheme Realized with a Chaotic Pseudo-Random Bit Generator J. of Engineering Science and Technolog Review 6 94 [] Andreatos AS and Leros A P 0 Secure Image Encrption Based on a Chua Chaotic Noise Generator Journal of Engineering Science and Technolog Review 6 90-0 [] Lian S, Sun J, Liu G and Wang 008 Efficient Video Encrption Scheme Based on Advanced Video Coding Multimed.Tools Appl. 8 75-89 [] Abdulkareem M and Abduljaleel I Q 0 Speech Encrption using Chaotic Map and Blowsh Algorithms Journal of Basrah Researches 9 68-76 [4] Feng J C and Tse C K 007 Reconstruction of Chaotic Signals with Applications to Chaos- Based Communications (Singapore: World Scientific Publishing Co. Pte. Ltd.) [5] Pehlivan I and Uaroglu Y 007 Rikitake Attractor and It s Snchronization Application for Secure Communication Sstems Journal of Applied Sciences 7-6 [6] Sambas A, Sanjaa W S M and Mamat M 05 Bidirectional Coupling Scheme of Chaotic Sstems and its Application in Secure Communication Sstem Journal of Engineering Science and Technolog Review 8 89-95 [7] Sanjaa W S M, Halimatussadiah, and Maulana D S 0 Bidirectional chaotic snchronization of non-autonomous chaotic circuit and its application for secure communication World Academ of Science Engineering and Technolog 56 067-07 [8] Pecora L M and Carroll T L 990 Snchronization in Chaotic Sstems Phsical Review Letters 64 8-85 [9] Lee T H and Park J H 00 Generalized functional projective snchronization of Chen-Lee chaotic sstems and its circuit implementation International Journal of the Phsical Sciences 5 7 8-90 [0] Vaidanathan S, Volos Ch K Pham V T and Madhavan K 05 Analsis, adaptive control and snchronization of a novel 4-D hperchaotic hperjerk sstem and its SPICE implementation. Archives of Control Sciences 5 5-58 [] Moore F C and Spiegel E A 966 A Thermall Excited Non-Linear Oscillator Astrophs Journal 4 87-887 [] Wolf A 986 Quantit Chaos with Lapunov Exponents Chaos, Princeton Universit Press 7-90 9