Secure Communications Based on the. Synchronization of the New Lorenz-like. Attractor Circuit

Transcription

1 Advanced Studies in Theoretical Physics Vol. 9, 15, no. 8, HIKARI Ltd, Secure Communications Based on the Synchronization of the New Lorenz-like Attractor Circuit Aceng Sambas 1,3, Mustafa Mamat 1,*, Mada Sanjaya WS,3, Zabidin Salleh and Fatma Susilawati Mohamad 5 1,5 Department of Information Technology Universiti Sultan Zainal Abidin, Kuala Terengganu, Malaysia * Corresponding author Department of Physics, Universitas Islam Negeri Sunan Gunung Djati Bandung, Indonesia 3 Bolabot Techno Robotic Institute Sanjaya Star Group Corp, Bandung, Indonesia Pusat Pengajian Informatik dan Matematik Gunaan Universiti Malaysia Terengganu, Malaysia Copyright 15 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 Synchronization of chaotic systems is important because it might be useful in some type of private communications. In this paper, the phenomenon of Chaos that produced in the case of autonomous New Lorenz-like attractor circuit, have been studied extensively. The initial study of this work also includes some of the most well-known tools of nonlinear dynamics, such as the Lyapunov exponents and the Poincaré map. Furthermore, the use of this type of chaotic circuit in the connection substitution synchronization method is presented in details. After conducting the analysis of the proposed synchronization scheme, the use of such an autonomous chaotic circuit as a signal modulation in secure communication systems, has been examined. Finally, numerical simulations by using MATLAB

2 38 Aceng Sambas et al. 1, as well as implementation of circuit simulations by using MultiSIM 1., has been performed in this work. Keywords: New Lorenz-like attractors, bifurcation diagram, secure communication system 1. Introduction Chaos theory has been established since the 197s due to its applications in many different research areas, such as electronic circuits [1], ecology [], biology [3], economy [], bit generators [5], psychology [], robotics [7], secure communication systems [8-11] etc. Generally, Chaos is an aperiodic, long-term behavior in a deterministic system that exhibits a sensitive dependence on initial conditions. Since the Dutch scientist Christiaan Huijgens in 15 noted the synchronizing behavior of pendulum clocks, many scientists have been investigating the synchronization of several dynamical systems. The landmark in the evolution of the chaotic synchronization s theory is the study of the interaction between coupled chaotic systems by Pecora and Carroll [1], in which two identical chaotic systems with different initial conditions can be synchronized. Till now, various types of synchronization have been reported, namely complete synchronization [13], phase synchronization [1], generalized synchronization [15], anti-synchronization and anti-phase synchronization [1], lag synchronization [17], anticipating synchronization [18], projective synchronization [19] and inverse π-lag synchronization []. So, the phenomenon of chaotic synchronization has been intensively and extensively investigated due to its potential applications in a variety of areas, such as in secure communications [1, ], encryption systems [3], chemical reactions [], neuronal systems [5, ] and economic models [7]. This paper focuses on the use of an autonomous New Lorenz-like chaotic attractor circuit in the case of a signal masking system. This new type of nonlinear system presents the same chaotic behavior as the well-known Lorenz system. Numerical simulations in MATLAB as well as analog circuit s simulations in MultiSIM confirm the expected chaotic behavior of the system. Next, this system is used in the connection substitution synchronization scheme which is applicated to a secure communication system. The success of the proposed method has been confirmed by the extended simulation results. The paper is organized as follows. In Section, numerical simulations in MATLAB 1, Lyapunov exponent s analysis, Poincaré map and MultiSIM 1. simulation results of the proposed nonlinear system, are obtained. The application of the connection substitution synchronization method in the case of coupled autonomous New Lorenz-like attractor circuits is presented in Section 3. In Section, the simulation in MATLAB 1 as well as in MultiSIM 1. of a chaotic masking communication system, by using this new type of nonlinear circuit, is described in details. Finally, Section 5 contains the conclusion remarks.

3 Secure communications based on the synchronization 381. The New Lorenz-like System In this work the new Lorenz-like system, which is presented by Li et al in 8 [8], is used. This is a three-dimensional autonomous nonlinear system that is described by the following system of ordinary differential equations: x = a(y - x) y = abx - axz z = xy - cz (1) Where ( x, y, z ) T R 3 is the state variables of the system, while a, b and c are the system s parameters. As it is shown in equation (1) there are six terms on the right-hand side of differential equations, but the system s nonlinearity only relies on the two quadratic terms xy and xz..1 Numerical Simulations In this section, all the numerical simulations are carried out using the MATLAB 1. The fourth-order Runge-Kutta method is used to solve numerically the system of differential equations (1). 1 8 max() c Figure 1: Bifurcation diagram of x vs. the control parameter c, for the specific values (a = 5, b = ), with MATLAB 1. A bifurcation occurs when a small change made in the parameter value (the bifurcation parameter) of a system causes a sudden qualitative or topological change in its dynamical behavior. In dynamical systems, a bifurcation diagram shows the possible long term values (equilibrium points, periodic orbits or chaotic behavior) of a system as a function of a bifurcation parameter. Figure 1 shows a possible bifurcation diagram for system (1) in the range of c. Specifically, for c 3.3, Figure displays a chaotic region, whereas for 3.3 c the system displays a periodic behavior.

4 38 Aceng Sambas et al. In Figures (a)-(c), the projections of the phase space orbit onto the x-y plane, the x-z plane and the y-z plane, are shown respectively. The parameters and initial conditions of the New Lorenz-like system (1) are chosen as (a, b, c) = (5,,.5) and (x, y, z ) = (.1,.1,.1), so that the system shows the expected chaotic behavior. So, it can be clearly observed that the phase portraits, especially onto the x-z plane are similar to that of the family of Lorenz systems. This is happened because the proposed system has all the characteristics of the above mentioned system s family, such as the symmetry and invariance under the transformation S: (x, y, z) ( x, y, z), the dissipativity, the diffeomorphism and the topological equivalence. Furthermore, the time-series of signals x, y and z, for the same set of parameters and initial conditions, are shown in Figures (d)-(f). Especially, the signal x has the well-known pattern of all the signals produced by the systems of Lorenz systems. In details, the signal x spirals outward from one of the symmetric equilibria P = ( bc, bc,b) and P = (- bc,- bc,b) for some times and then + switches to spiraling outward from the other equilibrium point. It is also known from the theory of nonlinear dynamics that for a three dimensional system, like this, there has been three Lyapunov exponents (λ 1, λ, λ 3 ). In more details, for a 3D 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 [9]. (λ 1, λ, λ 3 ) (,, ): a fixed point (λ 1, λ, λ 3 ) (,, ): a limit point (λ 1, λ, λ 3 ) (,, ): a -torus (λ 1, λ, λ 3 ) (+,, ): a strange attractor Therefore, the last configuration is the only possible third-order chaotic system. In this case, a positive Lyapunov exponent reflects a direction of stretching and folding and therefore determines chaos in the system. So, in Figure 3 the dynamics of the proposed system s Lyapunov exponents for the variation of the parameter c [, ], is shown. For c 3.3 a strange attractor is displayed as the system has one positive Lyapunov exponent, while for values of 3.3 c is a transition to a periodic behavior as the system has three negative Lyapunov exponents. Another useful tool in the study of nonlinear systems is the Poincaré map, which is simply a map that showing a pattern from its time-series data. It is not a time-series map, yet it allows transversal changes of time-series data in each time of iteration. In this way each element of displayed data can no longer be viewed differently from time-series that represent them. In fact, every time-series data has been inherent with the data that graphically represented [3]. Figures (a)-(c) shows the Poincaré section map using MATLAB 1, for a = 5, b =, c =.5. So, for the chosen value of c (c =.5), the system has a chaotic behavior. A single trajectory plotted in the phase plane intersects itself many times, and the portrait soon becomes very messy. However, if one plots the first returns on the

5 Secure communications based on the synchronization 383 Poincaré section, then a strange attractor is formed that demonstrates some underlying structure as shown in Figures (a)-(c). It must be noted that the chaotic attractor will have different forms on different Poincaré sections. This strange attractor has a fractal structure. 1 Phase Space New Lorenz-like 8 Phase Space New Lorenz-like 8 Phase Space New Lorenz-like signal y - signal z 3 signal z signal x signal x signal y (a) (b) (c) 8 Time Series signal New Lorenz-like 1 Time Series signal New Lorenz-like 8 Time Series signal New Lorenz-like 8 7 signal x signal y - signal z Time (s) Time (s) Time (s) (d) (e) (f) Figure : Numerical simulation results using MATLAB 1, for a = 5, b =, c =.5: (a) x-y plane, (b) x-z plane, (c) y-z plane, (d) time series of signal x, (e) time series of signal y and (f) time series of signal z Dynamics Lyapunov exponent x y z Lyapunov exponents c Figure 3: Lyapunov exponents versus the parameter c [, ], with MATLAB 1

6 38 Aceng Sambas et al..8 Poincare Map Analysis New lorenz-like Circuit.5 Poincare Map Analysis New lorenz-like Circuit 7. Poincare Map Analysis New lorenz-like Circuit x(n+1) 5.8 y(n+1).5 z(n+1) x(n) y(n) (a) (b) (c) z(n) Figure : A gallery of Poincaré maps for system (1), when a = 5, b =, c =.5: (a) show the plot given the maxima of x(n + 1) against those of x(n); (b) show the plot given the maxima of y(n + 1) against those of y(n); (c) show the plot given the maxima of z(n + 1) against those of z(n), with MATLAB 1.. Analog Circuit Simulations A circuit schematic that closely realizes the scaled Lorenz system is shown in Figure 5. The voltages at the nodes labeled x, y, and z correspond to the states of the equation system (1). The operational amplifiers and associated circuitry perform the basic operations of addition, subtraction, and integration. The nonlinear terms in the equation are implemented with the analog multipliers AD33 []. The occurrence of the chaotic attractor can be clearly seen in Figures 5(a) (c), for the same values of system s parameters. By comparing Figures (a)- (c) and Figures (a)-(c) a good qualitative agreement between the numerical integration of system (1) by using MATLAB 1, and the circuit s simulation by using MultiSIM 1., can be concluded. 3. Connection Substitution Chaotic Synchronization Scheme Based on Master-Slave approach, a New Lorenz-like equation before connection substitution synchronized method is: x 1 f z f (, z x ( x 1, y 1, 1 ) x, y ) () New Lorenz-like circuit equations in a synchronization scheme are as follows: x 1 f z f ( 1, z x ( x1, y1, 1 ) x, y ) (3)

7 Secure communications based on the synchronization 385 R1 R U1 R3 R5 R 5kΩ U R IC=V C1 signal U3 R8 R7 kω U R9 R1 R11 R1 U5 R1 R15 U7 R1 IC=V C Signal R13 3kΩ U kω U8 A1 R5 1mΩ 1 V/V V R17 3kΩ A R18 kω U9 R19 R 3kΩ U1 R R1 R3 U11 R IC=V C3 Z Signal U1 V1 V 1 V/V V Figure 5: Schematic of the proposed New Lorenz-Like circuit (a) (b) (c) Figure : Various projections of the chaotic attractor using MultiSIM in (a) x-y plane, (b) x-z plane and (c) y-z plane

8 38 Aceng Sambas et al. The following master-slave (connection substitution method) configuration, as described below: Master x 1 a( y1 x1 ) y 1 abx1 ax1z1 z 1 x1 y1 cz1 Slave () x a( y x1 ) y abx1 ax1z z x1 y cz The asymptotic synchronized situation is defined as: lim y1 ( t ) y ( t ) t (5) 3.1 Numerical Simulations Numerical simulations in MATLAB 1, by solving the coupling system () with the fourth-order Runge-Kutta method, are used to describe the dynamics of the connection substitution chaotic synchronization scheme. Figure 7 shows the chaotic synchronization phase portrait and the error numerical results. 1 Substitution Chaotic Synchronization 1 Error Chaotic Synchronization signal y - signal y signal y1 (a) Time (s) x 1 (b) Figure 7: (a) Synchronization phase portrait of y versus y 1 and (b) synchronization error (y y 1 ) versus time t, by using the connection substitution technique, in MATLAB 1 3. Analog Circuit Simulation in MultiSIM 1. Synchronization of chaotic systems is the key issue in symmetric chaos-based secure communication schemes. It is a phenomenon that may occur when two, or

9 Secure communications based on the synchronization 387 more, chaotic oscillators are coupled. This paper presents in this Section the study of circuit s simulation by using MultiSIM 1.. For this reason, the drive and response circuits were constructed. So, in Figure 8 the implementation of the connection substitution synchronization scheme of coupled New Lorenz-Like circuits, with MultiSIM 1., is displayed. Finally, in Figure 9 the simulation results of this scheme which confirms the case of chaotic synchronization are shown. R1 R U1 R3 R5 Master R 5kΩ U R C1 IC=V U3 R R7 U13 Slave R8 R9 5kΩ R3 U1 R31 C IC=V U15 R7 A1 R8 kω U R13 R9 R1 R11 3kΩ kω U R1 U5 R15 R1 U7 R1 C U8 IC=V R5 1mΩ R3 A3 R33 kω U1 R38 R3 R37 R3 3kΩ kω U18 R35 U17 R R39 U19 R1 C5 U IC=V R5 1mΩ 1 V/V V R17 3kΩ A R18 kω U9 R19 R 3kΩ U1 R R1 R3 U11 R C3 IC=V U1 V1 V 1 V/V V R 3kΩ A R3 kω U1 R R5 3kΩ U R7 R R8 U3 R9 C IC=V U V V 1 V/V V 1 V/V V Figure 8: Schematic of the connection substitution chaotic synchronization of coupled New Lorenz-Like circuits. Application to Secure Communication Systems A true square wave is a signal of periodic recurrence made up of an infinite number of odd harmonics of the fundamental frequency. The general equation of the square wave can be written [31]: m s ( t ) 1 sin ( n 1) x n 1 () n 1

10 388 Aceng Sambas et al. (a) (b) Figure 9: (a) Synchronization phase portrait y versus y 1 and (b) Time series of signal y, with MultiSIM 1. The message signal m s (t) is a square wave and x t. The sum of the signal m s (t) and the chaotic signal m NewLorenz Like (t), produced by the New Lorenz-Like circuit, is a new encryption signal m encryption, which is given by Eq.(7). m ( t) m (t) m (t) (7) Encryption S NewLorenzLike The signal m NewLorenz Like (t) is one of the parameters of equation (1). After finishing the encryption process the original signal can be recovered with the following procedure. m ( t) m (t) m (t) (8) New_Signal Encryption NewLorenzLike So, m New_Signal (t) is the original signal and must be the same with m s (t). Due to the fact that the input signal can be recovered from the output signal, it turns out that it is possible to implement a secure communication system using the proposed chaotic system..1 Numerical Simulations Figures 1 (a)-(c) show the MATLAB 1 numerical simulation results for the proposed chaotic masking communication scheme.

11 Secure communications based on the synchronization Information signal i(t) 15 chaotic masking transmitted signal S(t) 15 retrieved signal i (t) i(t) S(t) i (t) Time(s) Time(s) Time(s) (a) (b) (c) Figure 1: MATLAB 1 simulation of New Lorenz-Like circuit masking communication system when amplitude.5 V and frequency KHz: (a) Information signal, (b) Chaotic masking transmitted signal, (c) Retrieved signal. Analog Circuit Simulation in MultiSIM 1. The principle of a chaos-based secure communication scheme is the information signal, which is masked by a chaotic signal at the transmitter, and then sent it to the receiver by the public channel. Finally, the encrypted signal is decrypted at the receiver. In this scheme, the key issue is that the two identical chaos generators in the transmitter and the receiver end need to be synchronized [3]. Figure 11 shows the MultiSIM 1. simulation results for the masking signal communication system. (a) (b) (c) Figure 11: MultiSIM 1. outputs of New Lorenz-Like circuit masking communication systems when amplitude.5 V and frequency KHz: (a) Information signal, (b) Chaotic masking transmitted signal, (c) Retrieved signal

12 39 Aceng Sambas et al. Also, in the proposed masking scheme, the square wave signal of amplitude.5 V and frequency KHz is added to the synchronizing driving chaotic signal in order to regenerate the original driving signal at the receiver. Thus, as it can be shown from Fig.11(c), the message signal has been perfectly recovered by using the signal masking approach through the synchronization of chaotic New Lorenz- Like circuits. Furthermore, simulation results with Multisim 1. have shown that the performance of chaotic New Lorenz-Like circuits in chaotic masking and message recovery is very satisfactory. Finally, Figure 1 shows the circuit schematic of implementing the New Lorenz-Like circuit chaotic masking communication scheme. R1 R7 R U1 A1 1 V/V V R17 3kΩ R8 kω U A 1 V/V V R13 R18 kω U9 R3 R5 R1 MASTER R 5kΩ U R9 3kΩ U R11 kω R19 R1 U5 R R 3kΩ U1 R R1 C1 IC=V U3 R1 R15 U7 R3 U11 R R1 C U8 IC=V C3 IC=V U1 R5 1mΩ V1 V R R3 R7 U13 A3 1 V/V V R 3kΩ R33 kω U1 A 1 V/V V R38 R3 kω U1 R8 R3 SLAVE R37 R9 5kΩ U1 R3 R3 3kΩ kω U18 R R35 U17 R31 R5 3kΩ U R7 R C IC=V U15 R39 R U19 R8 U3 R9 R1 C5 U IC=V C IC=V U R5 1mΩ V V R5 R57 V -1mV 5mV.5msec msec i(t) R53 U5 R5 U8 Buffer Adder R 1. Substractor U7 R5 R59 R55 R58 U R1 U9 R S(t) i'(t) 5. Conclusion Figure 1: New Lorenz-Like circuit masking communication circuit. In this paper the chaotic synchronization of two identical New Lorenz-Like circuits system has been investigated by implementing connection substitution technique. The proposed method of synchronization between chaotic circuits can be applied successfully to a secure communication scheme. Chaos synchronization and chaos masking were realized by using MATLAB 1 and

13 Secure communications based on the synchronization 391 MultiSIM 1. programs. The comparison between MATLAB 1 and MultiSIM 1. simulation results demonstrate the effectiveness of the proposed secure communication scheme. References [1] M. Sanjaya W.S., D. S Maulana, M. Mamat, and Z. Salleh, Nonlinear Dynamics of Chaotic Attractor of Chua Circuit and Its Application for Secure Communication, J. Oto. Ktrl. Inst (J. Auto. Ctrl. Inst), 3 (1), 11, 1-1. [] 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, 1, [3] M. Sanjaya W. S, M. Mamat, Z. Salleh, and I. Mohd, Bidirectional Chaotic Synchronization of Hindmarsh-Rose Neuron Model, Applied Mathematical Sciences, 5 (5), 11, [] Ch. K. Volos, I. M. Kyprianidis, S. G. Stavrinides, I. N. Stouboulos, I. Magafas, and A. N. Anagnostopoulos, Nonlinear Dynamics of a Financial System from an Engineer s point of View, Journal of Engineering Science and Technology Review, (3), 11, [5] Ch. K. Volos, I. M. Kyprianidis, and I. N. Stouboulos, Fingerprint Images Encryption Process Based on a Chaotic True Bits Generator, International Journal of Multimedia Intelligence and Security, 1(), 1, [] J. C. Sprott., Dynamical models of love, Nonlinear Dyn. Psych. Life Sci., 8,, [7] Ch. K. Volos, I. M. Kyprianidis, and I. N. Stouboulos, Α Chaotic Path Planning Generator for Autonomous Mobile Robots, Robotics and Autonomous Systems,, 1, [8] A. Sambas., M. Sanjaya W.S. and Halimatussadiyah, Unidirectional Chaotic Synchronization of Rossler Circuit and Its Application for Secure Communication, WSEAS Transactions on Systems, 9(11), 1, [9] A. Sambas, M. Sanjaya W.S., M. Mamat and Halimatussadiyah, Design and Analysis Bidirectional Chaotic Synchronization of Rossler Circuit and its

14 39 Aceng Sambas et al. Application for Secure Communication. Applied Mathematical Sciences, 7(1), 13, [1] Aceng Sambas, Mada Sanjaya W. S., M. Mamat, N. V Karadimas and O. Tacha, Numerical Simulations in Jerk Circuit and It s Application in a Secure Communication System. Recent Advances in Telecommunications and Circuit Design. WSEAS 17th International Conference on Communications Rhodes Island, Greece July 1-19, 13, 19-19, ISBN: [11] Aceng Sambas, Mada Sanjaya W. S., M. Mamat and O. Tacha., 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. (), [1] L. M. Pecora and T. L. Carroll, Synchronization in Chaotic Systems, Physical Review Letters,, 199, [13] A. S. Pikovski, On the Interaction of Strange Attractors, Z Phys B: Condens Matter, 55, 198, [1] M. G. Rosenblum, A. S. Pikovski, and J. Kurths, Phase Synchronization of Chaotic Oscillators. Phys. Rev. Lett., 7, 199, [15] N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. Abarbanel, Generalized Synchronization of Chaos in Directionally Coupled Chaotic Systems, Phys. Rev. E., 51, 1995, [1] L.. Cao and. C. Lai, Antiphase Synchronism in Chaotic Systems, Phys. Rev. E, 58, 1998, [17] M. G. Rosenblum, A. S. Pikovski, and J. Kurths, From Phase to Lag Synchronization in Coupled Chaotic Oscillators, Phys. Rev. Lett., 78, 1997, [18] H. U. Voss, Anticipating Chaotic Synchronization, Phys. Rev. E, 1,, [19] R. Mainieri and J. Rehacek, Projective Synchronization in Three- Dimensional Chaotic Systems. Phys. Rev. Lett., 8, 1999, [] Ch.K. Volos, I.M. Kyprianidis, and I.N. Stouboulos, Various Synchroniza-

15 Secure communications based on the synchronization 393 tion Phenomena in Bidirectionally Coupled Double Scroll Circuits, Commun. Nonlinear Sc. Numer. Simulat., 1, 11, [1] I. Pehlivan and. Uyaroglu, Rikitake Attractor and It s Synchronization Application for Secure Communication Systems. Journal of Applied Science, 7 (), 7, [] F. Zhu, Observer-based synchronization of uncertain chaotic system and its application to secure communications. Chaos, Solitons and Fractals, 9, [3] Ch. K. Volos, Image Encryption Using the Coexistence of Chaotic Synchronization Phenomena, Journal of Applied Mathematics and Bioinformatics, 3(1), 13, [] K. Nakajima and. Sawada, Experimental studies on the weak coupling of oscillatory chemical reaction systems. J. Chem. Phys. 7(), 198, [5] M. Mamat, Z. Salleh, Mada Sanjaya W. S. and Ismail Mohd, Numerical Simulation Bidirectional Chaotic Synchronization FitzHugh-Nagumo Neuronal System. Applied Mathematical Sciences, (38), 1, [] I. M. Kyprianidis, V. Papachristou, I. N. Stouboulos and Ch. K. Volos, Dynamics of Coupled Chaotic Bonhoeffer van der Pol Oscillators, WSEAS Trans. Systems, 11(9), 1, 51 I 5. [7] Ch. K. Volos, I. M. Kyprianidis, and I. N. Stouboulos, Synchronization Phenomena in Coupled Nonlinear Systems Applied in Economic Cycles, WSEAS Trans. Systems, 11(1), 1, [8]. F. Li,. D. Chu, J. G. Zhang and.. Chang, Nonlinear dynamics and circuit implementation for a new Lorenz-like attractor. Chaos, Solitons and Fractals. 1, 9, [9] Q. H. Alsafasfeh and M. S. Al-Arni, A New Chaotic Behavior from Lorenz and Rossler Systems and Its Electronic Circuit Implementation. Circuits and Systems,, 11, [3] H. Situngkir and. Surya, Perception on Modified Poincaré Map of Financial Time Series Data, Applications of Physics in Financial Analysis (APFA) Europhysics Conference of European Physical Society, 3, 1-1.

16 39 Aceng Sambas et al. [31] S.. Wang, Simulation of Chaos Synchronization, Ph.D thesis, University of Western Ontario, London, [3] H. Zhang, Chaos Synchronization and Its Application to Secure Communication, PhD thesis, University of Waterloo, Canada, 1. Received: March 1, 15; Published: May 15, 15