See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/261074864 Implementation of Rössler chaotic system through inherent exponential nonlinearity of a diode with two-channel chaotic synchronization applications Conference Paper · June 2013 DOI: 10.1109/ICICIP.2013.6568179 CITATIONS READS 2 120 2 authors, including: Wimol San-Um Thai-Nichi Institute of Technology 85 PUBLICATIONS 152 CITATIONS SEE PROFILE All content following this page was uploaded by Wimol San-Um on 10 July 2016. The user has requested enhancement of the downloaded file. 2013 Fourth International Conference on Intelligent Control and Information Processing (ICICIP) June 9 – 11, 2013, Beijing, China Implementation of Rössler Chaotic System through Inherent Exponential Nonlinearity of a Diode with Two-Channel Chaotic Synchronization Applications S. Larptwee and W. San-Um Abstract—This paper presents a new Rössler chaotic system using exponential nonlinearity and its application to twochannel synchronization. The proposed chaotic system exhibits a chaotic attractor that resembles the original Rössler system with only six-term in three-dimensional ordinary equation systems using the exponential nonlinearity. Chaotic dynamics are described in terms of equilibria, Jacobian matrix, time domain waveforms, chaotic attractors, and bifurcation diagram. The circuit implementation is relatively compact and simple sine the exponential nonlinearity can be achieved by an inherent nonlinearity of single diode. An application to a twochannel secure communication are also demonstrated, showing a fast, low-error and robust synchronization processes. E I. INTRODUCTION DWARD LORENZ[1] encountered sensitively dependent initial conditions of an atmospheric convection model while performing numerical simulations in 1963 leading to the discovery of the Lorenz system with seven-terms in three-dimensional ordinary differential equations and two quadratic nonlinearities. In 1976, Rössler [2] proposed a chaotic system with seven terms and a single quadratic nonlinearity, which is algebraically simpler than Lorenz system. In addition, a single folded-band attractor of Rössler system is topologically simpler than a two-scroll Lorenz attractor. Such Lorenz and Rössler systems have consequently led to considerable research interests in searching for new chaotic systems with fewer terms in ODEs or more complex attractor topology. Several chaotic systems with fewer than seven terms and two quadratic nonlinearities continuously been reported as variants in Lorenz system family [3-7]. Complex three-scroll and four-scroll attractors based on Lorenz system have also been suggested through the use of three or more quadratic nonlinearities [8-10]. On the other hand, simple chaotic systems with a single nonlinearity similar to Rössler system are rarely found. In fact, Rössler himself had proposed another system with six terms and a single quadratic nonlinearity in 1979. In 1994, Sprott [4] found fourteen cases with six terms and a single quadratic nonlinearity through an intensive numerical computer search. Recently, many simple systems have been proposed in simple Jerk S. Larptawee is with the Master of Engineering Program and Basic Science Department, Faculty of Engineering, Thai-Nichi Institute of Technology, E-mail Address: sura.l@tni.ac.th. W. San-Um is with Intelligent Electronic Systems Research laboratory and Computer Engineering Department, Faculty of Engineering, Thai-Nichi Institute of Technology. Pattanakarn 37-39, Suanluang, Bangkok, Thailand, 10250. Tel: (+662) 763-2600 Ext.2926. Fax: (+662) 763-2700. E-mail Address: wimol@tni.ac.th. 978-1-4673-6249-8/13/$31.00 ©2013 IEEE equations with single quadratic or non-quadratic nonlinearities. Despite the fact that these simple Jerk chaotic systems with a single nonlinearity potentially resemble the single folded-band Rössler attractor ,the Kapelan-York [4] dimension (DKY) as a measure of complexity is somewhat lower than the original Rössler attractor that possesses the greatest value of DKY =2.1587. This leads to a question of whether the original Rössler system in dynamic forms can be simplified into fewer terms with simple nonlinearity, or modified for more complex attractor [12-15]. No simplifications of Rössler system has never been found so far. This paper therefore presents a new Rössler chaotic system using exponential nonlinearity and its application to two-channel synchronization. The proposed chaotic system exhibits a chaotic attractor that resembles the original Rössler system with only six-term in three-dimensional ordinary equation systems using the exponential nonlinearity. An application to a two-channel secure communication are also demonstrated, showing a fast, lowerror and robust synchronization processes. II. PROPOSED NEW RÖSSLER CHAOTIC SYSTEM USING EXPONENTIAL NONLINEARITY Based on the Rössler system proposed in 1979 [7], the first and the second equations, i.e. x = − y − z and y = x + ay , initiate a normal band of the attractor through an outward spiral motion into the x-y phase plane. Nonlinear interactions between x and z variables in the third equation, i.e. z = b + z ( x − c) , form an additional folded band to the overall attractor. It is noticeable that the folded band in Rössler attractor rises and returns exponentially in zdimension especially for positive values of x variable under the flows. This aspect implies that the third equation may be modified through the use of an exponential nonlinearity. Therefore, a new chaotic system is therefore presented in three-dimensional autonomous ODEs expressed as x = − y − z y = x + ay (1) z = − z + bF ( x) where ( x, y, z ) ∈ ℜ3 are dynamical variables, (a, b) ∈ ℜ+ are system parameters, and F(x) is a nonlinear function required for chaos. A particularly simple case of the nonlinear function F(x) is presented using exponential functions. In other words, 787 F ( x) = exp( x) (2) It can be considered from (2) such an exponential nonlinearity can be implemented through the use of a diode instead of using a complicated voltage multiplier circuit such as AD633 chip. III. DYNAMICS ANALYSIS The system (1) maintains the first and the second equations of Rössler system while the third equation has been simplified into two distinct terms, including the linear term –z and the nonlinear term bF(x). The existence of attractor can be described by the divergence of flows as ∇ .V = ∂x / ∂x + ∂y / ∂y + ∂z / ∂z = a − 1. (3) For a dissipative chaotic system, p<0 and therefore a is limited into the region 0<a<1. The exponential rate of contraction is dV/dt =exp(a-1) and hence a volume element V0 is contracted in time t by the flows into a volume element V0exp(−t). Each volume containing the system trajectories shrinks to zero as time t approaches +∞. All system orbits will be confined to a specific limit set of zero volume, and the asymptotic motion converges onto an attractor. It can be concluded that the existence of attractors is constant and independent to the nonlinear term bF(x). For the equilibrium analysis, linearizing (1) by setting the system of equation equals zero, i.e. Fig.1. Bifurcation diagrams exhibiting a route to chaos. 0 = −y − z 0 = x + ay 0 = − z + be x (4) The system (4) has a single equilibrium point at (0, 0, 0) and the Jacobian of the system is ⎡ 0 J = ⎢⎢ 1 ⎢⎣bF / −1 −1⎤ a 0 ⎥⎥ 0 −1⎥⎦ (5) Applying the equilibrium point P into this Jacobian matrix and analyzing |Iλ-J|=0 reveals a resulting characteristic polynomial as follows: λ 3 + (1 − a )λ 2 + (bF / − a + 1)λ + ( abF / − 1) = 0 Fig.2. Simulation Phase portraits with at a =0.30 and b=0.0007, LEs = (0.0638, 0, -0.8641), DKY =2.0738. (6) According to the Routh–Hurwitz [9] stability criterion, the / system (1) is unstable when F < (1 + (1-a)2)/(b-2ab). Note that dynamic behaviors depend on two parameters a and b, and can be characterized completely by the plot of parameter space without redundancy. For all particular values of a and b in the subsequent numerical analyses, the resulting eigenvalues 1 is a positive real number and 2 and 3 are a pair of complex conjugate with positive real parts, indicating that the equilibrium points are all saddle focus points. Fig.3. Time-domain waveforms showing chaotic behaviors. 788 Fig. 4. The block diagrams of the transmitter and receiver of two-channel chaotic synchronizations. Numerical simulations have been performed in MATLAB using the initial condition of (x0, y0, z0) = (1, 0, 1). In fact, the initial condition is not crucial, and can be selected from any point that lies in the basin of attractor. In order to find the control parameter that offers the maximum values of chaoticty and complexity, Fig. 1 shows the bifurcation diagram of the peak of z (z max) versus the parameters a and b. It is seen in Fig. 1 that the system exhibits a perioddoubling route to chaos. As for particular illustrations, the control parameter at a=0.35 and b=0.0007 is chosen in simulations of dynamical behaviors. The chaotic attractors are displayed in Figs. 2(a), 2(b), 2(c) and 2(d) for a threedimensional view, an x–y phase plane, an x–z phase plane and a y–z phase plane, respectively. It is apparent in Fig. 2 that the attractor of the proposed system has a single-scroll topology, and potentially resembles the existing Rössler attractors in all phase planes. Fig. 3 shows apparently chaotic waveforms in time domain. It can be seen that the three signals are random in both amplitudes and frequencies. The DC offset is zero since the equilibrium point is at (0, 0, 0). IV. APPLICATIONS TO TWO-CHANNEL SECURE COMMUNICATION SYSTEMS Fig 5. Circuit design of the proposed Rössler chaotic system using exponential nonlinearity. Fig 6. Simulink model of the proposed Rössler chaotic system using exponential nonlinearity. The chaotic synchronizations provide two input and two output messages [16-23]. Fig. 4 shows the block diagrams of the transmitter and receiver of two-channel chaotic synchronizations. At the transmitter, a modified Rössler attractor described in (1) can be used as a single drive system for a dual-channel transmitter independent of its response subsystem at the receiver as follows: x = − y − z y = x + 0.35 y (7) z = 0.0007 e x − z Based on such a modified rössler system using diode equation as shown in Fig. 4, the dual channel transmitter consists of two transmitter and receiver signals. The first transmitter signal is s1 (t) = xt (t) + i1 (t) where xt (t) is a chaotic signal and i1 (t) represents the first original input which is transmitted. The second transmitter signal is s2 (t) = zt (t) + i2 (t) where zt (t) is a chaotic masking signal and i2 (t) Fig 7. Circuit implementations of the proposed chaotic system and its synchronization systems. 789 Fig 8. Measured chaotic output signals Fig 10. Simulated and measured synchornized signals in two channels. Fig 11. Phase-space plots of two chanel signals showsing a highly correlated transmiteed and received sigansl. Similarly when the receiver synchronizes with s1 (t), then zr (t) ≈ zt (t). The input signal i2 (t) can be recovered as î2 (t) = s2 (t) – zr (t) = zt (t) + i2 (t) – xt (t) ≈ i2 (t). As for a simple example, the first transmitter input, i1 (t) = 0.1 sin (2πf1t), where the frequency f1 = 1000 Hz and the second transmitted input i2 (t) is a pulse-train rectangular wave form with an amplitude 0.1 and the frequency f2 = 1000 Hz. Selfsynchronize can be achieved over a wide range of initial condition e.g. at the transmitter [xt (0), yt (0), zt (0)] = [1, 0, 1] and at the receiver [xt (0), yt (0), zt (0)] = [3, 0, 3]. Fig 9. Simulated and measured chaotic attractors at different parameter values. represents the second original message which is transmitter. At the receiver, a modified rössler attractor described in (1) can be used as single response subsystem for dual-channel receiver as follow: x r = − y r − z r y r = x r + 0 .35 y r z r = 0.0007 e xr − z r (8) With reference to Fig.4, the dual-channel receiver consist of produces a cmasking signal xt (t) and zt (t), respectively when the receiver synchronizes with s2 (t), then xt (t) ≈ xr (t). The input signal i1 (t) can be recovered as î1 (t) = s1 (t) – xr (t) = xt (t) + i1 (t) – xt (t) ≈ i1 (t). V. EXPERIMENTAL RESULTS Fig.5 shows the circuit diagram of the proposed new Rössler chaotic system using exponential nonlinearity. The circuit consists of three integrator section, including opamps A1, A2 and A3. The inverting amplifiers are A4 and A5. All operational amplifiers are implemented by TL084CN with 9-V power supply. The diode is IN4001. The state equations from nodal analysis is given by where k1R =23.28 KΩ, and k2 R = 0.7 KΩ are scaling parameters. The value of the R = 10 KΩ and C=1nF. In order to verify in terms of block diagram, Fig 6 shows the Simulink model of the proposed Rössler chaotic system using exponential nonlinearity when a =0.35 and b = 0.0007. Fig 7 shows circuit implementations of the proposed chaotic system and its synchronization systems. Fig 8 depicts the measured chaotic output signals. Fig 9 shows the simulated and measured chaotic attractors at different parameter values. It 790 can be seen from Figs. 8 and 9 that the results from both simulations and experiments are closely resemble. Fig 10 shows the simulated and measured synchornized signals in two channels. Fig 11 shows phase-space plots of two chanel signals showsing a highly correlated transmiteed and received signals, indicating that the two signals are completely synchronized. CONCLUSIONS This paper has presented a new Rössler chaotic system using exponential nonlinearity and its application to twochannel synchronization. In comparisons to other existing implementation of Rössler-based chaotic system, the proposed chaotic system exhibits a chaotic attractor that resembles the original Rössler system with only six-term in three-dimensional ordinary equation systems using the exponential nonlinearity. All dynamical behaviors were investigated through equilibria, Jacobian matrix, time domain waveforms, chaotic attractors, and bifurcation diagram. Cost-effective implementations of chaotic circuits were based on linear op-amps, capacitors, resistors, and a single diode employed as a nonlinear component. An application to a two-channel secure communication was also demonstrated through sinusoidal and pulse signals. The synchronization could recover the transmitted signals with fast and robust synchronization processes. The proposed circuit offers a potential alternative to robust cost-effective nonlinear oscillators in communications and controls applications. 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