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Circuits and Systems, 2011, 2, 101-105 doi:10.4236/cs.2011.22015 Published Online April 2011 (http://www.SciRP.org/journal/cs) Copyright © 2011 SciRes. CS A New Chaotic Behavior from Lorenz and Rossler Systems and Its Electronic Circ uit Implementation Qais H. Alsafasfeh1, Mohammad S. Al-Arni2 1Electrical Engineering Department, Tafila Technical University, Tafila, Jordan 2Electrical Engineering Department, Tafila Technical University, Tafila, Jordan E-mail: qsafasfeh@ttu.edu.jo, m_alarni@yahoo.com Received February 15, 2011; revised March 9, 2011; accepted March 21, 2011 Abstract This paper presents a new three-dimensional continuous autonomous chaotic system with ten terms and three quadratic nonlinearities. The new system contains five variational parameters and exhibits Lorenz and Ross- ler like attractors in numerical simulations. The basic dynamical properties of the new system are analyzed by means of equilibrium points, eigenvalue structures. Some of the basic dynamic behavior of the system is explored further investigation in the Lyapunov Exponent. The new system examined in Matlab-Simulink and Orcad-PSpice. An electronic circuit realization of the proposed system is presented using analog electronic elements such as capacitors, resistors, operational amplifiers and multipliers. Keywords: Chaos, Lorenz System, Rossler System, Lyapunov Exponent, Bifurcation 1. Introduction The science of nonlinear dynamics and chaos theory has sparked many researchers to develop mathematical models that simulate vector fields of nonlinear chaotic physical systems. Nonlinear phenomena arise in all fields of engi- neering, physics, chemistry, biology, economics, and soci- ology. Examples of nonlinear chaotic systems include pla- netary climate prediction models, neural network models, data compression, turbulence, nonlinear dynamical eco- nomics, information processing, preventing the collapse of power systems, high-performance circuits and devices, and liquid mixing with low power consumption [1-3]. The Lorenz system of differential equations arose from the work of meteorologist/mathematician Edward N. Lo- renz, who was studying thermal variations in an air cell underneath a thunderhead. The Lorenz equations are a fairly simple model in which to study chaos [3]. xyx yrxyxz zxy z (1) The arbitrary parameters , r and β > 0 and for this example are = 10, r = 28 and β = 8/3. The Rossler system has only one quadratic nonlinearity xz numerical integration shows that this system has a strange attractor for a = b= 0.2, c = 5.7 [2]. x yz yxay zbzxc (2) This paper propose a new chaotic system based on add- ing two chaotic system (Lorenz and Rosslere) and it com- pares the results with the chaotic system and an electronic circuit realization of the proposed system is presented using analog electronic elements, The remainder of the paper is organized as follows: Section 2 discusses the proposal of a new chaotic system and its analysis, section 3 present deal with circuit realization of the new attractor and section 4 discusses and examines a new scheme [4]. 2. A New Chaotic System and Its Analysis Most researchers developed a new chaotic system depend- ing on one chaotic system like Lorenz or Rossler systems the proposed scheme in this paper based on merging two chaotic systems Lorenz chaotic system and Rossler chaotic system. Therefore will be added two chaotic systems in (1) and (2), a new system is shown in (3). xyxyz yrxyxzxay zxy zbzxc (3) Q. H. ALSAFASFEH ET AL. Copyright © 2011 SciRes. CS 102 We note after adding the two chaotic systems, it is no- ticed (1) and (2) that the control parameter increased from three (δ, r, β) to six (δ, r, β, b, a, c) but to check the new system is suitable for achieving the chaotic re- quirements, by plot phase plane for a new system we note a new system loss chaotic behavior shown in Figure 1. Therefore we try to manipulate the above equation to achieve a chaotic behavior, so we will add cuomo Cir- cuit shown in (4) (linear transformation of Lorenz equ- ations with a new scale) [2,5] to Rossler equations after changing z instead x in last equation of Rossler system, the final system is shown below in (5) and (6). 20 5 uvu vruv uw wuvw (4) 20 5 xyxyz yrxy xzxay zxy zbxzc (5) 1 11 20 5 xyxz yr xayxz zxyzbxzcx (6) To check that the new system has a chaotic behavior or not, no definition of the term chaos has been universally accepted yet but most researchers agree on the three in- gredients used in following definition “Chaos is aperiodic long term behavior in a deterministic system that exhibits dependence on initial condition” [1-4,6]. Even though the definition of chaos has not been agreed upon by mathe- maticians, two properties that are generally agreed to characterize it are sensitivity to initial conditions and the presence of period-doubling cycles leading to chaos. -4-3 -2 -10 1 2 34 -6 -4 -2 0 2 4 6 8Phase plane plot Figure 1. Phase plane at adding Lorenz with Rossler sys- tems. The new system has six terms, two quadratic nonlin- earities (xz, xy) and six real constant parameters (δ, r, a, b, β and c). The state variables of the system are x, y, and z. The new system equations have one equilibrium point. This point which satisfies this requirement is found by setting x, y, z = 0, in (5), and solving for x, y and z: 0 020 05 yx yz rxyxzx ay x yzbxzc (7) The fixed point just we have one point (0,0,0), The Ja- cobian of the system is: 11 1201 20 55 rza x yzcxx J For the case when the fixed point is (x*,y*,z*) = (0,0,0), the Jacobian becomes 11 110 0 ra cb J The eigenvalues are found by solving the characteristic equation, 0 JI , for which is yielding eigenvalues λ1 =18.4561, λ2 = –30.6770 and λ3 = –8.279 the equilib- rium points are unstable and this implies chaos. Thus, the system orbits around the unstable equilibrium point. Us- ing a Matlab-Simulink model as shown in Figure 2. The xy, xz, and yz phase portraits of the new system achieved are shown in Figure 3, Figure 4, and Figure 5, also the time series for the new chaotic system is shown in Figure 6. Figure 2. The Matlab-Simulink model of the new system. – 4 –3 –2 –1 0 1 2 3 4 – 6 – 4 – 2 0 2 4 6 8 Q. H. ALSAFASFEH ET AL. Copyright © 2011 SciRes. CS 103 -3 -2 -1 0 1 2 3 4 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 x z Figure 3. xz phase portrait of the new system. -3 -2 -101234 -3 -2 -1 0 1 2 3 4 5 6 x y Figure 4. xy phase portrait of the new system. -3 -2 -1 0 12 3 4 56 -0. 5 0 0.5 1 1.5 2 2.5 3 3.5 y z Figure 5. yz phase portrait of the new system. Another important test is the Lyapunov exponents, which measures the exponential rates of divergence and convergence of nearby trajectories in state space, and the Lyapunov exponent spectrum provides additional useful information about the system as shown in Figure 7. A positive and zero Lyapunov exponent indicates chaos, two zero Lyapunov exponents indicate a bifurcation, and a zero and a negative Lyapunov exponent indicates perio- dicity, however as noticed from Lyapunov exponent the sum of the Lyapunov exponents must be negative. A pos- itive Lyapunov exponent reflects a “direction” of stret- Figure 6. The time series for new chaotic system. 020 4060 80 100 120 140 160 180200 -20 -15 -10 -5 0 5Dynami cs of Ly apunov expo nent s Time Ly ap unov e xp onents Figure 7. Dynamics Lyapunov exponent of the new syste m . ching and folding and therefore determines chaos in the system, 3D continuous dissipative (λ1,λ2,λ3) ,(+,0,–)—A strange attractor; (0,0,–)—A two-torus; (0,–,–)—A limit cycle; (–,–,–)—A fixed point [7-11]. 3. Circuit Realization of the New Attractor A simple electronic circuit is designed, so that it can be used to study chaotic phenomena. The circuit employs simple electronic elements, such as resistors, and opera- tional amplifiers, the operational amplifiers and associ- ated circuitry perform the operations of addition, sub- traction, and integration. Analog multipliers implement the nonlinear terms in the circuit equations, and is easy to construct [12]. Circuit schematic for implementing the new chaotic system in (6). By applying standard node analysis techniques to the circuit of Figure 8, a set of state equations that govern the dynamical behavior of the circuit can be obtained. This set of equations is given by –3 –2 –1 0 1 2 3 4 –0.5 –1 –2 –3 –3 –2 –1 0 1 2 3 4 –3 –2 –1 0 1 2 3 4 5 6 –0.5 –5 –10 –15 –20 x x y z y z Q. H. ALSAFASFEH ET AL. Copyright © 2011 SciRes. CS 104 Figure 8. The electronic circuit schematic of the new chaotic system. (8): 11 311141 52 8262 9272 10 311312 313 314 3 1111 1111 1 11111 xyxz RCRCRCRC yxyxz RC RCRC RCRC zxyzdxzx) RCRCRCRCRC (8) For the chosen component value is equivalent to after rescaling time by a factor of 1500. An electronic circuit of the new chaotic system is implemented with parame- ters of (δ = 20, r = 20,a = 9, β = 8.5, b = 0 and c = 8 ) and initial conditions x0 = 0.0010, y0 = 0.001, z0 = 0.1 LM741 opamps, and the analog multipliers are used with R1 = R2 = R5 =20 K, R3 = R4 = R6 = R8 = R12 = 400 K, R9 = 44.44 K, R10=8 K, R11 = 47.06, R7 = 2 K, R13 = 40 K and R14 = 50 K and C1 = C2 = C3 = 1 nF. The output voltage is the products of the inputs multiplied by 10 V. PSpice simulations of the new chaotic system are also attained in Figure 9, Figure 10, and Figure 11 for xy, xz, and yz attractors, respectively. In this simulation, the parameters (δ,r,a,β,b and c) are set at a value of 20,20,9,8.5,0 and 8. Figure 9. PSpice simulation result of the new chaotic sys- tem’s electronic oscillator (Figure 3) for xz strange attrac- tor. 4. Conclusions In this paper, we have displayed a three-dimensional continuous autonomous chaotic system modified from the Lorenz system and Rössler system, which the first equation has not non-linear cross-product term but the second equation has one non-linear cross-product term Q. H. ALSAFASFEH ET AL. Copyright © 2011 SciRes. CS 105 Figure 10. PSpice simulation result of the new chaotic sys- tem’s electronic oscillator (Figure 4) for xy strange attrac- tor. Figure 11. PSpice simulation result of the new chaotic sys- tem’s electronic oscillator (Figure 5) for yz strange attrac- tor. and the third one has two non-linear cross-product term. Part of the basic dynamic behavior of the system is ex- plored further investigation in the Lyapunov Exponent and bifurcation diagrams. Moreover, this was the new system also physically realized using analogue electronic circuits. 5. References [1] G. Chen and X. 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