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AROB Jimukyoku Journal Style AROB Jimukyoku Journal of Advances in Artificial Life Robotics, Vol. 1, No. 3 (December 2020) 132-136 Analysis and Circuit Design of a Novel 4D Chaotic System Yan Sun Department of Automation, Tianjin University of Science and Technology, 1038 Dagunanlu Road, Hexi District, Tianjin 300222, PR China Yongchao Zhang Department of Automation, Tianjin University of Science and Technology, 1038 Dagunanlu Road, Hexi District, Tianjin 300222, PR China Jiaqi Chen Department of Automation, Tianjin University of Science and Technology, 1038 Dagunanlu Road, Hexi District, Tianjin 300222, PR China E-mail: [email protected]; www.tust.edu.cn Abstract In this paper, a novel 4D chaotic system is proposed. First, the basic dynamic characteristics of this system are analyzed theoretically. Second, dynamical properties of the system are investigated by phase portraits, Poincaré sections, Lyapunov exponent spectrum and bifurcation diagram. In addition, an analog circuit of the system is designed and simulated by Multisim, a circuit simulation software. The experimental results show that the circuit simulation results are consistent with the numerical simulation results. Therefore, the existence of chaotic attractor in the system is verified from the physical level. Keywords: Novel 4D chaotic system; Dynamic characteristics; Simulation analysis; Analog circuit design. Through chaos theory, we can not only further 1. Introduction investigate the nonlinear characteristics and their Since 1963, when Lorenz discovered the butterfly generation mechanism in these fields, but also provide a effect [1], more and more scholars have studied chaos. In theoretical basis for the construction of new chaotic recent decades, with the development of mathematical systems. theory, chaos research has also made great progress In this paper, a novel 4D chaotic system is proposed. with emerging a number of newly built systems. For Firstly, the dissipation and equilibrium of the system are example, in 1993, Chua et al. designed Chua circuits in theoretically analyzed. Secondly, simulation analysis is which the mathematical model became a very classical carried out through phase portraits, Poincaré section chaotic system [2]. And Sprott J et al. proposed Sprott diagrams, Lyapunov exponent spectrum and bifurcation system in the following year [3]. In 1999, Chen system diagram to analyze the dynamic characteristics of the was successfully constructed by Chen et al. [4] In the system in detail. Then, in order to illustrate the early 21century, Lü et al. discovered the Lü system [5]. A existence of chaotic attractor in the system from the three-dimensional continuous autonomous chaotic physical level, the circuit design and simulation are system was constructed by Liu et al. called Liu system carried out with the circuit simulation software [6]. Qi et al. proposed a four-dimensional chaos system, Multisim. The existence of chaotic attractor in the namely Qi system [7]. Except to studying how to system is further verified in the experiment. construct a system with better dynamic performance, on the basis of existing systems, many scholars also 2. 4D System Model investigate and realize chaotic system through In this paper, a novel 4D chaotic system is proposed. numerical analysis, analog circuit implementation, The system equation of motion is as follows: digital circuit implementation and other methods [8,9,10,11]. In recent years, chaos theory has been applied in many fields, including information studying encryption and secure communication [12,13,14], astrophysics [15], electromagnetic mechanics [16] and economics [17]. Copyright © 2020, the Authors. Published by AROB Jimukyoku This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). 132 AROB Jimukyoku Journal of Advances in Artificial Life Robotics, Vol. 1, No. 3 (December 2020) 132-136 Yan Sun, Yongchao Zhang, Jiaqi Chen x =−++ cx ay xz + yw As can be seen from Fig.1, chaos is generated in the system with the selected parameters and initial values. y =−+ ax bw In order to further illustrate that the state is chaotic 22 (1) z =−+ xw indeed, the Poincaré mapping diagrams under the same w =−− xy by − zw + u parameters and initial values are given. w = 0 and y = 0 In formula (1), x, y, z, w are the system state variables, are the selected sections, and the results are shown in a, b, c are the system adjustable parameters, u is the Fig.2(a) and Fig.2(b). 150 150 external input of the system, and all four numbers are 100 100 positive. 50 50 0 0 2.1 Dissipation -50 -50 The divergence of formula (1) is: -100 -100 -150 -150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 ∂xyzw∂∂∂ ∇V= + + + =−+−czz =−c (2) (a) (b) ∂xy∂∂∂zw Fig.2 Poincaré sections of system (1): (a) w = 0; (b) y = 0. The parameter c is positive, so the divergence of system (1) is negative. This indicates the system (1) is a As can be seen from the Poincaré sections, the map of dissipative system. system (1) under these two sections are many points with dense distribution and fractal structure. This 2.2 Equilibrium point indicates the chaotic behavior occurs in this system. Let all the left sides of equation (1) are 0, then equation (1) becomes: 2.3.2 Lyapunov exponent spectrum and bifurcation diagram 0 =−++cx ay xz + yw The most common way to determine whether a =−+ 0 ax bw system is chaotic is to determine the number of positive 22 (3) 0 =−+xw Lyapunov exponent of a system. For a 4D system, if 0 =−−xy by − zw + u there are two positive Lyapunov exponents, the system is hyperchaotic. If there is one positive Lyapunov Through formula (3), calculate the equilibrium point exponent, the system is chaotic. As analyzing this of system (1). It is found that there is no equilibrium chaotic system, the fixed parameter is a = 50, b = 25, u point in the system. A chaotic attractor generated by a = 70, and the initial values are (1,1,1,1). The Lyapunov system without equilibrium point is a hidden chaotic exponent spectrum and bifurcation diagram of system attractor. (1) changing with the parameter c in the range (0.3,1) are made. The results are as shown in Fig.3(a) and 2.3 Basic dynamic analysis Fig.3(b) respectively. 2.3.1Phase portraits and Poincaré mapping diagram Fig.3.Lyapunov exponent spectrum of system (1) The parameters and external input are a = 50, b = 25, c = 0.47, u = 70, the initial values are (1,1,1,1), the phase portraits of system (1) are shown in Fig.1. 100 150 80 100 60 50 40 0 20 w -50 0 y (a) (b) -100 -20 -150 -40 100 50 100 -60 50 Fig.3 (a): Lyapunov exponent spectrum and (b): 0 0 -80 -50 -50 -100 z -100 -150 x -100 bifurcation diagram of system (1). -100 -50 0 50 100 x (a) (b) By combining the Lyapunov exponent spectrum and Fig.1 Phase portraits of system (1) with a = 50, b = 25, c = bifurcation diagram, it can be analyzed that when 0.47, u = 70: (a) xy− ; (b) xzw−− . c ∈ (0.3,0.74) , system (1) is chaotic; When c ∈ (0.74, 133 AROB Jimukyoku Journal of Advances in Artificial Life Robotics, Vol. 1, No. 3 (December 2020) 132-136 Analysis and Circuit Design 0.783) , system (1) is quasi-periodic. When c ∈ (0.783, x = 10X, y = 10Y, z = 10Z, w = 10W (4) 1) , system (1) is periodic. Select c = 0.47, c = 0.7224, c Put equation (4) into equation (1) and replace X, Y, Z = 0.76 and c = 0.9 of system (1) in these three ranges and W with x, y, z and w again, then: respectively. Make the corresponding phase portraits, x =−++ cx ay10 xz + 10 yw the results are as shown in Fig.4. When c = 0.47, the y =−+ ax bw four Lyapunov exponents are LE1 = 2.5001, LE2 = 0, LE3 22 (5) = –0.2192, LE4 = –2.8070, which accord with the feature z =−+10 xw 10 of chaotic state. When c = 0.7224, the four Lyapunov w =−−−10 xy by 10 zw + u exponents are LE1 = 1.2961, LE2 = 0, LE3 = –0.2706, Now the circuit is designed with the improved LE4 = –1.7442, which accord with the feature of chaotic method. The circuit is standardized in advance, and the state. When c = 0.76, the four Lyapunov exponents are parameters are applied into equation (5). Then, equation LE1 = 0, LE2 = 0, LE3 = –0.3503, LE4 = –0.4908, which (5) becomes: accord with the feature of quasi-periodic state. When c =−−−−−−− = 0.9, the four Lyapunov exponents are LE1 = 0, LE2 = – x 0.47 x 50( y ) 10( xz ) 10( yw ) 0.2664, LE3 = –0.3201, LE4 = –0.3752, which accord yx =−−−50 25( w ) (6) with the feature of periodic state. These features 2 z =−−−10 x 10( ww ) indicate that the dynamic behavior of the system is very w =−10 xy − 25 y − 10 zw −− ( 70) divers. 150 150 In order to observe the phase portraits of system (1) 100 100 in the oscilloscope, time scale transformation is required. 50 50 Let tT=ττ00, = 100 and use t to replace T, so: z 0 0 z -50 -50 x =−−−−−−−47 x 5000( y ) 1000( xz ) 1000( yw ) -100 -100 yx =−−−5000 2500( w ) -150 (7) -100 -50 0 50 100 -150 -100 -50 0 50 100 150 2 x x z =−−−1000 x 1000( ww ) (a) (b) w =−1000 xy − 2500 y − 1000 zw −− ( 7000) 100 80 80 60 60 The circuit model built according to equation (7) is 40 40 shown in Fig.5. The four parts of the circuit correspond 20 20 0 z to four first nonlinear differential equations respectively, z 0 -20 -20 and the circuit design is realized by using operational -40 -40 amplifiers—LF347N chips, multipliers—AD633 chips, -60 -60 [18,19] -80 resistors and capacitors .
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