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Transfer function approximation in frequency domain is not only one of common numerical analysis methods studying portraits of fractional-order chaotic systems, but also a main method to design their chaotic circuits. According to it, in this paper we first investigate the chaotic characteristics of the fractional-order Lorenz system, find some more complex dynamics by analyzing Lyapunov exponents diagrams, bifurcation diagrams and phase portraits, that is, we display the chaotic characteristics as well as periodic characteristics of the system when changing fractional-order from 0.7 to 0.9 in steps of 0.1, and show that the chaotic motion exists in the a lower-dimensional fractional-order Lorenz system. Then, according to transfer function approximation and the approach to designing integer-order chaotic circuits, we also design an analog circuit to implement the fractional-order system. The resistors and capacitors in the circuit are selected according to the system parameters and transfer function approximation in frequency domain. Some phase portraits including chaotic attractors and periodic attractors are observed by oscilloscope, which are coincident well with numerical simulations, and the chaotic characteristics of the fractional-order Lorenz system are further proved by the physical implementation.
[1] Hartley T T, Lorenzo C F, Qammer H K 1995 IEEE Trans. Circuits Syst.-I: Fundamental Theory and Applications 42 485
[2] Li C G, Chen G R 2004 Physica A 341 55
[3] Grigorenko I, Grigorenko E 2003 Phys. Rev. Lett. 91 034101
[4] Ichise M, Nagayanagi Y, Kojima T 1971 J. Electroanal. Chem. 33 253
[5] Bagley R L, Calico R A 1991 J. Guid. Contr. Dyn. 14 304
[6] Sugimoto N 1991 J. Fluid Mech. 25 631
[7] Torvik P J, Bagley R L 1984 J. Appl. Mech. Trans. ASMF 51 294
[8] Lu J G, Chen G R 2006 Chaos, Solitons and Fractals 27 685
[9] Li C P, Guo J P 2004 Chaos, Solitons and Fractals 22 443
[10] Li C G, Chen G R 2004 Chaos, Solitons and Fractals 22 549
[11] Lu J G 2006 Phys. Lett. A 354 305
[12] Huang X, Zhao Z, Wang Z, Li Y X 2012 Neurocomputing 94 13
[13] Ge Z M, Qu C Y 2007 Chaos, Solitons and Fractals 34 262
[14] Hu J B, Xiao J, Zhao L D 2011 Acta Phys. Sin. 60 110515 (in Chinese) [胡建兵, 肖建, 赵灵东 2011 60 110515]
[15] Wu C J, Zhang Y B, Yang N N 2011 Chin. Phys. B 20 060505
[16] Chen L P, Chai Y, Wu R W, Sun J, Ma T D 2012 Phys. Lett. A 376 2381
[17] Wang Z, Huang X, Zhao Z 2012 Nonlinear Dyn. 69 999
[18] Li H Q, Liao X F, Lou M W 2012 Nonlinear Dyn. 68 137
[19] Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese) [刘崇新2007 56 6865]
[20] Chen X R, Liu C X, Wang F Q, Li Y X 2008 Acta Phys. Sin. 57 1416 (in Chinese) [陈向荣, 刘崇新, 王发强, 李永勋 2008 57 1416]
[21] Yu Y G, Li H X, Wang S, Yu J Z 2009 Chaos, Solitons and Fractals 1181
[22] Yu S M, L J H, Chen G R 2007 Phys. Lett. A 364 244
[23] Yang X S, Li Q D, Chen G R 2003 Int. J. Circ. Theor. Appl. 31 637
[24] Li Y X, Tang W K S, Chen G R 2005 Int. J. Circ. Theor. Appl. 33 235
[25] Jia H Y, Chen Z Q, Yuan Z Z 2009 Acta Phys. Sin. 58 4469 (in Chinese) [贾红艳, 陈增强, 袁著祉 2009 58 4469]
[26] Wang G Y, He H L 2008 Chin. Phys. B 17 4014
[27] Wang G Y, Liu J B, Zheng X 2007 Chin. Phys. 16 2278
[28] Zhang Z X, Yu S M 2009 Chin. Phys. B 18 119
[29] Yu S M, Yu Z D 2008 Acta Phys. Sin. 57 6859 (in Chinese) [禹思敏, 禹之鼎 2008 57 6859]
[30] Liu Y Z 2008 Acta Phys. Sin. 57 1439 (in Chinese) [刘扬正 2008 57 1439]
[31] Liu Y Z, Lin C S, Li X C 2011 Acta Phys. Sin. 60 060507 (in Chinese) [刘扬正, 林长圣, 李心朝 2011 60 060507]
[32] Charef A, Sun Y Y, Tsao Y Y 1992 IEEE Trans. Autom. Control 37 1465
[33] Ahmad W M, Sprott J C 2003 Chaos, Solitons and Fractals 16 339
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[1] Hartley T T, Lorenzo C F, Qammer H K 1995 IEEE Trans. Circuits Syst.-I: Fundamental Theory and Applications 42 485
[2] Li C G, Chen G R 2004 Physica A 341 55
[3] Grigorenko I, Grigorenko E 2003 Phys. Rev. Lett. 91 034101
[4] Ichise M, Nagayanagi Y, Kojima T 1971 J. Electroanal. Chem. 33 253
[5] Bagley R L, Calico R A 1991 J. Guid. Contr. Dyn. 14 304
[6] Sugimoto N 1991 J. Fluid Mech. 25 631
[7] Torvik P J, Bagley R L 1984 J. Appl. Mech. Trans. ASMF 51 294
[8] Lu J G, Chen G R 2006 Chaos, Solitons and Fractals 27 685
[9] Li C P, Guo J P 2004 Chaos, Solitons and Fractals 22 443
[10] Li C G, Chen G R 2004 Chaos, Solitons and Fractals 22 549
[11] Lu J G 2006 Phys. Lett. A 354 305
[12] Huang X, Zhao Z, Wang Z, Li Y X 2012 Neurocomputing 94 13
[13] Ge Z M, Qu C Y 2007 Chaos, Solitons and Fractals 34 262
[14] Hu J B, Xiao J, Zhao L D 2011 Acta Phys. Sin. 60 110515 (in Chinese) [胡建兵, 肖建, 赵灵东 2011 60 110515]
[15] Wu C J, Zhang Y B, Yang N N 2011 Chin. Phys. B 20 060505
[16] Chen L P, Chai Y, Wu R W, Sun J, Ma T D 2012 Phys. Lett. A 376 2381
[17] Wang Z, Huang X, Zhao Z 2012 Nonlinear Dyn. 69 999
[18] Li H Q, Liao X F, Lou M W 2012 Nonlinear Dyn. 68 137
[19] Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese) [刘崇新2007 56 6865]
[20] Chen X R, Liu C X, Wang F Q, Li Y X 2008 Acta Phys. Sin. 57 1416 (in Chinese) [陈向荣, 刘崇新, 王发强, 李永勋 2008 57 1416]
[21] Yu Y G, Li H X, Wang S, Yu J Z 2009 Chaos, Solitons and Fractals 1181
[22] Yu S M, L J H, Chen G R 2007 Phys. Lett. A 364 244
[23] Yang X S, Li Q D, Chen G R 2003 Int. J. Circ. Theor. Appl. 31 637
[24] Li Y X, Tang W K S, Chen G R 2005 Int. J. Circ. Theor. Appl. 33 235
[25] Jia H Y, Chen Z Q, Yuan Z Z 2009 Acta Phys. Sin. 58 4469 (in Chinese) [贾红艳, 陈增强, 袁著祉 2009 58 4469]
[26] Wang G Y, He H L 2008 Chin. Phys. B 17 4014
[27] Wang G Y, Liu J B, Zheng X 2007 Chin. Phys. 16 2278
[28] Zhang Z X, Yu S M 2009 Chin. Phys. B 18 119
[29] Yu S M, Yu Z D 2008 Acta Phys. Sin. 57 6859 (in Chinese) [禹思敏, 禹之鼎 2008 57 6859]
[30] Liu Y Z 2008 Acta Phys. Sin. 57 1439 (in Chinese) [刘扬正 2008 57 1439]
[31] Liu Y Z, Lin C S, Li X C 2011 Acta Phys. Sin. 60 060507 (in Chinese) [刘扬正, 林长圣, 李心朝 2011 60 060507]
[32] Charef A, Sun Y Y, Tsao Y Y 1992 IEEE Trans. Autom. Control 37 1465
[33] Ahmad W M, Sprott J C 2003 Chaos, Solitons and Fractals 16 339
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