分数阶Lorenz系统的分析及电路实现

贾红艳; 陈增强; 薛薇

doi:10.7498/aps.62.140503

摘要

频域传递函数近似方法不仅是常用的分数阶混沌系统相轨迹的数值分析方法之一, 而且也是设计分数阶混沌系统电路的主要方法. 应用该方法首先研究了分数阶Lorenz系统的混沌特性, 通过对Lyapunov指数图、分岔图和数值仿真分析, 发现了其较为丰富的动态特性, 即当分数阶次从0.7到0.9以步长0.1变化时, 该分数阶Lorenz系统既存在混沌特性, 又存在周期特性, 从数值分析上说明了在更低维的Lorenz系统中存在着混沌现象. 然后又基于该方法和整数阶混沌电路的设计方法, 设计了一个模拟电路实现了该分数阶Lorenz系统, 电路中的电阻和电容等数值是由系统参数和频域传递函数近似确定的. 通过示波器观测到了该分数阶Lorenz系统的混沌吸引子和周期吸引子的相轨迹图, 这些电路实验结果与数值仿真分析是一致的, 进一步从物理实现上说明了其混沌特性.

关键词:

Abstract

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.

Keywords:

作者及机构信息

1.
天津科技大学自动化系, 天津 300222;

2.
南开大学自动化系, 天津 300071

基金项目: 国家自然科学基金青年科学基金(批准号: 11202148); 国家自然科学基金(批准号: 61174094);高等学校博士学科点专项科研基金(批准号: 20090031110029)和天津科技大学科学研究基金(批准号: 20110124)资助的课题.

Authors and contacts

1.
Department of Automation, Tianjin University of Science and Technology, Tianjin 300222, China;

2.
Department of Automation, Nankai University, Tianjin 300071, China

Funds: Project supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11202148), the Natural Science Foundation of China (Grant No. 61174094), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20090031110029), and the Research Fund of Tianjin University of Science and Technology, China (Grant No. 20110124).

参考文献

[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

施引文献

[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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