新Chua多涡卷混沌吸引子的产生及应用

贾美美; 蒋浩刚; 李文静

doi:10.7498/aps.68.20182183

摘要

本文采用对数函数序列构造了一个新Chua多涡卷混沌系统, 分析了该系统的非线性动力学行为, 主要包括对称性、不变性、平衡点、最大李雅普诺夫指数等. 然后, 设计递归反步控制器控制Chua多涡卷混沌系统中的混沌行为. 最后, 利用Chua多涡卷混沌系统检测了多频微弱周期信号. 结果表明, 对数函数序列与新Chua双涡卷混沌系统相结合能够产生丰富的Chua多涡卷混沌吸引子. 产生机制为指标2的鞍焦平衡点用于产生涡卷, 指标1的鞍焦平衡点用于连接涡卷. 递归反步控制器能够将Chua多涡卷混沌系统控制到不动点或给定的正弦函数. Chua多涡卷混沌系统与递归反步控制器相结合的新微弱信号检测方法能够检测出淹没在高斯噪声背景下多频微弱周期信号的各频率.

关键词:

Abstract

Chaos has great potential applications in engineering fields, such as secure communication and digital encryption. Since the double-scroll Chua’s circuit was developed first by Chua, it has quickly become a paradigm to study the double-scroll chaotic attractors. Compared with the conventional double-scroll chaotic attractors, the multi-scroll chaotic attractors have complex structures and rich nonlinear dynamical behaviors. The multi-scroll chaotic attractors have been applied to various chaos-based information technologies, such as secure communication and chaotic cryptanalysis. Hence, the generation of the multi-scroll chaotic attractors has become a hot topic in research field of chaos at present. In this paper, a novel Chua multi-scroll chaotic system is constructed by using a logarithmic function series. The nonlinear dynamical behaviors of the novel Chua multi-scroll chaotic system are analyzed, including symmetry, invariance, equilibrium points, the largest Lyapunov exponent, etc. The existence of chaos is confirmed by theoretical analyses and numerical simulations. The results show that the rich Chua multi-scroll chaotic attractors can be generated by combining the logarithmic function series with the novel Chua double-scroll chaotic system. The generation mechanism of the Chua multi-scroll chaotic attractors is that the saddle-focus equilibrium points of index 2 are used to generate the scrolls, and the saddle-focus equilibrium points of index 1 are used to connect these scrolls. Then, three recursive back-stepping controllers are designed to control the chaotic behavior in the novel Chua multi-scroll chaotic system. The recursive back-stepping controllers can control the novel Chua multi-scroll chaotic system to a fixed point or a given sinusoidal function. Finally, a new method of detecting a weak signal embedded in the Gaussian noise is proposed on the basis of the novel Chua multi-scroll chaotic system and the recursive back-stepping controllers. The immunity of the novel Chua multi-scroll chaotic system to the Gaussian noise with the zero mean is analyzed by using the stochastic differential equation theory. The results show that the proposed new method of detecting the weak signal can detect the frequencies of the multi-frequency weak periodic signal embedded in the Gaussian noise. In addition, the novel Chua multi-scroll chaotic system has strong immunity to any Gaussian noise with the zero mean. The proposed method provides a new thought for detecting the weak signal.

Keywords:

作者及机构信息

内蒙古工业大学电力学院自动化系, 呼和浩特　010080

通信作者: 贾美美, meimeijia14@163.com

基金项目: 内蒙古工业大学校基金(批准号: ZD201520)和内蒙古自治区自然科学基金(批准号: 2017BS0603)资助的课题.

Authors and contacts

Department of Automation, College of Electric Power, Inner Mongolia University of Technology, Hohhot 010080, China

Corresponding author: Jia Mei-Mei, meimeijia14@163.com

Funds: Project supported by the Inner Mongolia University of Technology Foundation, China (Grant No. ZD201520) and the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 2017BS0603).

文章全文 : translate this paragraph

参考文献

[1]	Chua L O, Komuro M, Matsumoto T 1986 IEEE T. Circuits 33 1072 Google Scholar
[2]	Suykens J A K, Van de walle J 1993 IEEE T. Circuits-I 40 861 Google Scholar
[3]	Lü J H, Chen G R, Yu X H, Leung H 2004 IEEE T. Circuits-I 51 2476 Google Scholar
[4]	Lü J H, Han F L, Yu X H, Chen G R 2004 Automatica 40 1677 Google Scholar
[5]	陈仕必, 曾以成, 徐茂林, 陈家胜 2011 60 020507 Google Scholar Chen S B, Zeng Y C, Xu M L, Chen J S 2011 Acta Phys. Sin. 60 020507 Google Scholar
[6]	艾星星, 孙克辉, 贺少波 2014 63 040503 Google Scholar Ai X X, Sun K H, He S B 2014 Acta Phys. Sin. 63 040503 Google Scholar
[7]	Hong Q H, Xie Q G, Xiao P 2017 Nonlinear Dyn. 87 1015 Google Scholar
[8]	Zhang G T, Wang F Q 2018 Chin. Phys. B 27 018201 Google Scholar
[9]	Xu F, Yu P 2010 J. Math. Anal. Appl. 362 252 Google Scholar
[10]	Chen Z, Wen G L, Zhou H A, Chen J Y 2017 Optik 130 594 Google Scholar
[11]	Wang C H, Luo X W, Wan Z 2014 Optik 125 6716 Google Scholar
[12]	Lü J H, Murali K, Sinha S, Leung H, Aziz-Alaoui M A 2008 Phys. Lett. A 372 3234 Google Scholar
[13]	Yuan F, Wang G Y, Wang X W 2016 Chaos 26 073107 Google Scholar
[14]	Wang C H, Liu X M, Xia H 2017 Chaos 27 033114 Google Scholar
[15]	Hu X Y, Liu C X, Liu L, Yao Y P, Zheng G C 2017 Chin. Phys. B 26 110502 Google Scholar
[16]	Wang C H, Xia H, Zhou L 2017 Int. J. Bifurcat. Chaos 27 1750091 Google Scholar
[17]	肖利全, 段书凯, 王丽丹 2018 67 090502 Google Scholar Xiao L Q, Duan S K, Wang L D 2018 Acta Phys. Sin. 67 090502 Google Scholar
[18]	Wang G Y, Yuan F, Chen G R, Zhang Y 2018 Chaos 28 013125 Google Scholar
[19]	Ott E, Grebogi C, Yorke J A 1990 Phys. Rev. Lett. 64 1196 Google Scholar
[20]	Yang C H, Ge Z M, Chang C M, Li S Y 2010 Nonlinear Anal-real. 11 1977 Google Scholar
[21]	Danaca M F, Fečkan M 2019 Commun. Nonlinear Sci. 74 1 Google Scholar
[22]	Litak G, Syta A, Borowice M 2007 Chaos Soliton. Fract. 32 694 Google Scholar
[23]	Gamal Mahmoud M, Ayman A A, Tarek M A, Emad E M 2017 Chaos Soliton. Fract. 104 680 Google Scholar
[24]	Shen Y J, Wen S F, Yang S P, Guo S Q, Li L R 2018 Int. J. Nonlin. Mech. 98 173 Google Scholar
[25]	Mfoumou G S, Kenmoé G D, Kofané T C 2019 Mech. Syst. Signal Pr. 119 399 Google Scholar
[26]	Harb A, Zaher A, Zohdy M 2002 Proceedings of the American Control Conference Anchorage, Ak, USA, May 8-10, 2002 p2251
[27]	Laoye J A, Vincent U E, Kareem S O 2009 Chaos Soliton. Fract. 39 356 Google Scholar
[28]	Njah A N, Sunday O D 2009 Chaos Soliton. Fract. 41 2371 Google Scholar
[29]	Njah A N 2010 Nonlinear Dyn. 61 1 Google Scholar
[30]	Birx D L, Pipenberg S J 1992 International Joint Conference on Neural Networks Baltimore, MD, USA, June 7-11, 1992 p881
[31]	徐艳春, 杨春玲 2010 哈尔滨工业大学学报 42 446 Google Scholar Xu Y C, Yang C L 2010 Journal of Harbin Institute of Technology 42 446 Google Scholar
[32]	Xu Y C, Yang C L, Qu X D 2010 Chin. Phys. B 19 030516 Google Scholar
[33]	Li G Z, Zhang B 2017 IEEE T. Ind. Electron. 64 2255
[34]	Zhong G Q, Man K F, Chen G R 2002 Int. J. Bifurcat. Chaos 12 2907 Google Scholar
[35]	Yu S M, Lü J H, Chen G R 2007 Int. J. Bifurcat. Chaos 17 1785 Google Scholar
[36]	Christopher P S 1993 IEEE T. Circuits-I 40 675 Google Scholar
[37]	Cafagna D, Grassi G 2003 Int. J. Bifurcat. Chaos 13 2889 Google Scholar
[38]	李月, 杨宝俊, 石要武 2003 52 526 Google Scholar Li Y, Yang B J, Shi Y W 2003 Acta Phys. Sin. 52 526 Google Scholar
[39]	钱勇, 黄成军, 陈陈, 江秀臣 2007 中国电机工程学报 27 89 Qian Y, Huang C J, Chen C, Jiang X C 2007 Proceedings of the CSEE 27 89

施引文献

图 1 对数函数

Fig. 1. Logarithmic function.

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图 2 新Chua双涡卷混沌系统的电路图

Fig. 2. Circuit diagram of the novel Chua double-scroll chaotic system.

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图 3 $\left| {{v_{{C_1}}}} \right|$的电路图

Fig. 3. Circuit diagram of $\left| {{v_{{C_1}}}} \right|$.

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图 8 $g\left( {{v_{{C_1}}}} \right) = \log _2^{\frac{{{{\rm{e}}^{ - 4\left| {{v_{{C_1}}}} \right|}} + 1}}{2}} \cdot {\rm{sgn}} ( - {v_{{C_1}}})$的电路图

Fig. 8. Circuit diagram of $g\left( {{v_{{C_1}}}} \right) = \log _2^{\frac{{{{\rm{e}}^{ - 4\left| {{v_{{C_1}}}} \right|}} + 1}}{2}}\times$ $ {\rm{sgn}} ( - {v_{{C_1}}})$.

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图 4 ${{\rm{e}}^{4\left| {{v_{{C_1}}}} \right|}}$的电路图

Fig. 4. Circuit diagram of ${{\rm{e}}^{4\left| {{v_{{C_1}}}} \right|}}$.

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图 5 ${{\rm{e}}^{ - 4\left| {{v_{{C_1}}}} \right|}}$的电路图

Fig. 5. Circuit diagram of ${{\rm{e}}^{ - 4\left| {{v_{{C_1}}}} \right|}}$.

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图 6 $\log _2^{\frac{{{{\rm{e}}^{ - 4\left| {{v_{{C_1}}}} \right|}} + 1}}{2}}$的电路图

Fig. 6. Circuit diagram of $\log _2^{\frac{{{{\rm{e}}^{ - 4\left| {{v_{{C_1}}}} \right|}} + 1}}{2}}$.

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图 7 ${\rm{sgn}} \left( { - {v_{{C_1}}}} \right)$的电路图

Fig. 7. Circuit diagram of ${\rm{sgn}} \left( { - {v_{{C_1}}}} \right)$.

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图 9 系统[(20)式]的李雅普诺夫指数

Fig. 9. Lyapunov exponents of system (Equation(20)).

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图 10 新Chua双涡卷混沌系统　(a) $x\text{ - }y$平面的相图; (b) x方向的时域图

Fig. 10. Novel Chua double-scroll chaotic system: (a) Phase diagram on the $x\text{ - }y$ plane; (b) time domain diagram in the x direction.

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图 11 多分段对数函数序列[(24)式], 取M = 1

Fig. 11. Multi-segment logarithmic function series(Equation(24)) with M = 1.

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图 12 多分段对数函数序列[(25)式], 取N = 1

Fig. 12. Multi-segment logarithmic function series(Equation(25)) with N = 1.

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图 13 x-y平面4-涡卷混沌吸引子的相图

Fig. 13. Phase diagram of the 4-scroll chaotic attractor on the x-y plane.

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图 14 x-z平面12-涡卷混沌吸引子的相图

Fig. 14. Phase diagram of the 12-scroll chaotic attractor on the x-z plane.

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图 15 最大李雅普诺夫指数

Fig. 15. Largest Lyapunov exponent.

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图 16 x-y平面的庞加莱映射

Fig. 16. Poincaré mapping on the x-y plane.

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图 17 状态变量和期望值$\left[ {\sin \left( t \right),{\rm{0}},{\rm{0}}} \right]$的时域图　(a) x, x_d; (b) y, y_d; (c) z, z_d

Fig. 17. Time domain diagram of state variables and desired values $\left[ {\sin \left( t \right),{\rm{0}},{\rm{0}}} \right]$: (a) x, x_d; (b) y, y_d; (c) z, z_d.

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图 18 状态变量和期望值$\left( {{\rm{0}},{\rm{0}},{\rm{0}}} \right)$的时域图　(a) x, x_d; (b) y, y_d; (c) z, z_d

Fig. 18. Time domain diagram of state variables and desired values$\left( {{\rm{0}},{\rm{0}},{\rm{0}}} \right)$: (a) x, x_d; (b) y, y_d; (c) z, z_d.

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图 19 检测原理图

Fig. 19. Detection schematic diagram.

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图 20 系统[(52)式]的相图

Fig. 20. Phase diagram of system [Equation(52)].

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图 21 检测系统[(53)式]的相图

Fig. 21. Phase diagram of the detection system [Equation(53)]

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图 22 控制信号${U_1}\left( t \right)$、${U_2}\left( t \right)$、${U_3}\left( t \right)$的时域图　(a)原始图; (b)放大图, $ t = [0,100] {\rm{s}} $

Fig. 22. Time domain diagram of control signals ${U_1}\left( t \right)$、${U_2}\left( t \right)$、${U_3}\left( t \right)$: (a) Original diagram (b) enlarging diagram, $t = [0,100]{\rm{ s}}$.

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图 23 待测信号的频谱图

Fig. 23. Frequency spectrum of the signal to be detected.

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表 1 12-涡卷混沌吸引子的平衡点、特征值和平衡点的类型

Table 1. Equilibrium points, eigenvalues and types of equilibrium points for the 12-scroll chaotic attractor.

平衡点	特征值	平衡点的类型
${Q_0}\left( {0,0,0} \right)$	$67.2809$,$ - 0.5730 \pm {\rm{i3}}{\rm{.9544}}$	Ⅰ
${Q_{1,2}}\left( { \pm 10,0,0} \right)$	$67.2809$,$ - 0.5730 \pm {\rm{i3}}{\rm{.9544}}$	Ⅰ
${Q_{3,4}}\left( { \pm 20,0,0} \right)$	$67.2809$,$ - 0.5730 \pm {\rm{i3}}{\rm{.9544}}$	Ⅰ
${Q_{5,6}}\left( { \pm 30,0,0} \right)$	$67.2809$,$ - 0.5730 \pm {\rm{i3}}{\rm{.9544}}$	Ⅰ
${Q_{7,8}}\left( { \pm 40,0,0} \right)$	$67.2809$,$ - 0.5730 \pm {\rm{i3}}{\rm{.9544}}$	Ⅰ
${Q_{9,10}}\left( { \pm 50,0,0} \right)$	$67.2809$,$ - 0.5730 \pm {\rm{i3}}{\rm{.9544}}$	Ⅰ
${Q_{11,12}}\left( { \pm 5,0,0} \right)$	$ - 6.2777$,$0.1389 \pm {\rm{i3}}{\rm{.5671}}$	Ⅱ
${Q_{13,14}}\left( { \pm 15,0,0} \right)$	$ - 6.2777$,$0.1389 \pm {\rm{i3}}{\rm{.5671}}$	Ⅱ
${Q_{15,16}}\left( { \pm 25,0,0} \right)$	$ - 6.2777$,$0.1389 \pm {\rm{i3}}{\rm{.5671}}$	Ⅱ
${Q_{17,18}}\left( { \pm 35,0,0} \right)$	$ - 6.2777$,$0.1389 \pm {\rm{i3}}{\rm{.5671}}$	Ⅱ
${Q_{19,20}}\left( { \pm 45,0,0} \right)$	$ - 6.2777$,$0.1389 \pm {\rm{i3}}{\rm{.5671}}$	Ⅱ
${Q_{2{\rm{1}},2{\rm{2}}}}\left( { \pm 55,0,0} \right)$	$ - 6.2777$,$0.1389 \pm {\rm{i3}}{\rm{.5671}}$	Ⅱ

下载: 导出CSV

map

[1]	Chua L O, Komuro M, Matsumoto T 1986 IEEE T. Circuits 33 1072 Google Scholar
[2]	Suykens J A K, Van de walle J 1993 IEEE T. Circuits-I 40 861 Google Scholar
[3]	Lü J H, Chen G R, Yu X H, Leung H 2004 IEEE T. Circuits-I 51 2476 Google Scholar
[4]	Lü J H, Han F L, Yu X H, Chen G R 2004 Automatica 40 1677 Google Scholar
[5]	陈仕必, 曾以成, 徐茂林, 陈家胜 2011 60 020507 Google Scholar Chen S B, Zeng Y C, Xu M L, Chen J S 2011 Acta Phys. Sin. 60 020507 Google Scholar
[6]	艾星星, 孙克辉, 贺少波 2014 63 040503 Google Scholar Ai X X, Sun K H, He S B 2014 Acta Phys. Sin. 63 040503 Google Scholar
[7]	Hong Q H, Xie Q G, Xiao P 2017 Nonlinear Dyn. 87 1015 Google Scholar
[8]	Zhang G T, Wang F Q 2018 Chin. Phys. B 27 018201 Google Scholar
[9]	Xu F, Yu P 2010 J. Math. Anal. Appl. 362 252 Google Scholar
[10]	Chen Z, Wen G L, Zhou H A, Chen J Y 2017 Optik 130 594 Google Scholar
[11]	Wang C H, Luo X W, Wan Z 2014 Optik 125 6716 Google Scholar
[12]	Lü J H, Murali K, Sinha S, Leung H, Aziz-Alaoui M A 2008 Phys. Lett. A 372 3234 Google Scholar
[13]	Yuan F, Wang G Y, Wang X W 2016 Chaos 26 073107 Google Scholar
[14]	Wang C H, Liu X M, Xia H 2017 Chaos 27 033114 Google Scholar
[15]	Hu X Y, Liu C X, Liu L, Yao Y P, Zheng G C 2017 Chin. Phys. B 26 110502 Google Scholar
[16]	Wang C H, Xia H, Zhou L 2017 Int. J. Bifurcat. Chaos 27 1750091 Google Scholar
[17]	肖利全, 段书凯, 王丽丹 2018 67 090502 Google Scholar Xiao L Q, Duan S K, Wang L D 2018 Acta Phys. Sin. 67 090502 Google Scholar
[18]	Wang G Y, Yuan F, Chen G R, Zhang Y 2018 Chaos 28 013125 Google Scholar
[19]	Ott E, Grebogi C, Yorke J A 1990 Phys. Rev. Lett. 64 1196 Google Scholar
[20]	Yang C H, Ge Z M, Chang C M, Li S Y 2010 Nonlinear Anal-real. 11 1977 Google Scholar
[21]	Danaca M F, Fečkan M 2019 Commun. Nonlinear Sci. 74 1 Google Scholar
[22]	Litak G, Syta A, Borowice M 2007 Chaos Soliton. Fract. 32 694 Google Scholar
[23]	Gamal Mahmoud M, Ayman A A, Tarek M A, Emad E M 2017 Chaos Soliton. Fract. 104 680 Google Scholar
[24]	Shen Y J, Wen S F, Yang S P, Guo S Q, Li L R 2018 Int. J. Nonlin. Mech. 98 173 Google Scholar
[25]	Mfoumou G S, Kenmoé G D, Kofané T C 2019 Mech. Syst. Signal Pr. 119 399 Google Scholar
[26]	Harb A, Zaher A, Zohdy M 2002 Proceedings of the American Control Conference Anchorage, Ak, USA, May 8-10, 2002 p2251
[27]	Laoye J A, Vincent U E, Kareem S O 2009 Chaos Soliton. Fract. 39 356 Google Scholar
[28]	Njah A N, Sunday O D 2009 Chaos Soliton. Fract. 41 2371 Google Scholar
[29]	Njah A N 2010 Nonlinear Dyn. 61 1 Google Scholar
[30]	Birx D L, Pipenberg S J 1992 International Joint Conference on Neural Networks Baltimore, MD, USA, June 7-11, 1992 p881
[31]	徐艳春, 杨春玲 2010 哈尔滨工业大学学报 42 446 Google Scholar Xu Y C, Yang C L 2010 Journal of Harbin Institute of Technology 42 446 Google Scholar
[32]	Xu Y C, Yang C L, Qu X D 2010 Chin. Phys. B 19 030516 Google Scholar
[33]	Li G Z, Zhang B 2017 IEEE T. Ind. Electron. 64 2255
[34]	Zhong G Q, Man K F, Chen G R 2002 Int. J. Bifurcat. Chaos 12 2907 Google Scholar
[35]	Yu S M, Lü J H, Chen G R 2007 Int. J. Bifurcat. Chaos 17 1785 Google Scholar
[36]	Christopher P S 1993 IEEE T. Circuits-I 40 675 Google Scholar
[37]	Cafagna D, Grassi G 2003 Int. J. Bifurcat. Chaos 13 2889 Google Scholar
[38]	李月, 杨宝俊, 石要武 2003 52 526 Google Scholar Li Y, Yang B J, Shi Y W 2003 Acta Phys. Sin. 52 526 Google Scholar
[39]	钱勇, 黄成军, 陈陈, 江秀臣 2007 中国电机工程学报 27 89 Qian Y, Huang C J, Chen C, Jiang X C 2007 Proceedings of the CSEE 27 89

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