基于Julia分形的多涡卷忆阻混沌系统

肖利全; 段书凯; 王丽丹

doi:10.7498/aps.67.20172761

摘要

忆阻器作为一种非线性电子元件，能用作混沌系统中的非线性项，从而提高系统的复杂度.分形与混沌是密切相连的，分别对两者的研究都已成熟，却鲜有将分形过程应用到混沌系统中，以产生丰富的混沌吸引子.为了探索将分形与混沌系统相结合的可能性，本文首先提出了一个新的忆阻混沌系统，并从对称性、耗散性、平衡点稳定性、功率谱、Lyapunov指数和分数维等方面探讨了系统的动力学特性；紧接着，把经典的Julia分形过程应用到该忆阻混沌系统中，产生了新的混沌吸引子，并将几种由Julia分形衍生的变形Julia分形过程应用于文中提出的忆阻混沌系统，获得了丰富的混沌吸引子；最后，讨论了分形过程中的复常数对系统的影响.从仿真结果可以看出，分形过程与混沌系统的结合能产生丰富的多涡卷混沌吸引子.这不仅为产生多涡卷混沌吸引子提供了一种新方法，还弥补了使用功能函数方法造成混沌系统不光滑的不足.

关键词:

Abstract

A memristor can be used in chaotic system as a nonlinear term, and thus enhancing the complexity of the chaotic system. Fractal theory is a leading and important branch of nonlinear science, and has been widely studied in many fields in the past few decades. The fractal and chaos are bound tightly and their relevant researches are well-established, but few of them focus on the research of the possibility of combining the fractal and the chaotic system. In order to obtain a multi scroll chaotic attractor, the fractal process is novelty introduced into the memristive chaotic system. In this paper, at first, a new memristive chaotic system is proposed. Then, the dynamic characteristics of the system are discussed from the aspects of symmetry, dissipation, stabilization of equilibrium points, power spectrum, Lyapunov exponent and fractional dimension. A mapping relationship based on classical Julia fractal is established. Through this mapping relationship, a multi-scroll memristive chaotic system based on the Julia fractal is obtained. Moreover, several deformed Julia fractal processes are applied to the memristive chaotic system, and abundant chaotic attractors are obtained. For example, the square term of the Julia fractal expression is multiplied by a coefficient, and according to the difference in coefficient, the resulting chaotic attractors have the same shape but different sizes. The exponent of the square term in the Julia fractal is changed into a variable, and the chaotic attractor of different scroll numbers is obtained according to the difference in power exponent. In addition, a rich multi-scroll chaotic attractor is obtained by using the fractal expression in the form of weighted sum polynomial. Finally, the influence of a complex constant in the fractal process on the system is discussed. The simulation results show that the combination of fractal process and chaotic system can obtain rich chaotic attractors, such as multi-scroll chaotic attractors. In general, compared with the single-scroll chaotic attractor, the multi-scroll chaotic attractor has a higher complexity and more adjustability. In addition, compared with other multi-scroll chaotic system, the proposed multi-scroll chaotic system is easy to adjust the number of the scrolls. To summarize, this work not only provides a new method of generating multi-scroll chaotic attractors, but also makes up for the lack of smoothness of the chaotic system caused by using functional methods.

Keywords:

作者及机构信息

1.
西南大学电子信息工程学院, 重庆 400715;

2.
非线性电路与智能信息处理重庆市重点实验室, 重庆 400715

通信作者: 段书凯, duansk@swu.edu.cn

基金项目: 国家自然科学基金（批准号：61372139，61571372，61672436）、中央高校基本科研业务费专项资金（批准号：XDJK2016A001，XDJK2014A009）和重庆市基础科学与前沿技术研究（批准号：cstc2017jcyjBX0050）资助的课题.

Authors and contacts

1.
School of Electronic and Information Engineering, Southwest University, Chongqing 400715, China;

2.
Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, Chongqing 400715, China

Corresponding author: Duan Shu-Kai, duansk@swu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61372139, 61571372, 61672436), the Fundamental Research Funds for the Central Universities of Ministry of Education of China (Grant Nos. XDJK2016A001, XDJK2014A009), and the Chongqing Basic Science and Frontier Technology Research, China (Grant No. cstc2017jcyjBX0050).

参考文献

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[3]	Zhou J, Huang D 2012 Chin. Phys. B 21 048401
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[6]	Mandelbrot B B 1967 Science 156 636
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[8]	Li H Q, Wang F Q 1999 Fractal Theory and Its Application in Molecular Science (Beijing: Science Press) p33 (in Chinese) [李后强, 汪富泉 1999分形理论及其在分子科学中的应用(北京:科学出版社) 第33页]
[9]	Lorenz E N 1963 J. Atmos. Sci. 20 130
[10]	Chua L O, Komuro M, Matsumoto T 1986 IEEE Trans. Circ. Syst. 33 1072
[11]	Chen G R 1999 Int. J. Bifurcat. Chaos 9 1465
[12]	Wang L D, Drakakis E, Duan S K, He P F, Liao X F 2012 Int. J. Bifurcat. Chaos 22 1250205
[13]	Muthuswamy B, Kokate P P 2009 IETE Tech. Rev. 26 417
[14]	Zhou Z W, Su Y L, Wang W D, Yan Y 2017 J. Petrol. Explor. Prod. Technol. 7 487
[15]	Bouallegue K 2015 Int. J. Bifurcat. Chaos 25 1530002
[16]	Chua L O, Roska T 1993 IEEE Trans. Circ. Syst. I 40 147
[17]	Yalcin M, Suykens J, Vandewalle J, Ozoguz S 2002 Int. J. Bifurcat. Chaos 12 23
[18]	Tang W K S, Zhong G Q, Chen G, Man K F 2001 IEEE Trans. Circ. Syst. I 48 1369
[19]	Zarei A 2015 Nonlinear Dyn. 81 585
[20]	More C, Vlad R, Chauveau E 2010 Nonlinear Dyn. 59 45
[21]	Huan S M, Li Q D, Yang X S 2012 Nonlinear Dyn. 69 1915
[22]	L J H, Yu X H, Chen G R 2003 IEEE Trans. Circ. Syst. I 50 198
[23]	Yalcin M, Suykens J, van de Walle J 2005 Chaos Modeling and Control Systems Design (Singapore: World Scientific) p59
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施引文献

[1]	Chua L O 1971 IEEE Trans. Circ. Theor. 18 507
[2]	Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80
[3]	Zhou J, Huang D 2012 Chin. Phys. B 21 048401
[4]	Wang L D, Li H F, Duan S K, Huang T W, Wang H M 2016 Neurccomputing 171 23
[5]	Min G Q, Wang L D, Duan S K 2015 Acta Phys. Sin. 64 210507 (in Chinese) [闵国旗, 王丽丹, 段书凯 2015 物理学报 64 210507]
[6]	Mandelbrot B B 1967 Science 156 636
[7]	Mandelbrot B B 1975 Fractals: Form, Chance and Dimension (San Francisco: WH Freeman and Company) pp35-37
[8]	Li H Q, Wang F Q 1999 Fractal Theory and Its Application in Molecular Science (Beijing: Science Press) p33 (in Chinese) [李后强, 汪富泉 1999分形理论及其在分子科学中的应用(北京:科学出版社) 第33页]
[9]	Lorenz E N 1963 J. Atmos. Sci. 20 130
[10]	Chua L O, Komuro M, Matsumoto T 1986 IEEE Trans. Circ. Syst. 33 1072
[11]	Chen G R 1999 Int. J. Bifurcat. Chaos 9 1465
[12]	Wang L D, Drakakis E, Duan S K, He P F, Liao X F 2012 Int. J. Bifurcat. Chaos 22 1250205
[13]	Muthuswamy B, Kokate P P 2009 IETE Tech. Rev. 26 417
[14]	Zhou Z W, Su Y L, Wang W D, Yan Y 2017 J. Petrol. Explor. Prod. Technol. 7 487
[15]	Bouallegue K 2015 Int. J. Bifurcat. Chaos 25 1530002
[16]	Chua L O, Roska T 1993 IEEE Trans. Circ. Syst. I 40 147
[17]	Yalcin M, Suykens J, Vandewalle J, Ozoguz S 2002 Int. J. Bifurcat. Chaos 12 23
[18]	Tang W K S, Zhong G Q, Chen G, Man K F 2001 IEEE Trans. Circ. Syst. I 48 1369
[19]	Zarei A 2015 Nonlinear Dyn. 81 585
[20]	More C, Vlad R, Chauveau E 2010 Nonlinear Dyn. 59 45
[21]	Huan S M, Li Q D, Yang X S 2012 Nonlinear Dyn. 69 1915
[22]	L J H, Yu X H, Chen G R 2003 IEEE Trans. Circ. Syst. I 50 198
[23]	Yalcin M, Suykens J, van de Walle J 2005 Chaos Modeling and Control Systems Design (Singapore: World Scientific) p59
[24]	L J H, Chen G R, Yu X H, Leung H 2004 IEEE Trans. Circ. Syst. I 51 2476
[25]	L J H, Yu S M, Leung H, Chen G R 2006 IEEE Trans. Circ. Syst. I 53 149

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