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## Approximations of chaotic attractors and its circuit design based on the parameter switching algorithm

Luo Shao-Xuan, He Bo-Xia, Qiao Ai-Min, Wang Yan-Chun
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• #### 摘要

基于参数切换算法和离散混沌系统, 设计一种新的混沌系统参数切换算法, 给出了两算法的原理. 采用混沌吸引子相图观测法, 研究了不同算法下统一混沌系统和Rössler混沌系统参数切换结果, 最后引入方波发生器, 设计了Rössler混沌系统参数切换电路. 结果表明, 采用参数切换算法可以近似出指定参数下的系统, 其吸引子与该参数下吸引子一致; 基于离散系统的参数切换结果更为复杂, 当离散序列分布均匀时, 只可近似得到指定参数下的系统; 相比传统切换混沌电路, 参数切换电路不用修改原有系统电路结构, 设计更为简单, 输出结果受方波频率影响, 通过加入合适频率的方波发生器, 数值仿真与电路仿真结果一致.

#### Abstract

Based on the parameter switching algorithm and the discrete chaotic system, a new chaotic system based parameter switching algorithm is proposed. The principles of parameter switching algorithm and chaotic system based parameter switching algorithm are presented in detail by means of flow chart and step description. By applying phase diagram observation method, chaotic attractor approximation of the unified chaotic system is investigated based on parameter switching algorithm and chaotic system based parameter switching algorithm. It shows that chaos can be obtained by switching two periodic parameters and periodic state can be observed by switching two chaotic parameters. Thus the formulas chaos+ chaos = periodic and period+ period = chaos are proved to be workable in this paper. Chaotic attractor approximation of Rössler chaotic system is also studied by employing the two switching methods. Two cases are investigated. Firstly, a chaotic switching system is obtained by switching a chaotic parameter and a periodic parameter. Then a more complex switching scheme is carried out. Periodic system is switched by two periodic parameters and a chaotic parameter. So, the formulas chaos+ periodic = chaos and periodic+ period+ chaos = periodic are proved to be workable. It shows that the switching system is the approximation of the original system under specified parameter, and the attractor is in accordance with the attractor of the targeting system. The outputs of the Logistic map based parameter switching algorithm are more complex than those of existing parameter switching algorithm. As the distribution of logistic map is not uniform, the approximate attractor does not consist of the targeting system and shows more complicated structure. But approximate attractors can be obtained when the distribution of discrete sequence is uniform. In addition, the chaotic map based parameter switching algorithm has larger secret key space since it has the initial values and parameter of the chaotic map. Finally, the parameter switching circuit of Rössler system is designed by introducing a square wave generator. Compared with the traditional switching chaotic circuit (switching between different systems), the design of parameter switch circuit is simpler as it does not need to change the original structure of the system. The output is affected by the frequency of the square wave. By adding an appropriate frequency square wave generator, the circuit simulation agrees with the numerical simulation. It presents a theoretical and experimental base for the practical application of the parameter switching chaotic systems.

#### 作者及机构信息

###### 1. 蚌埠学院机械与电子工程系, 蚌埠 233030; 2. 南京理工大学, 南京 210094
• 基金项目: 国家自然科学基金 (批准号: 51175276)、安徽省高等学校省级自然科学重点研究项目(批准号: KJ2013Z193)和安徽省高等学校优秀青年人才基金 (批准号: 2012SQRL212)资助的课题.

#### Authors and contacts

###### 1. Mechanical and Electronic Engineering Department, Bengbu College, Bengbu 233030, China; 2. Nanjing University of Science and Technology, Nanjing 210094, China
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51175276), the Key Program of the Natural Science Foundation of the Higher Education Institutions of Anhui Province, China (Grant No. KJ2013Z193), and the Foundation for Outstanding Young Teachers in University of Anhui Province, China (Grant No. 2012SQRL212).

#### 参考文献

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#### 施引文献

•  [1] Duan S M, Wu G Z 2013 Appl. Mech. Mater. 392 232 [2] Xiao X, Zhou L, Zhang Z 2014 Commun. Nonl. Sci. Num. Simul. 19 2039 [3] Wang Z, Huang X, Li Y X 2013 Chin. Phys. B 22 010504 [4] Jamal S S, Shah T, Hussain I 2013 Nonl. Dyn. 73 1469 [5] Zhang Y, Chen T L 2006 J. Electr. Sci. Tech. 34 763 (in Chinese) [张勇, 陈天麒 2006 电子科技大学学报 34 763] [6] Corporation H P 2014 J. Nonl. Dyn. 2014 918586 [7] Sun C C, Xu Q C, Ying S 2013 Chin. Phys. B 22 030507 [8] Liu Y Z, Jiang C S, Ling C S 2007 Acta Phys. Sin. 56 3107 (in Chinese) [刘扬正, 姜长生, 林长圣 2007 物理学报 56 3107] [9] Liu Y Z, Jiang C S, Ling C S 2007 Acta Phys. Sin. 56 707 (in Chinese) [刘扬正, 姜长生, 林长圣 2007 物理学报 56 707] [10] Qi A, Han C, Wang G 2010 Communications, Circuits and Systems (ICCCAS) International Conference on. IEEE 2010 p417 [11] Chen Z Y, Xue Z H, Zhang C 2014 Acta Phys. Sin. 63 010504 (in Chinese) [陈章耀, 雪增红, 张春 2014 物理学报 63 010504] [12] Almeida J, Peralta-Salas D, Romera M 2005 Physica D 200 124 [13] Danca M F 2013 Commu. Nonl. Sci. Num. Simul. 18 500 [14] Danca M F, Romera M, Pastor G 2012 Nonl. Dyn. 67 2317 [15] Mao Y, Tang W K S, Danca M 2010 Appl. Math. Comp. 217 355 [16] Foias C, Jolly M S 1995 Nonlinearity 8 295 [17] Lu J, Wu X, Han X, L J 2004 Phys. Lett. A 329 327 [18] Rössler O E 1976 Phys. Lett. A 57 397
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##### 出版历程
• 收稿日期:  2015-03-23
• 修回日期:  2015-06-08
• 刊出日期:  2015-10-05

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