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## Algorithm for finding horseshoes in three-dimensional hyperchaotic maps and its application

Li Qing-Du, Tang Song
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• #### 摘要

拓扑马蹄理论是严格研究混沌的重要理论,然而却很少用在超混沌的研究中. 主要原因是超混沌系统不仅相空间维数比普通混沌高,而且存在的拉伸方向数也较多, 导致拓扑马蹄的寻找难度很大.为此,本文针对三维超混沌映射,提出一种实用的拓扑马蹄寻找算法. 超混沌系统通常有较大的负Lyapunov指数,其吸引子会靠向某一曲面.基于这种特性, 本文首先沿着系统收缩方向进行降维,得出二维平面投影系统;接着在新系统中搜索二维拉伸的投影马蹄; 最后利用投影马蹄升维构造出原三维系统拓扑马蹄.为了验证算法的有效性, 本文以经典Lorenz超混沌系统和著名Saito超混沌电路为例,利用数值计算, 在它们的Poincare映射中找出了具有二维拉伸的三维拓扑马蹄.

#### Abstract

Topological horseshoe theory is fundamental for studying chaos rigorously, which, however, has rarely applied to hyperchaos. The reason is that it is too hard to find a topological horseshoe in a hyperchaotic system, due to the high dimension of the system and the multiple expansion directions in the state space. Therefore, in this paper a practical algorithm for three-dimensional (3D) hyperchaotic maps is proposed. Usually, a hyperchaotic system has a large negative Lyapunov exponent, its attractor is often contracted closely to a certain surface. Based on this feature, the algorithm first deducts the dimension along the direction of contraction to obtain a 2D projective system; then it detects a projective horseshoe with 2D expansion; finally, it constructs a 3D horseshoe for the original system. In order to verify the validity of the algorithm, it is applied to the classic hyperchaotic Lorenz system and the famous Saito hyperchaotic circuit, and their horseshoes with 2D expansion are successfully found from the Poincaré mapping.

#### 作者及机构信息

###### 1. 重庆邮电大学工业物联网与网络化控制教育部重点实验室, 重庆 400065; 2. 重庆邮电大学非线性系统研究所, 重庆 400065
• 基金项目: 国家自然科学基金(批准号: 10972082, 61104150)、 重庆市科委基金(批准号: cstcjjA40044) 和重庆邮电大学博士启动金(批准号: A2009-12)资助的课题.

#### Authors and contacts

###### 1. Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China; 2. Institute for Nonlinear Circuits and Systems, Chongqing University Posts and Telecommunications, Chongqing 400065, China
• Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 10972082, 61104150), the Natural Science Foundation Project of Chongqing (Grant No. cstcjjA40044), and the Doctoral Fund of CQUPT (Grant No. A2009-12).

#### 参考文献

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#### 施引文献

•  [1] Rossler O 1979 Physics Letters A 71 155 [2] Zheng J 2011 Computers & Mathematics with Applications 61 2000 [3] Yu H, Cai G, Li Y 2012 Nonlinear Dynamics 67 2171 [4] Sheikhan M, Shahnazi R, Garoucy S 2011 Neural Computing & Applications 20 1 [5] Vaidyanathan S, Sampath S 2012 Advances in Computer Science and Information Technology. Computer Science and Engineering 85 257 [6] Uchida A, Amano K, Inoue M 2008 Nature Photonics 2 728 [7] Sun L, Jiang D P 2006 Acta Phys. Sin. 55 3288 (in Chinese) [孙琳, 姜德平 2006 物理学报 55 3283] [8] Wang J, Jiang G P 2011 Acta Phys. Sin. 60 60503 (in Chinese) [王晶, 蒋国平 2011 物理学报 60 60503] [9] Kennedy J, Kocak S, Yorke J A 2001 The American Mathematical Monthly 108 411 [10] Kennedy J, Yorke J A 2001 Transactions of the American Mathematical Society 353 2513 [11] Yang X S 2004 Chaos, Solitons & Fractals 20 1149 [12] Szymczak A 1996 Topology 35 287 [13] Plumecoq J, Lefranc M 2000 Physica D: Nonlinear Phenomena 144 231 [14] Zgliczyński P, Gidea M 2004 Journal of Differential Equations 202 32 [15] Li Q, Yang X S 2006 Journal of Physics a-Mathematical and General 39 9139 [16] Yang F, Li Q, Zhou P 2007 International Journal of Bifurcation and Chaos 17 4205 [17] Li Q, Yang X S 2007 Discrete Dynamics in Nature and Society 2007 16239 [18] Li Q 2008 Physics Letters A 372 2989 [19] Li Q, Yang X S 2008 International Journal of Circuit Theory and Applications 36 19 [20] Li Q, Yang X S, Chen S 2011 International Journal of Bifurcation and Chaos 21 1719 [21] Yang X S 2009 International Journal of Bifurcation and Chaos 19 1127 [22] Yang X S, Li H, Huang Y 2005 Journal of Physics A: Mathematical and General 38 4175 [23] Li Q, Yang X S 2010 International Journal of Bifurcation and Chaos 20 467 [24] Wang X Y, Wang M J 2007 Acta Phys. Sin. 56 5136 (in Chinese) [王兴元, 王明军 2007 物理学报 56 5136] [25] Saito T 1990 Circuits and Systems, IEEE Transactions on 37 399
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##### 出版历程
• 收稿日期:  2012-07-15
• 修回日期:  2012-08-21
• 刊出日期:  2013-01-05

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