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## Algorithm for calculating the Lyapunov exponents of switching system and its application

Li Qing-Du, Guo Jian-Li
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• #### 摘要

Lyapunov指数是判定系统非线性行为的重要工具，然而目前的大多算法并不适用于切换系统. 在传统Jacobi法的基础上，提出了一种新算法，可以直接计算得到n维切换系统的n个Lyapunov 指数. 首先，根据切换面处相邻轨线的动态变化规律，从相空间几何推导出切换面处轨线变化的Jacobi 矩阵；然后，对该矩阵进行QR分解，从而利用R的对角线元素实现Lyapunov指数的切换补偿；最后，将新算法应用到平面双螺旋混沌系统、Glass网络和航天器供电系统三个实例中，并将计算结果与Poincaré映射方法的计算结果进行比较，对新算法的有效性进行验证.

#### Abstract

Lyapunov characteristic exponent is significant for analyzing nonlinear dynamics. However, most algorithms are not applicable for the switching system. According to the traditional Jacobi method, in this paper we propose a new algorithm which can be used to compute n Lyapunov exponents for an n-dimensional switching system. We first study the geometric dynamics of two adjacent trajectories near the switching manifold, and obtain a compensation Jacobi matrix caused by switching. Then with QR-decomposition of this matrix, we compensate for the diagonal vector of R to realize the Lyapunov exponent expansion. Finally, we use the algorithm in a two-dimensional double-scrolls system, the Glass network and a spacecraft power system, and show its correctness and effectiveness by comparing the results with the Poincaré-map method.

#### 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
• 基金项目: 国家自然科学基金（批准号：61104150）、重庆市杰出青年科学基金（批准号：cstc2013jcyjjq40001）和重庆市教育委员会科学技术研究计划（批准号：KJ130517）资助的课题.

#### Authors and contacts

###### 1. Key Laboratory of Industrial Internet of Things and Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61104150), the Science Fund for Distinguished Young Scholars of Chongqing, China (Grant No. cstc2013jcyjjq40001), and the Science and Technology Research Program of Education Committee of Chongqing, China (Grant No. KJ130517).

#### 参考文献

 [1] Yang X S 2009 Int. J. Bifurcat. Chaos 19 1127 [2] Li Q D, Yang X S 2010 Int. J. Bifurcat. Chaos 20 467 [3] Li Q D, Tang S 2013 Acta Phys. Sin. 62 020510 (in Chinese) [李清都, 唐宋 2013 物理学报 62 020510 [4] Kaczyński T, Mischaikow K M, Mrozek M 2004 Comput. Homol. 157 100 [5] Neumann N, Sattel T, Wallaschek J 2007 J. Vib. Control 13 1393 [6] Yang F Y, Hu M, Yao S P 2013 Acta Phys. Sin. 62 100501 (in Chinese) [杨芳艳, 胡明, 姚尚平 2013 物理学报 62 100501] [7] Li Q D, Tan Y L, Yang F Y 2011 Acta Phys. Sin. 60 030206 (in Chinese) [李清都, 谭宇玲, 杨芳艳 2011 物理学报 60 030206] [8] Li Q D, Zhou H W, Yang X S 2012 Acta Phys. Sin. 61 040503 (in Chinese) [李清都, 周红伟, 杨晓松 2012 物理学报 61 040503] [9] Zhang H G, Fu J, Ma T D, Tong S C 2009 Chin. Phys. B 18 969 [10] Wu L F, Guan Y, Liu Y 2013 Acta Phys. Sin. 62 110510 (in Chinese) [吴立峰, 关永, 刘勇 2013 物理学报 62 110510] [11] Ji Y, Bi Q S 2010 Acta Phys. Sin. 59 7612 (in Chinese) [季颖, 毕勤胜 2010 物理学报 59 7612 ] [12] Zhang X F, Chen X K, Bi Q S 2013 Acta Phys. Sin. 62 010502 (in Chinese) [张晓芳, 陈小可,毕勤胜 2013 物理学报 62 010502 ] [13] Gao C, Bi Q S, Zhang Z D 2013 Acta Phys. Sin. 62 020504 (in Chinese) [高超, 毕勤胜, 张正娣 2013 物理学报 62 020504 ] [14] Lin C S, Xiong X, Shi L, Liu Y Z, Jiang C S 2007 Acta Phys. Sin. 56 3107 [林长圣, 熊星, 石磊, 刘扬正, 姜长生 2007 物理学报 56 3107] [15] Li S R, Jian J G, Geng Y F 2009 J. Henan Normal Univ. (Nat. Sci. Ed.) 5 14 (in Chinese) [李圣荣, 蹇继贵, 耿艳峰 2009 河南师范大学学报 (自然学科版) 5 14] [16] Yu Y G, Li H X, Duan J 2009 Chaos Solitons Fract. 41 457 [17] Chen W H, Guan Z H, Lu X M 2008 Asian J. Control 7 135 [18] Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285 [19] Galvanetto U 2000 Comput. Phys. Commun. 131 1 [20] Stefański A, Kapitaniak T 2003 Chaos Solitons Fract. 15 233 [21] Stefański A 2000 Chaos Solitons Fract. 11 2443 [22] Stefański A, Kapitaniak T 2000 Discrete Dyn. Nat. Soc. 4 207 [23] de Souza S L T, Caldas I L 2004 Chaos Solitons Fract. 19 569 [24] Li Q D, Yang X S 2005 Acta Electron. Sin. 33 1299 (in Chinese) [李清都, 杨晓松 2005 电子学报 33 1299] [25] Kappler K, Edwards R, Glass L 2003 Signal Process. 83 789 [26] Li Q D, Yang X S 2006 Chaos 16 033101 [27] Lim Y H, Hamill D C 1999 Electron. Lett. 35 510 [28] Li Q, Yang X S, Chen S 2011 Int. J. Bifurcat. Chaos 21 1719

#### 施引文献

•  [1] Yang X S 2009 Int. J. Bifurcat. Chaos 19 1127 [2] Li Q D, Yang X S 2010 Int. J. Bifurcat. Chaos 20 467 [3] Li Q D, Tang S 2013 Acta Phys. Sin. 62 020510 (in Chinese) [李清都, 唐宋 2013 物理学报 62 020510 [4] Kaczyński T, Mischaikow K M, Mrozek M 2004 Comput. Homol. 157 100 [5] Neumann N, Sattel T, Wallaschek J 2007 J. Vib. Control 13 1393 [6] Yang F Y, Hu M, Yao S P 2013 Acta Phys. Sin. 62 100501 (in Chinese) [杨芳艳, 胡明, 姚尚平 2013 物理学报 62 100501] [7] Li Q D, Tan Y L, Yang F Y 2011 Acta Phys. Sin. 60 030206 (in Chinese) [李清都, 谭宇玲, 杨芳艳 2011 物理学报 60 030206] [8] Li Q D, Zhou H W, Yang X S 2012 Acta Phys. Sin. 61 040503 (in Chinese) [李清都, 周红伟, 杨晓松 2012 物理学报 61 040503] [9] Zhang H G, Fu J, Ma T D, Tong S C 2009 Chin. Phys. B 18 969 [10] Wu L F, Guan Y, Liu Y 2013 Acta Phys. Sin. 62 110510 (in Chinese) [吴立峰, 关永, 刘勇 2013 物理学报 62 110510] [11] Ji Y, Bi Q S 2010 Acta Phys. Sin. 59 7612 (in Chinese) [季颖, 毕勤胜 2010 物理学报 59 7612 ] [12] Zhang X F, Chen X K, Bi Q S 2013 Acta Phys. Sin. 62 010502 (in Chinese) [张晓芳, 陈小可,毕勤胜 2013 物理学报 62 010502 ] [13] Gao C, Bi Q S, Zhang Z D 2013 Acta Phys. Sin. 62 020504 (in Chinese) [高超, 毕勤胜, 张正娣 2013 物理学报 62 020504 ] [14] Lin C S, Xiong X, Shi L, Liu Y Z, Jiang C S 2007 Acta Phys. Sin. 56 3107 [林长圣, 熊星, 石磊, 刘扬正, 姜长生 2007 物理学报 56 3107] [15] Li S R, Jian J G, Geng Y F 2009 J. Henan Normal Univ. (Nat. Sci. Ed.) 5 14 (in Chinese) [李圣荣, 蹇继贵, 耿艳峰 2009 河南师范大学学报 (自然学科版) 5 14] [16] Yu Y G, Li H X, Duan J 2009 Chaos Solitons Fract. 41 457 [17] Chen W H, Guan Z H, Lu X M 2008 Asian J. Control 7 135 [18] Wolf A, Swift J B, Swinney H L, Vastano J A 1985 Physica D 16 285 [19] Galvanetto U 2000 Comput. Phys. Commun. 131 1 [20] Stefański A, Kapitaniak T 2003 Chaos Solitons Fract. 15 233 [21] Stefański A 2000 Chaos Solitons Fract. 11 2443 [22] Stefański A, Kapitaniak T 2000 Discrete Dyn. Nat. Soc. 4 207 [23] de Souza S L T, Caldas I L 2004 Chaos Solitons Fract. 19 569 [24] Li Q D, Yang X S 2005 Acta Electron. Sin. 33 1299 (in Chinese) [李清都, 杨晓松 2005 电子学报 33 1299] [25] Kappler K, Edwards R, Glass L 2003 Signal Process. 83 789 [26] Li Q D, Yang X S 2006 Chaos 16 033101 [27] Lim Y H, Hamill D C 1999 Electron. Lett. 35 510 [28] Li Q, Yang X S, Chen S 2011 Int. J. Bifurcat. Chaos 21 1719
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##### 出版历程
• 收稿日期:  2013-12-09
• 修回日期:  2014-01-09
• 刊出日期:  2014-05-05

## 切换系统Lyapunov指数的算法及应用

• 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
基金项目:

国家自然科学基金（批准号：61104150）、重庆市杰出青年科学基金（批准号：cstc2013jcyjjq40001）和重庆市教育委员会科学技术研究计划（批准号：KJ130517）资助的课题.

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