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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.
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Keywords:
- switching systems /
- Lyapunov exponents /
- Jacobi matrix /
- switching manifold
[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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[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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