Search

Article

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

Highly accurate computation of finite-time Lyapunov exponent

Cao Xiao-Qun Song Jun-Qiang Ren Kai-Jun Leng Hong-Ze Yin Fu-Kang

Citation:

Highly accurate computation of finite-time Lyapunov exponent

Cao Xiao-Qun, Song Jun-Qiang, Ren Kai-Jun, Leng Hong-Ze, Yin Fu-Kang
PDF
Get Citation

(PLEASE TRANSLATE TO ENGLISH

BY GOOGLE TRANSLATE IF NEEDED.)

  • Aiming at the shortcomings of current method of calculating finite-time Lyapunov exponent (FTLE), such as low accuracy, inability to obtain boundary values, etc., a method of highly accurately computing FTLE is proposed based on dual number theory. Firstly, the weakness and disadvantages of the finite difference method used widely for computing FTLE are described. Secondly, the dual number theory is introduced to evaluate the derivatives accurately and efficiently, and its distinct virtues are also presented. The computation of Cauchy-Green deformation tensors for a dynamical system is transformed into a numerical integration problem of solving the nonlinear ordinary differential equation in dual number space by the new method. Finally, the proposed method is applied to typical pendulum system and nonlinear Duffing oscillator separately. The results of simulation experiments indicate that the new method is effective, convenient and accurate for computing the field of FTLE, from which Lagrangian coherent structures can be identified successfully.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 41475094, 41105063, 41375105) and the Young Innovation Science Foundation of CHREO (Grant No. GFZX04060103-5-19).
    [1]

    Wu H, Hou W, Wang W X, Yan P C 2013 Acta Phys. Sin. 62 129204(in Chinese)[吴浩, 侯威, 王文祥, 颜鹏程 2013 物理学报 62 129204]

    [2]

    Zhang W C, Tan S C, Gao P Z 2013 Acta Phys. Sin. 62 060502(in Chinese)[张文超, 谭思超, 高璞珍 2013 物理学报 62 060502]

    [3]

    Yao T L, Liu H F, Xu J L, Li W F 2012 Acta Phys. Sin. 61 234704(in Chinese)[姚天亮, 刘海峰, 许建良, 李伟锋 2012 物理学报 61 234704]

    [4]

    Chen B H, Li J P, Ding R Q 2006 Sci. China D 36 1068

    [5]

    Ding R Q, Li J P 2007 Phys. Lett. A 364 396

    [6]

    Lorenz E N 1963 J. Atmos. Sci. 20 130

    [7]

    Haller G 2001 Physica D 149 248

    [8]

    Haller G 2002 Phys. Fluids A 14 1851

    [9]

    Farazmand M, Haller G 2012 Chaos 22 013128

    [10]

    Tang W, Mathur M, Haller G, Hahn D C 2010 J. Atmos. Sci. 67 2307

    [11]

    Sapsis T, Haller G 2009 J. Atmos. Sci. 66 2481

    [12]

    Sapsis T, Peng J, Haller G 2011 Bull. Math. Biol. 73 1841

    [13]

    Tang W, Haller G, Baik J J, Ryu Y H 2009 Phys. Fluids 21 043302

    [14]

    Mathur M, Haller G, Peacock T 2007 Phys. Rev. Lett. 98 144502

    [15]

    Green M A, Rowley C W, Haller G 2007 J. Fluid Mech. 572 111

    [16]

    Lekien F, Coulliette C, Mariano A J 2005 Physica D. 210 1

    [17]

    Shadden S C, Lekien F, Marsden J E 2005 Physica D 212 271

    [18]

    Shadden S C, Dabiri J O, Marsden J E 2006 Phys. Fluids 18 047105

    [19]

    Shadden S C, Katija K, Rosenfeld M 2007 J. Fluid Mech. 593 315

    [20]

    Pan C, Wang J J, Zhang C 2009 Sci. Sin. G: Phys. Mech. Astronom. 39 627(in Chinese)[潘翀, 王晋军, 张草 2009 中国科学G辑 物理学 力学 天文学 39 627]

    [21]

    Yang A L, Jia L B, Yin X Z 2012 J. Exp. Mech. 27 677(in Chinese)[杨岸龙, 贾来兵, 尹协振 2012 实验力学 27 677]

    [22]

    Lei P F, Zhang J Z, Wang Z P, Chen J H 2014 Acta Phys. Sin. 63 084702(in Chinese)[雷鹏飞, 张家忠, 王琢璞, 陈嘉辉 2014 物理学报 63 084702]

    [23]

    Gawlik E S, Du Toit P C, Campagnola S 2009 Celest. Mech. Dyn. Astron. 103 227

    [24]

    Qi R, Xu S J 2013 Aerospace Control and Application 39 6(in Chinese)[祁瑞, 徐世杰 2013 空间控制技术与应用 39 6]

    [25]

    Ali S, Shah M 2007 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Washington DC, USA, May 13-15, 2007 pp1-6

    [26]

    Clifford W K 1871 Proceedings of the London M athematical Society London, UK, April 13-15, 1871 p381

    [27]

    Study E 1981 Mathematische Annalen 39 441

    [28]

    He J H, Lee E W M 2009 Phys. Lett. A 373 1644

    [29]

    He J H 2007 Comput. Math. Appl. 54 881

    [30]

    Brodsky V, Shoham M 1999 Mechanism and Machine Theory 34 693

    [31]

    Wang J Y, Liang H Z, Sun Z W 2010 J. Astronaut. 31 1711(in Chinese)[王剑颖, 梁海朝, 孙兆伟 2010 宇航学报 31 1711]

    [32]

    Spall R, Yu W 2013 J. Fluids Engineer. 135 014501

    [33]

    Yu W B, Blair M 2013 Comput. Phys. Commun. 184 1446

    [34]

    Cao X Q, Song J Q, Zhang W M, Zhao J 2011 Acta Phys. Sin. 60 070511(in Chinese)[曹小群, 宋君强, 张卫民, 赵军 2011 物理学报 60 070511]

    [35]

    Cao X Q, Song J Q, Zhang W M, Zhu X Q 2011 Acta Phys. Sin. 60 080401(in Chinese)[曹小群, 宋君强, 张卫民, 朱小谦 2011 物理学报 60 080401]

    [36]

    He J H 2008 Int. J. Modern. Phys. B 22 3487

    [37]

    He J H 2001 Int. J. Nonlin. Sci. Numer. 2 309

    [38]

    Wu G C 2012 Chin. Phys. B 21 120504

    [39]

    Wu G C, Dumitru B 2013 Appl. Math. Model. 37 6183

    [40]

    Cao X Q, Song J Q, Zhu X Q 2012 Chin. Phys. B 21 020203

  • [1]

    Wu H, Hou W, Wang W X, Yan P C 2013 Acta Phys. Sin. 62 129204(in Chinese)[吴浩, 侯威, 王文祥, 颜鹏程 2013 物理学报 62 129204]

    [2]

    Zhang W C, Tan S C, Gao P Z 2013 Acta Phys. Sin. 62 060502(in Chinese)[张文超, 谭思超, 高璞珍 2013 物理学报 62 060502]

    [3]

    Yao T L, Liu H F, Xu J L, Li W F 2012 Acta Phys. Sin. 61 234704(in Chinese)[姚天亮, 刘海峰, 许建良, 李伟锋 2012 物理学报 61 234704]

    [4]

    Chen B H, Li J P, Ding R Q 2006 Sci. China D 36 1068

    [5]

    Ding R Q, Li J P 2007 Phys. Lett. A 364 396

    [6]

    Lorenz E N 1963 J. Atmos. Sci. 20 130

    [7]

    Haller G 2001 Physica D 149 248

    [8]

    Haller G 2002 Phys. Fluids A 14 1851

    [9]

    Farazmand M, Haller G 2012 Chaos 22 013128

    [10]

    Tang W, Mathur M, Haller G, Hahn D C 2010 J. Atmos. Sci. 67 2307

    [11]

    Sapsis T, Haller G 2009 J. Atmos. Sci. 66 2481

    [12]

    Sapsis T, Peng J, Haller G 2011 Bull. Math. Biol. 73 1841

    [13]

    Tang W, Haller G, Baik J J, Ryu Y H 2009 Phys. Fluids 21 043302

    [14]

    Mathur M, Haller G, Peacock T 2007 Phys. Rev. Lett. 98 144502

    [15]

    Green M A, Rowley C W, Haller G 2007 J. Fluid Mech. 572 111

    [16]

    Lekien F, Coulliette C, Mariano A J 2005 Physica D. 210 1

    [17]

    Shadden S C, Lekien F, Marsden J E 2005 Physica D 212 271

    [18]

    Shadden S C, Dabiri J O, Marsden J E 2006 Phys. Fluids 18 047105

    [19]

    Shadden S C, Katija K, Rosenfeld M 2007 J. Fluid Mech. 593 315

    [20]

    Pan C, Wang J J, Zhang C 2009 Sci. Sin. G: Phys. Mech. Astronom. 39 627(in Chinese)[潘翀, 王晋军, 张草 2009 中国科学G辑 物理学 力学 天文学 39 627]

    [21]

    Yang A L, Jia L B, Yin X Z 2012 J. Exp. Mech. 27 677(in Chinese)[杨岸龙, 贾来兵, 尹协振 2012 实验力学 27 677]

    [22]

    Lei P F, Zhang J Z, Wang Z P, Chen J H 2014 Acta Phys. Sin. 63 084702(in Chinese)[雷鹏飞, 张家忠, 王琢璞, 陈嘉辉 2014 物理学报 63 084702]

    [23]

    Gawlik E S, Du Toit P C, Campagnola S 2009 Celest. Mech. Dyn. Astron. 103 227

    [24]

    Qi R, Xu S J 2013 Aerospace Control and Application 39 6(in Chinese)[祁瑞, 徐世杰 2013 空间控制技术与应用 39 6]

    [25]

    Ali S, Shah M 2007 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Washington DC, USA, May 13-15, 2007 pp1-6

    [26]

    Clifford W K 1871 Proceedings of the London M athematical Society London, UK, April 13-15, 1871 p381

    [27]

    Study E 1981 Mathematische Annalen 39 441

    [28]

    He J H, Lee E W M 2009 Phys. Lett. A 373 1644

    [29]

    He J H 2007 Comput. Math. Appl. 54 881

    [30]

    Brodsky V, Shoham M 1999 Mechanism and Machine Theory 34 693

    [31]

    Wang J Y, Liang H Z, Sun Z W 2010 J. Astronaut. 31 1711(in Chinese)[王剑颖, 梁海朝, 孙兆伟 2010 宇航学报 31 1711]

    [32]

    Spall R, Yu W 2013 J. Fluids Engineer. 135 014501

    [33]

    Yu W B, Blair M 2013 Comput. Phys. Commun. 184 1446

    [34]

    Cao X Q, Song J Q, Zhang W M, Zhao J 2011 Acta Phys. Sin. 60 070511(in Chinese)[曹小群, 宋君强, 张卫民, 赵军 2011 物理学报 60 070511]

    [35]

    Cao X Q, Song J Q, Zhang W M, Zhu X Q 2011 Acta Phys. Sin. 60 080401(in Chinese)[曹小群, 宋君强, 张卫民, 朱小谦 2011 物理学报 60 080401]

    [36]

    He J H 2008 Int. J. Modern. Phys. B 22 3487

    [37]

    He J H 2001 Int. J. Nonlin. Sci. Numer. 2 309

    [38]

    Wu G C 2012 Chin. Phys. B 21 120504

    [39]

    Wu G C, Dumitru B 2013 Appl. Math. Model. 37 6183

    [40]

    Cao X Q, Song J Q, Zhu X Q 2012 Chin. Phys. B 21 020203

  • [1] Cao Xiao-Qun, Huang Qun-Bo, Liu Bai-Nian, Zhu Meng-Bin, Yu Yi. A new data assimilation method based on dual-number theory. Acta Physica Sinica, 2015, 64(13): 130502. doi: 10.7498/aps.64.130502
    [2] Zhang Yao-Li, Wu Bao-Wei, Wang Yue-E, Han Xiao-Xia. Finite-time stability for switched singular systems. Acta Physica Sinica, 2014, 63(17): 170205. doi: 10.7498/aps.63.170205
    [3] Yu Wan-Bo. Chaotic characteristics of three-dimensional function determined by cross-section geometric shape. Acta Physica Sinica, 2014, 63(12): 120501. doi: 10.7498/aps.63.120501
    [4] Tang Chuan-Sheng, Dai Yue-Hong. Finite-time stability control of permanent magnet synchronous motor chaotic system with parameters uncertain. Acta Physica Sinica, 2013, 62(18): 180504. doi: 10.7498/aps.62.180504
    [5] Shi Lan-Fang, Mo Jia-Qi. Solution of a class of rotational relativistic rotation dynamic equation using the generalized variational iteration theory. Acta Physica Sinica, 2013, 62(4): 040203. doi: 10.7498/aps.62.040203
    [6] Zhao Jian-Li, Wang Jing, Wang Hui. The study of finite-time stability active control method for Lorenz-Haken laser chaotic system. Acta Physica Sinica, 2012, 61(11): 110209. doi: 10.7498/aps.61.110209
    [7] Gao Zai-Rui, Shen Yan-Xia, Ji Zhi-Cheng. Uniform finite-time stability of discrete-time switched descriptor systems. Acta Physica Sinica, 2012, 61(12): 120203. doi: 10.7498/aps.61.120203
    [8] Shi Pei-Ming, Han Dong-Ying, Li Ji-Zhao, Jiang Jin-Shui, Liu Bin. Lyapunov-Schmidt reduction and singularity analysis of a high-dimensional relative-rotation nonlinear dynamical system. Acta Physica Sinica, 2012, 61(19): 194501. doi: 10.7498/aps.61.194501
    [9] Tao Wei-Jun, Huan Shi. Study on Lagrangian analysis for solving the stress gradually along the time. Acta Physica Sinica, 2012, 61(20): 200703. doi: 10.7498/aps.61.200703
    [10] Cao Xiao-Qun, Song Jun-Qiang, Zhang Wei-Min, Zhao Jun, Zhu Xiao-Qian. The modified variational iteration method for air-sea coupled dynamical system. Acta Physica Sinica, 2012, 61(3): 030203. doi: 10.7498/aps.61.030203
    [11] Zhao Jian-Li, Wang Jing, Wei Wei. Approximate finite-time stable control of Lorenz Chaos system. Acta Physica Sinica, 2011, 60(10): 100203. doi: 10.7498/aps.60.100203
    [12] Zhao Ling-Dong, Hu Jian-Bing, Bao Zhi-Hua, Zhang Guo-An, Xu Chen, Zhang Shi-Bing. A finite-time stable theorem about fractional systems and finite-time synchronizing fractional super chaotic Lorenz systems. Acta Physica Sinica, 2011, 60(10): 100507. doi: 10.7498/aps.60.100507
    [13] Wang Jiao-Jiao, Yan Hua, Wei Ping. Anticipating projective response in coupled dynamical systems. Acta Physica Sinica, 2010, 59(11): 7635-7643. doi: 10.7498/aps.59.7635
    [14] Chen Xu, Qiu Shui-Sheng. Precise planement of all Lyapunov exponents for discrete-time dynamical systems. Acta Physica Sinica, 2010, 59(11): 7630-7634. doi: 10.7498/aps.59.7630
    [15] Li Qing-Du, Yang Xiao-Song. A new algorithm for computation of two-dimensional unstable manifolds and its applications. Acta Physica Sinica, 2010, 59(3): 1416-1422. doi: 10.7498/aps.59.1416
    [16] Mo Jia-Qi, Wang Hui, Lin Wan-Tao. Approximate analytic solution of land-air couplind dynamical system. Acta Physica Sinica, 2006, 55(2): 485-489. doi: 10.7498/aps.55.485
    [17] Lei Min, Meng Guang, Feng Zheng-Jin. Detecting the nonlinearity for time series sampled from continuous dynamic systems. Acta Physica Sinica, 2005, 54(3): 1059-1063. doi: 10.7498/aps.54.1059
    [18] Hu Yin-Qiao. Ordered structure and system development of the force dissipation system(Ⅱ),Minimal value principle of generalized energy and system development. Acta Physica Sinica, 2003, 52(6): 1354-1359. doi: 10.7498/aps.52.1354
    [19] LIU HAI-FENG, ZHAO YAN-YAN, DAI ZHENG-HUA, GONG XIN, YU ZUN-HONG. CALCULATION OF THE LARGEST LYAPUNOV EXPONENT IN THE DISCRETE DYNAMICAL SYSTEM WITH WAVELET ANALYSIS. Acta Physica Sinica, 2001, 50(12): 2311-2317. doi: 10.7498/aps.50.2311
    [20] YANG ZHI-AN, CHEN SHI-GANG, WANG GUANG-RUI. ANALYSIS OF TIME SERIES RECONSTRUCTION FOR DYNAMIC SYSTEM. Acta Physica Sinica, 1996, 45(6): 904-911. doi: 10.7498/aps.45.904
Metrics
  • Abstract views:  6123
  • PDF Downloads:  730
  • Cited By: 0
Publishing process
  • Received Date:  13 February 2014
  • Accepted Date:  16 May 2014
  • Published Online:  05 September 2014

/

返回文章
返回