有限时间Lyapunov指数的高精度计算新方法

曹小群; 宋君强; 任开军; 冷洪泽; 银福康

doi:10.7498/aps.63.180504

摘要

针对目前有限时间Lyapunov指数（FTLE）计算方法准确度不高和无法获得边界值的问题，基于对偶数理论提出了一种新的高精度计算方法. 首先描述了基于有限空间差分方法计算FTLE 的缺点和问题；其次介绍了基于对偶数理论的高精度导数计算方法及其显著优点，并将动力系统的柯西-格林形变张量计算问题转化为对偶数空间中非线性微分方程数值求解问题；最后对单摆和非线性Duffing振子两个典型物理动力系统进行了数值实验. 结果表明：基于对偶数理论的新方法能有效、方便和高精度地计算出有限时间Lyapunov指数场，并成功识别出所包含的拉格朗日相关结构.

关键词:

Abstract

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.

Keywords:

作者及机构信息

1.
国防科学技术大学计算机学院, 长沙 410073;

2.
国防科学技术大学, 计算机学院并行与分布处理重点实验室, 长沙 410073

基金项目: 国家自然科学基金（批准号：41475094，41105063，41375105）和高分青年创新基金项目（批准号：GFZX04060103-5-19）资助的课题.

Authors and contacts

1.
School of Computer Science, National University of Defense Technology, Changsha 410073, China;

2.
Science and Technology on Parallel and distributed Processing Laboratory, National University of Defense Technology, Changsha 410073, China

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).

参考文献

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施引文献

[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 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 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 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 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 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 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

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