-
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:
- finite-time Lyapunov exponent /
- dual number /
- dynamical system /
- Lagrangian coherent structure
[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
-
[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
Catalog
Metrics
- Abstract views: 7535
- PDF Downloads: 746
- Cited By: 0
下载:
