Computation of invariant manifolds with self-adaptive parameter and trajectories continuation method

Jia Meng; Fan Yang-Yu; Li Hui-Min

doi:10.7498/aps.59.7686

Abstract

Most work on manifold study focuses on two-dimensional manifolds and there have been proposed some good computing methods. However, the computation of two-dimensional manifold is still a hot research field. In this paper the two-dimensional manifold of hyperbolic equilibria for vector fields is computed by combining self-adaptive parameter with trajectories continuation, approximating the local manifold with an ellipse around the equilibria, extending the trajectory with equal distance, and adjusting the trajectory with self-adaptive parameter. This method is more accurate than the "trajectories and arc-length method", and better shows the trend of the manifolds than the "box covering method".

Keywords:

Authors and contacts

(1)Department of Electronics and Information, Northwestern Polytechnical University, Xi'an 710072, China; (2)Department of Electronics and Information, Northwestern Polytechnical University, Xi'an 710072, China;Department of Electrical Engineering, Xinxiang College, Xinxiang 453000, China

References

[1]	Liu Y Z, Chen L Q, Cheng G, Ge X S 2000 Advance Mechanics 30 351 (in Chinese ) [刘延柱、陈立群、成功、戈新生 2000 力学进展 30 351]
[2]	Krauskopf B, Osinga H M 1998 Comput. Phys. 146 406
[3]	Doedel, Auto E J 1981 Congr. Numer 30 265
[4]	Nils Berglund http: //www.math.ethz.ch/~berglund 2001
[5]	Krauskopf B, Osinga H M 1999 Chaos 9 768
[6]	Johnson M E, Jolly M S, Kevrekidis I G 1997 Numer Algorithms 14 125
[7]	Guckenheimer J, Vladimirsky A 2004 Appl. Dyn. Sys. 3 232
[8]	Dellnitz M, Hobmann A 1997 Num. Math. 75 293
[9]	Krauskopf B, Osinga H M, Doedel E J 2005 Bifur. Chaos. Appl. Sci. Engrg 15 763
[10]	Krauskopf B, Osinga H M 2003 Appl. Dyn. Sys. 2 546
[11]	Guckenheimer J, Worfolk P 1993 Dynamical systems (Ithaca: Kluwer Academic) p241
[12]	Ni F, Xu W, Fang T, Yue X 2010 Chin. Phys. B 19 010510-1
[13]	Liu Y L, Zhu J, Luo X S 2009 Chin. Phys. B 18 3772
[14]	Jiang G R, Xu B G, Yang Q G 2009 Chin. Phys. B 18 5235
[15]	Hobson D 1993 Comput. Phys. 104 14
[16]	Henderson M E 2005 Appl. Dyn. Sys. 4 832
[17]	Liang C X, Tang J S 2008 Chin. Phys. B 17 135
[18]	Zhang Y, Lei Y M, Fang T 2009 Acta Phys. Sin. 58 3799 (in Chinese)[张莹、雷佑铭、方同 2009 58 3799]
[19]	Jiang G R, Yang Q G 2008 Chin. Phys. B 17 4114
[20]	Zuo H L, Xu J X, Jiang J 2008 Chin. Phys. B 17 117
[21]	Xu Q, Tian Q 2009 Chin. Phys. B 18 2469

Cited By

[1]	Liu Y Z, Chen L Q, Cheng G, Ge X S 2000 Advance Mechanics 30 351 (in Chinese ) [刘延柱、陈立群、成功、戈新生 2000 力学进展 30 351]
[2]	Krauskopf B, Osinga H M 1998 Comput. Phys. 146 406
[3]	Doedel, Auto E J 1981 Congr. Numer 30 265
[4]	Nils Berglund http: //www.math.ethz.ch/~berglund 2001
[5]	Krauskopf B, Osinga H M 1999 Chaos 9 768
[6]	Johnson M E, Jolly M S, Kevrekidis I G 1997 Numer Algorithms 14 125
[7]	Guckenheimer J, Vladimirsky A 2004 Appl. Dyn. Sys. 3 232
[8]	Dellnitz M, Hobmann A 1997 Num. Math. 75 293
[9]	Krauskopf B, Osinga H M, Doedel E J 2005 Bifur. Chaos. Appl. Sci. Engrg 15 763
[10]	Krauskopf B, Osinga H M 2003 Appl. Dyn. Sys. 2 546
[11]	Guckenheimer J, Worfolk P 1993 Dynamical systems (Ithaca: Kluwer Academic) p241
[12]	Ni F, Xu W, Fang T, Yue X 2010 Chin. Phys. B 19 010510-1
[13]	Liu Y L, Zhu J, Luo X S 2009 Chin. Phys. B 18 3772
[14]	Jiang G R, Xu B G, Yang Q G 2009 Chin. Phys. B 18 5235
[15]	Hobson D 1993 Comput. Phys. 104 14
[16]	Henderson M E 2005 Appl. Dyn. Sys. 4 832
[17]	Liang C X, Tang J S 2008 Chin. Phys. B 17 135
[18]	Zhang Y, Lei Y M, Fang T 2009 Acta Phys. Sin. 58 3799 (in Chinese)[张莹、雷佑铭、方同 2009 58 3799]
[19]	Jiang G R, Yang Q G 2008 Chin. Phys. B 17 4114
[20]	Zuo H L, Xu J X, Jiang J 2008 Chin. Phys. B 17 117
[21]	Xu Q, Tian Q 2009 Chin. Phys. B 18 2469

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