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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".
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Keywords:
- self-adaptive parameter /
- trajectories continuation /
- manifolds /
- non-liner system
[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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[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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