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This paper proposes a new algorithm for computing two-dimensional （un）stable manifolds in time continuous systems. With this algorithm, one can not only get a picture of a manifold efficiently, but also has many information of its every point, which are very useful for investigating the global dynamics of a system geometrically, such as features of stability region, evolution of the system flow and so on. The algorithm is mainly by finding many well distributed trajectories by solving initial value problems. An example on Lorenz system suggests this algorithm is very convenient. In addition, we study the chaotic dynamic of a three-dimensional neural network by detecting a heteroclinic orbit.
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Keywords:
- stable manifold /
- unstable manifold /
- dynamic systems /
- Lorenz system
[1] ［1]Chiang H D, Hirsch M W, Wu F F 1988 IEEE Trans. Automat. Contr. 33 16
[2] ［2]Yu H, Zhu S J, Liu S Y 2008 Acta Phys. Sin. 57 2761 （in Chinese）［俞翔、朱石坚、刘树勇 2008 57 2761］
[3] ［3]Johnson M E, Jolly M S, Kevrekidis I G 1997 Numerical Algorithms 14 125
[4] ［4]Doedel E J, Champneys A R 1997 ftp://ftp.cs.concordia.ca/pub/doedel/auto/［2008-05-30］
[5] ［5]Krauskopf B, Osinga H M 2003 SIAM J. Appl. Dyn. Sys. 2 546
[6] ［6]Guckenheimer J, Vladimirsky A A 2004 SIAM J. Appl. Dyn. Sys. 3 232
[7] ［7]Henderson M 2005 SIAM J. Appl. Dyn. Sys. 4 832
[8] ［8]Krauskopf B, Singa H M 2005 Int. J. Bifurcation and Chaos 15 763
[9] ［9]Li Q D, Yang X S 2005 Computational Physics 22 549 （in Chinese）［李清都、杨晓松 2005 计算物理 22 549］
[10] ［10]Moore R E 1979 Methods and Applications of Interval Analysis （Philadelphia: Society for Industrial Mathematics）
[11] ［11]Zgliczyński P 2002 Foundations of Computational Mathematics 2 429
[12] ［12]Yu J Z, Su N, Vincent T L 1998 Acta Phys. Sin. 47 397 （in Chinese）［余建祖、苏楠、T.L.Vincent 1998 47 397］
[13] ［13]Li S H, Tian Y P 2003 Chin. Phys. 12 590
[14] ［14]Giuseppe G 2008 Chin. Phys. B 17 3247
[15] ［15]Chen G P, Hao J B 2009 Acta Phys. Sin. 58 2914 （in Chinese）［陈光平、郝加波 2009 58 2914］
[16] ［16]Hao J HG, Sun Z H, Xu H B 2007 Acta Phys. Sin. 56 6857（in Chinese）［郝建红、孙志华、许海波 2007 56 6857］
[17] ［17]Cang S J, Chen Z Q, Wu W J 2009 Chin. Phys. B 18 1792
[18] ［18]Wang G Y, Zheng N, Liu J B 2007 Acta Phys. Sin. 56 3113 （in Chinese）［王光义、郑艳、刘敬彪 2007 56 3113］
[19] ［19]Yang X S, Li Q D 2006 Int J Bifurcation and Chaos 16 157
[20] ［20]Yang X S, Li Q D 2007 Chaotic Systems and Chaotic Circuits （Beijing: Science Press） （in Chinese）［杨晓松 李清都 2007 混沌系统与混沌电路（北京:科学出版社）］
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[1] ［1]Chiang H D, Hirsch M W, Wu F F 1988 IEEE Trans. Automat. Contr. 33 16
[2] ［2]Yu H, Zhu S J, Liu S Y 2008 Acta Phys. Sin. 57 2761 （in Chinese）［俞翔、朱石坚、刘树勇 2008 57 2761］
[3] ［3]Johnson M E, Jolly M S, Kevrekidis I G 1997 Numerical Algorithms 14 125
[4] ［4]Doedel E J, Champneys A R 1997 ftp://ftp.cs.concordia.ca/pub/doedel/auto/［2008-05-30］
[5] ［5]Krauskopf B, Osinga H M 2003 SIAM J. Appl. Dyn. Sys. 2 546
[6] ［6]Guckenheimer J, Vladimirsky A A 2004 SIAM J. Appl. Dyn. Sys. 3 232
[7] ［7]Henderson M 2005 SIAM J. Appl. Dyn. Sys. 4 832
[8] ［8]Krauskopf B, Singa H M 2005 Int. J. Bifurcation and Chaos 15 763
[9] ［9]Li Q D, Yang X S 2005 Computational Physics 22 549 （in Chinese）［李清都、杨晓松 2005 计算物理 22 549］
[10] ［10]Moore R E 1979 Methods and Applications of Interval Analysis （Philadelphia: Society for Industrial Mathematics）
[11] ［11]Zgliczyński P 2002 Foundations of Computational Mathematics 2 429
[12] ［12]Yu J Z, Su N, Vincent T L 1998 Acta Phys. Sin. 47 397 （in Chinese）［余建祖、苏楠、T.L.Vincent 1998 47 397］
[13] ［13]Li S H, Tian Y P 2003 Chin. Phys. 12 590
[14] ［14]Giuseppe G 2008 Chin. Phys. B 17 3247
[15] ［15]Chen G P, Hao J B 2009 Acta Phys. Sin. 58 2914 （in Chinese）［陈光平、郝加波 2009 58 2914］
[16] ［16]Hao J HG, Sun Z H, Xu H B 2007 Acta Phys. Sin. 56 6857（in Chinese）［郝建红、孙志华、许海波 2007 56 6857］
[17] ［17]Cang S J, Chen Z Q, Wu W J 2009 Chin. Phys. B 18 1792
[18] ［18]Wang G Y, Zheng N, Liu J B 2007 Acta Phys. Sin. 56 3113 （in Chinese）［王光义、郑艳、刘敬彪 2007 56 3113］
[19] ［19]Yang X S, Li Q D 2006 Int J Bifurcation and Chaos 16 157
[20] ［20]Yang X S, Li Q D 2007 Chaotic Systems and Chaotic Circuits （Beijing: Science Press） （in Chinese）［杨晓松 李清都 2007 混沌系统与混沌电路（北京:科学出版社）］
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