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In this paper we present an algorithm of computing two-dimensional (2D) stable and unstable manifolds of hyperbolic fixed points of nonlinear maps. The 2D manifold is computed by covering it with orbits of one-dimensional (1D) sub-manifolds. A generalized Foliation condition is proposed to measure the growth of 1D sub-manifolds and eventually control the growth of the 2D manifold along the orbits of 1D sub-manifolds in different directions. At the same time, a procedure for inserting 1D sub-manifolds between adjacent sub-manifolds is presented. The recursive procedure resolves the insertion of new mesh point, the searching for the image (or pre-image), and the computation of the 1D sub-manifolds following the new mesh point tactfully, which does not require the 1D sub-manifolds to be computed from the initial circle and avoids the over assembling of mesh points. The performance of the algorithm is demonstrated with hyper chaotic three-dimensional (3D) Hnon map and Lorenz system.
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Keywords:
- nonlinear map /
- stable and unstable manifold /
- 3D Hnon map /
- Lorenz system
[1] Xu, Liu C X, Yang T 2010 Acta Phys. Sin. 59 131 (in Chinese) [许, 刘崇新, 杨韬 2010 59 131]
[2] Johnson M E, Jolly M S, Kevrekidis I G 1997 Numer. Algorithms. 14 125
[3] Guckenheimer J, Worfolk P 1993 Kluwer Academic Publishers 241
[4] Henderson M E 2005 SIAM J. Appl. Dyn. Syst. 4 832
[5] Krauskopf B, Osinga H M 1999 Chaos 9 768
[6] Krauskopf B, Osinga H M, Doedel E J, Henderson M E, Guckenheimer J, Vladimirsky A, Dellnitz M, Junge O 2005 Bifur. Chaos. Appl. Sci. Engrg. 15 763
[7] Guckenheimer J, Vladimirsky A 2004 SIAM J. Appl. Dyn. Syst. 3 232
[8] Li Q D, Yang X S 2010 Acta Phys. Sin. 59 1416 (in Chinese) [李清都, 杨晓松 2010 59 1416]
[9] Jia M, Fan Y Y, Li HM2010 Acta Phys. Sin. 59 7686 (in Chinese) [贾蒙, 樊养余, 李慧敏 2010 59 7686]
[10] Li Q D, Yang X S 2010 Acta Phys. Sin. 59 1416 (in Chinese) [李清都, 杨晓松 2010 59 1416]
[11] Hobson D 1991 J. Comput. Phys. 104 14
[12] You Z, Kostelich E J, Yorke J A 1991 Int. J. Bifurc. Chaos Appl. Sci. Eng. 1 605
[13] Sim′o C 1989 Benest D, Froeschlé C (eds.) Les Méthodes Modernes de la Mécanique Céleste 285
[14] Parker T S, Chua L O 1989 Practical Numerical Algorithms for Chaotic Systems (Berlin: Springer)
[15] Krauskopf B, Osinga H M 1998 J. Comput. Phys. 146 406
[16] England J P, Krauskopf B, Osinga H M 2004 SIAM J. Appl. Dyn. Syst. 3 161
[17] Krauskopf B, Osinga H M 1998 Int. J. Bifurc. Chaos 8 483
[18] Krauskopf B, Osinga H M 1999 Chaos. 9 768
[19] Dellnitz M, Hohmann A 1997 Numer. Math 75 293
[20] Li Q D, Zhou L, Zhou HW2010 Journal of Chongqing University of Posts and Telecommunications(Natural Science Edition) 22 339 (in Chinese) [李清都, 周丽, 周红伟 2010 重庆邮电大学学报(自然科学版) 22 339]
[21] Palis J, Melo W D 1982 Geometric Theory of Dynamical Systems (New York: Springer-Verlag)
[22] Gonchenko S V, Ovsyannikov I I, Simo C, Turaev D 2005 Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15 3493
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[1] Xu, Liu C X, Yang T 2010 Acta Phys. Sin. 59 131 (in Chinese) [许, 刘崇新, 杨韬 2010 59 131]
[2] Johnson M E, Jolly M S, Kevrekidis I G 1997 Numer. Algorithms. 14 125
[3] Guckenheimer J, Worfolk P 1993 Kluwer Academic Publishers 241
[4] Henderson M E 2005 SIAM J. Appl. Dyn. Syst. 4 832
[5] Krauskopf B, Osinga H M 1999 Chaos 9 768
[6] Krauskopf B, Osinga H M, Doedel E J, Henderson M E, Guckenheimer J, Vladimirsky A, Dellnitz M, Junge O 2005 Bifur. Chaos. Appl. Sci. Engrg. 15 763
[7] Guckenheimer J, Vladimirsky A 2004 SIAM J. Appl. Dyn. Syst. 3 232
[8] Li Q D, Yang X S 2010 Acta Phys. Sin. 59 1416 (in Chinese) [李清都, 杨晓松 2010 59 1416]
[9] Jia M, Fan Y Y, Li HM2010 Acta Phys. Sin. 59 7686 (in Chinese) [贾蒙, 樊养余, 李慧敏 2010 59 7686]
[10] Li Q D, Yang X S 2010 Acta Phys. Sin. 59 1416 (in Chinese) [李清都, 杨晓松 2010 59 1416]
[11] Hobson D 1991 J. Comput. Phys. 104 14
[12] You Z, Kostelich E J, Yorke J A 1991 Int. J. Bifurc. Chaos Appl. Sci. Eng. 1 605
[13] Sim′o C 1989 Benest D, Froeschlé C (eds.) Les Méthodes Modernes de la Mécanique Céleste 285
[14] Parker T S, Chua L O 1989 Practical Numerical Algorithms for Chaotic Systems (Berlin: Springer)
[15] Krauskopf B, Osinga H M 1998 J. Comput. Phys. 146 406
[16] England J P, Krauskopf B, Osinga H M 2004 SIAM J. Appl. Dyn. Syst. 3 161
[17] Krauskopf B, Osinga H M 1998 Int. J. Bifurc. Chaos 8 483
[18] Krauskopf B, Osinga H M 1999 Chaos. 9 768
[19] Dellnitz M, Hohmann A 1997 Numer. Math 75 293
[20] Li Q D, Zhou L, Zhou HW2010 Journal of Chongqing University of Posts and Telecommunications(Natural Science Edition) 22 339 (in Chinese) [李清都, 周丽, 周红伟 2010 重庆邮电大学学报(自然科学版) 22 339]
[21] Palis J, Melo W D 1982 Geometric Theory of Dynamical Systems (New York: Springer-Verlag)
[22] Gonchenko S V, Ovsyannikov I I, Simo C, Turaev D 2005 Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15 3493
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