Algrithm for detecting homoclinic orbits of time-continuous dynamical system and its application

Yang Fang-Yan; Hu Ming; Yao Shang-Ping

doi:10.7498/aps.62.100501

Abstract

Detecting homoclinic orbits is a key problem in nonlinear dynamical systems, especially in the study of bifurcation and chaos. In this paper, we propose a new method to solve the problem with trajectory optimization. By defining a distance between a saddle point and its near trajectories, the problem becomes a common problem in unconstrained nonlinear optimization to minimize the distance. A subdivision algorithm is also proposed in this paper to improve the integrity of results. By applying the algorithm to the Lorenz system, the Shimizu-Morioka system and the hyperchaotic Lorenz system, we successfully find many homoclinic orbits with the corresponding parameters, which suggests that the method is effective.

Keywords:

Authors and contacts

1.
Key Laboratory of Industrial Internet of Things and Networked Control of Ministry of Education, Chongqing University of Postsand Telecommunications, Chongqing 400065, China;
2.
Institute for Nonlinear Circuits and Systems, Chongqing University Posts and Telecommunications, Chongqing 400065, China
Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61104150), the Natural Science Foundation Project of Chongqing, China (Grant No. cstcjjA40044) and the Doctoral Fund of Chongqing University of Posts and Telecommunication, China (Grant No. A2009-12).

References

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[20]	Chen P, Xiao L 2010 E. J. Qualitative Theory Diff. Equ. 72 1
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[22]	Rodríguez-Luis A J, Freire E, Ponce E 1990 Continuation and Bifurcations: Numerical Techniques and Applications (Leuven:Kluwer Academic Publishers) 197
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[25]	Friedman M J, Doedel E J 1993 J. Dyn. Different. Equations 5 37
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[27]	Bao J H, Yang Q G 2011 Appl. Math.Comput. 217 6526
[28]	De Witte V, Govaerts W, Kuznetsov Y A, Friedman M 2012 ACM T. Math. Software 38 34
[29]	Lenci S, Rega G 2011 Nonlinear Dyn. 63 83
[30]	Tigan G 2010 Appl. Math. Inf. Sci. 4 383
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[35]	Shimizu T, Morioka N 1980 Phys. Lett. A 76 201
[36]	Wang X Y, Wang M J 2007 Acta Phys. Sin. 56 5136 (in Chinese) [王兴元, 王明军 2007 56 5136]

Cited By

[1]	Zhang Q F, Tang X H 2012 Appl. Math. Comput. 218 7164
[2]	Li X H, Bi Q S 2012 Acta Phys. Sin. 61 020504 (in Chinese) [李向红, 毕勤胜 2012 61 020504]
[3]	Jia B, Gu H G, Li Y Y 2011 Chin. Phys. Lett. 28 090507
[4]	Feng J J, Zhang Q C, Wang W 2011 Chin. Phys. B 20 090202
[5]	Huan S M, Li Q D, Yang X S 2012 Nonlinear Dyn. 69 1915
[6]	Yang X L, Xu W, Sun Z K 2006 Acta Phys. Sin. 55 1678 (in Chinese) [杨晓丽,徐伟, 孙中奎 2006 55 1678]
[7]	Li W Y, Zhang Q C, Wang W 2010 Chin. Phys. B 19 060510
[8]	Li Q D, Zhou L, Zhou H W 2010 J. Chongqing Univ. Posts and Telecommun. (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]
[9]	Guo K M, Li Q D 2008 J. Chongqing Univ. Posts Telecommun. (Natural Science Edition) 20 221 (in Chinese) [郭克敏, 李清都 2008 重庆邮电大学学报 (自然科学版) 20 221]
[10]	Melnikov V 1963 Trans. Moscow Math. Soc. 12 1
[11]	Di G H, Xu Y, Xu W, Gu R C 2011 Acta Phys. Sin. 60 020504 (in Chinese) [狄根虎, 许勇, 徐伟, 顾仁财2011 60 020504]
[12]	Tian R L, Yang X W, Cao Q J, Wu Q L 2012 Chin. Phys. B 21 020503
[13]	Li Q D, Yang X S 2010 Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 467
[14]	Li Q D 2008 Phys. Lett. A 372 2989
[15]	Li Q D, Tang S 2013 Acta Phys. Sin. 62 020510 (in Chinese) [李清都, 唐宋2013 62 020510]
[16]	El-Dessoky M M,Yassen M T, Aly E S 2012 Appl. Math. Comput. 218 11859
[17]	Li J B, Chen F J 2011 Internat. J. Bifur. Chaos Appl. Sci. Engrg. 21 3305
[18]	Li X Y, Wang H J 2011 Internat. J. Bifur. Chaos Appl. Sci. Engrg. 21 2695
[19]	Li T C, Chen G T, Chen G R 2006 Internat. J. Bifur. Chaos Appl. Sci. Engrg. 16 3035
[20]	Chen P, Xiao L 2010 E. J. Qualitative Theory Diff. Equ. 72 1
[21]	Lorenz E N 1963 J. Atmosph. Sci. 20 130
[22]	Rodríguez-Luis A J, Freire E, Ponce E 1990 Continuation and Bifurcations: Numerical Techniques and Applications (Leuven:Kluwer Academic Publishers) 197
[23]	Freire E, Pizarro L, Rodrigeuz-Luis A J 1999 IMA J. Numer. Anal. 19 51
[24]	Beyn W J 1990 IMA J. Numer. Anal. 10 379
[25]	Friedman M J, Doedel E J 1993 J. Dyn. Different. Equations 5 37
[26]	Sandstede B 1997 IMA J. Numer. Anal. 17 437
[27]	Bao J H, Yang Q G 2011 Appl. Math.Comput. 217 6526
[28]	De Witte V, Govaerts W, Kuznetsov Y A, Friedman M 2012 ACM T. Math. Software 38 34
[29]	Lenci S, Rega G 2011 Nonlinear Dyn. 63 83
[30]	Tigan G 2010 Appl. Math. Inf. Sci. 4 383
[31]	Li Q D, Yang X S 2010 Acta Phys. Sin. 59 1416 (in Chinese) [李清都, 杨晓松 2010 59 1416]
[32]	Li H M, Fan Y Y, Sun H Y, Zhang J, Jia M 2012 Acta Phys. Sin. 61 029501 (in Chinese) [李慧敏, 樊养余, 孙恒义, 张菁, 贾蒙 2012 61 029501]
[33]	Li Q D, Tan Y L, Yang F Y 2011 Acta Phys. Sin. 60 030206 (in Chinese) [李清都, 谭宇铃, 杨芳艳 2011 60 030206]
[34]	Li Q D, Zhou H W, Yang X S 2012 Acta Phys. Sin. 61 040503 (in Chinese) [李清都, 周红伟, 杨晓松 2012 61 040503]
[35]	Shimizu T, Morioka N 1980 Phys. Lett. A 76 201
[36]	Wang X Y, Wang M J 2007 Acta Phys. Sin. 56 5136 (in Chinese) [王兴元, 王明军 2007 56 5136]

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Vol. 62, Issue 10 - 2013-05-20

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