Please wait a minute...
Chin. Phys. B, 2018, Vol. 27(8): 080501    DOI: 10.1088/1674-1056/27/8/080501
GENERAL Prev   Next  

Topological classification of periodic orbits in Lorenz system

Chengwei Dong(董成伟)
Department of Physics, North University of China, Taiyuan, China
Abstract  We systematically investigate the periodic orbits of the Lorenz flow up to certain topological length. As an alternative to Poincaré section map analysis, we propose a new approach for establishing one-dimensional symbolic dynamics based on the topological structure of the orbit. A newly designed variational method is stable numerically for cycle searching, and two orbital fragments can be used as basic building blocks for initialization. The topological classification based on the entire orbital structure is revealed to be effective. The deformation of periodic orbits with the change of parameters provides a chart to the periods of cycles. The current research may provide a methodology for finding and systematically classifying periodic orbits in other similar chaotic flows.
Keywords:  Lorenz equations      periodic orbit      variational method      symbolic dynamics  
Received:  25 April 2018      Revised:  25 May 2018      Accepted manuscript online: 
PACS:  05.45.-a (Nonlinear dynamics and chaos)  
  05.45.Ac (Low-dimensional chaos)  
  02.60.Cb (Numerical simulation; solution of equations)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11647085, 11647086, and 11747106), the Applied Basic Research Foundation of Shanxi Province, China (Grant No. 201701D121011), and the Natural Science Research Fund of North University of China (Grant No. XJJ2016036).
Corresponding Authors:  Chengwei Dong     E-mail:

Cite this article: 

Chengwei Dong(董成伟) Topological classification of periodic orbits in Lorenz system 2018 Chin. Phys. B 27 080501

[1] Lorenz E N 1963 J. Atmos. Sci. 20 130
[2] Mischaikow K and Mrozek M 1998 Math. Comput. 67 1023
[3] Stewart I 2000 Nature 406 948
[4] Tucker W 1999 C. R. Acad. Sci. Paris Ser. I Math. 328 1197
[5] Viswanath D 2003 Nonlinearity 16 1035
[6] Zheng Y and Zhang X D 2010 Chin. Phys. B 19 010505
[7] Li R H and Chen W S 2013 Chin. Phys. B 22 040503
[8] Doedel E J, Krauskopf B and Osinga H M 2006 Nonlinearity 19 2947
[9] Postlethwaite C M and Silber M 2007 Phys. Rev. E 76 056214
[10] Munmuangsaen B and Srisuchinwong B 2018 Chaos, Solitons and Fractals 107 61
[11] Pinsky T 2017 P. Roy. Soc. A-Math. Phys. 473 20170374
[12] Zhang F and Zhang G 2016 Complexity 21 440
[13] Sparrow C 1982 The Lorenz equations: bifurcations, chaos, and strange attractors (New York: Springer Verlag)
[14] Artuso R, Aurell E and Cvitanović P 1990 Nonlinearity 3 325
[15] Artuso R, Aurell E and Cvitanović P 1990 Nonlinearity 3 361
[16] Christiansen F, Cvitanović P and Putkaradze V 1997 Nonlinearity 10 55
[17] Strogatz S H 2000 Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering (New York: Perseus Books Publishing) pp. 312-313
[18] Cvitanović P, Artuso R, Mainieri R, Tanner G, Vattay G, Whelan N and Wirzba A 2012 Chaos: Classical and Quantum (Copenhagen: Niels Bohr Institute)
[19] Guckenheimer J and Holmes P 1983 Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (New York: Springer Verlag)
[20] Lan Y and Cvitanović P 2004 Phys. Rev. E 69 016217
[21] Press W H, Teukolsky S A, Veterling W T and Flannery B P 1992 Numerical Recipes in Fortran 77. The Art of Scientific Computing (New York: Cambridge) pp. 34-40
[22] Dong C and Lan Y 2014 Commun. Nonlinear Sci. Numer. Simul. 19 2140
[23] Dong C 2018 Mod. Phys. Lett. B 32 1850155
[24] Dong C 2018 Int. J. Mod. Phys. B 32 1850227
[25] Dong C, Wang P, Du M, Uzer T and Lan Y 2016 Mod. Phys. Lett. B 30 1650183
[26] Chen G and Ueta T 1999 Int. J. Bifurcation Chaos 9 1465
[27] Lü J and Chen G 2002 Int. J. Bifurcation Chaos 12 1789
[28] Hao B L and Zheng W M 1998 Applied Symbolic Dynamics and Chaos (Singapore: World Scientific) pp. 6-10
[1] Variational approximation methods for long-range force transmission in biopolymer gels
Haiqin Wang(王海钦), and Xinpeng Xu(徐新鹏). Chin. Phys. B, 2022, 31(10): 104602.
[2] Propagation dynamics of dipole breathing wave in lossy nonlocal nonlinear media
Jian-Li Guo(郭建丽), Zhen-Jun Yang(杨振军), Xing-Liang Li(李星亮), and Shu-Min Zhang(张书敏). Chin. Phys. B, 2022, 31(1): 014203.
[3] An analytical variational method for the biased quantum Rabi model in the ultra-strong coupling regime
Bin-Bin Mao(毛斌斌), Maoxin Liu(刘卯鑫), Wei Wu(吴威), Liangsheng Li(李粮生), Zu-Jian Ying(应祖建), Hong-Gang Luo(罗洪刚). Chin. Phys. B, 2018, 27(5): 054219.
[4] Resonances for positron-helium and positron-lithium systems in kappa-distribution plasma
Zi-Shi Jiang(姜子实), Ya-Chen Gao(高亚臣), Sabyasachi Kar, Kurunathan Ratnavelu. Chin. Phys. B, 2018, 27(12): 123402.
[5] Odd-even harmonic emission from asymmetric molecules: Identifying the mechanism
Jianguo Chen(陈建国), Shujuan Yu(于术娟), Yanpeng Li(李雁鹏), Shang Wang(王赏), Yanjun Chen(陈彦军). Chin. Phys. B, 2017, 26(9): 094209.
[6] Properties of strong-coupling magneto-bipolaron qubit in quantum dot under magnetic field
Xu-Fang Bai(白旭芳), Ying Zhang(张颖), Wuyunqimuge(乌云其木格), Eerdunchaolu(额尔敦朝鲁). Chin. Phys. B, 2016, 25(7): 077804.
[7] Start-up phase plasma discharge design of a tokamak via control parameterization method
Guo Shan (郭珊), Xu Ke (许珂), Xu Chao (许超), Ren Zhi-Gang (任志刚), Xiao Bing-Jia (肖炳甲). Chin. Phys. B, 2015, 24(3): 035202.
[8] Research on the discrete variational method for a Birkhoffian system
Liu Shi-Xing (刘世兴), Hua Wei (花巍), Guo Yong-Xin (郭永新). Chin. Phys. B, 2014, 23(6): 064501.
[9] Studies of phase return map and symbolic dynamics in a periodically driven Hodgkin–Huxley neuron
Ding Jiong (丁炯), Zhang Hong (张宏), Tong Qin-Ye (童勤业), Chen Zhuo (陈琢). Chin. Phys. B, 2014, 23(2): 020501.
[10] Elastic fields around a nanosized elliptichole in decagonal quasicrystals
Li Lian-He (李联和), Yun Guo-Hong (云国宏). Chin. Phys. B, 2014, 23(10): 106104.
[11] Noise destroys the coexisting of periodic orbits of a piecewise linear map
Wang Can-Jun (王参军), Yang Ke-Li (杨科利), Qu Shi-Xian (屈世显). Chin. Phys. B, 2013, 22(3): 030502.
[12] Study of electronic structures and absorption bands of BaMgF4 crystal with F colour centre
Kang Ling-Ling(康玲玲), Liu Ting-Yu(刘廷禹), Zhang Qi-Ren(张启仁), Xu Ling-Zhi(徐灵芝), and Zhang Fei-Wu(张飞武) . Chin. Phys. B, 2011, 20(4): 047101.
[13] A method of recovering the initial vectors of globally coupled map lattices based on symbolic dynamics
Sun Li-Sha(孙丽莎), Kang Xiao-Yun(康晓云), Zhang Qiong(张琼), and Lin Lan-Xin(林兰馨) . Chin. Phys. B, 2011, 20(12): 120507.
[14] A complexity measure approach based on forbidden patterns and correlation degree
Wang Fu-Lai(王福来). Chin. Phys. B, 2010, 19(6): 060515.
[15] The energy levels of a two-electron two-dimensional parabolic quantum dot
Li Wei-Ping(李伟萍), Xiao Jing-Lin(肖景林),Yin Ji-Wen(尹辑文), Yu Yi-Fu(于毅夫), and Wang Zi-Wu(王子武). Chin. Phys. B, 2010, 19(4): 047102.
No Suggested Reading articles found!