Topological classification of periodic orbits in Lorenz system

doi:10.1088/1674-1056/27/8/080501

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: dongchengwei@tsinghua.org.cn

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

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