中国物理B ›› 2018, Vol. 27 ›› Issue (8): 80501-080501.doi: 10.1088/1674-1056/27/8/080501

• GENERAL • 上一篇    下一篇

Topological classification of periodic orbits in Lorenz system

Chengwei Dong(董成伟)   

  1. Department of Physics, North University of China, Taiyuan, China
  • 收稿日期:2018-04-25 修回日期:2018-05-25 出版日期:2018-08-05 发布日期:2018-08-05
  • 通讯作者: Chengwei Dong E-mail:dongchengwei@tsinghua.org.cn
  • 基金资助:
    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).

Topological classification of periodic orbits in Lorenz system

Chengwei Dong(董成伟)   

  1. Department of Physics, North University of China, Taiyuan, China
  • Received:2018-04-25 Revised:2018-05-25 Online:2018-08-05 Published:2018-08-05
  • Contact: Chengwei Dong E-mail:dongchengwei@tsinghua.org.cn
  • Supported by:
    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).

摘要: 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.

关键词: Lorenz equations, periodic orbit, variational method, symbolic dynamics

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.

Key words: Lorenz equations, periodic orbit, variational method, symbolic dynamics

中图分类号:  (Nonlinear dynamics and chaos)

  • 05.45.-a
05.45.Ac (Low-dimensional chaos) 02.60.Cb (Numerical simulation; solution of equations)