中国物理B ›› 2011, Vol. 20 ›› Issue (1): 10502-010502.doi: 10.1088/1674-1056/20/1/010502

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Horseshoe and entropy in a fractional-order unified system

周平1, 陈述2, 李清都3   

  1. (1)Institute for Nonlinear Systems, Chongqing University of Posts and Telecommunications, Chongqing 400065, China; (2)Key Laboratory of Networked Control and Intelligent Instrument of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China; (3)Key Laboratory of Networked Control and Intelligent Instrument of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China;Institute for Nonlinear Systems, Chongqing University of Posts and Telecommunications, C
  • 收稿日期:2010-06-02 修回日期:2010-07-14 出版日期:2011-01-15 发布日期:2011-01-15
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 10926072 and 10972082), Chongqing Municipal Education Commission (Grant No. KJ080515) and Natural Science Foundation Project of CQ CSTC, China (Grant No. 2008BB2409).

Horseshoe and entropy in a fractional-order unified system

Li Qing-Du(李清都)a)b)†, Chen Shu(陈述)a), and Zhou Ping(周平)b)   

  1. a Key Laboratory of Networked Control and Intelligent Instrument of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China; b Institute for Nonlinear Systems, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
  • Received:2010-06-02 Revised:2010-07-14 Online:2011-01-15 Published:2011-01-15
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 10926072 and 10972082), Chongqing Municipal Education Commission (Grant No. KJ080515) and Natural Science Foundation Project of CQ CSTC, China (Grant No. 2008BB2409).

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摘要： This paper studies chaotic dynamics in a fractional-order unified system by means of topological horseshoe theory and numerical computation. First it finds four quadrilaterals in a carefully-chosen Poincar'e section, then shows that the corresponding map is semiconjugate to a shift map with four symbols. By estimating the topological entropy of the map and the original time-continuous system, it provides a computer assisted verification on existence of chaos in this system, which is much more convincible than the common method of Lyapunov exponents. This new method can potentially be used in rigorous studies of chaos in such a kind of system. This paper may be a start for proving a given fractional-order differential equation to be chaotic.

关键词: chaos, topological horseshoe, fractional-order system, generalised Lorenz system

Abstract: This paper studies chaotic dynamics in a fractional-order unified system by means of topological horseshoe theory and numerical computation. First it finds four quadrilaterals in a carefully-chosen Poincar'e section, then shows that the corresponding map is semiconjugate to a shift map with four symbols. By estimating the topological entropy of the map and the original time-continuous system, it provides a computer assisted verification on existence of chaos in this system, which is much more convincible than the common method of Lyapunov exponents. This new method can potentially be used in rigorous studies of chaos in such a kind of system. This paper may be a start for proving a given fractional-order differential equation to be chaotic.

Key words: chaos, topological horseshoe, fractional-order system, generalised Lorenz system

中图分类号:  (Nonlinear dynamics and chaos)

  • 05.45.-a