|
|
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) |
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 |
|
|
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.
|
Received: 02 June 2010
Revised: 14 July 2010
Accepted manuscript online:
|
PACS:
|
05.45.-a
|
(Nonlinear dynamics and chaos)
|
|
Fund: 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). |
Cite this article:
Li Qing-Du(李清都), Chen Shu(陈述), and Zhou Ping(周平) Horseshoe and entropy in a fractional-order unified system 2011 Chin. Phys. B 20 010502
|
[1] |
Kilbas A, Srivastava H and Trujillo J 2006 Theory and Applications of Fractional Differential Equations (Amsterdam: Elsevier Science Ltd)
|
[2] |
L"u J 2002 International Journal of Bifurcation and Chaos 12 2917
|
[3] |
Li C and Peng G 2004 Chaos, Solitons and Fractals 22 443
|
[4] |
Grigorenko I and Grigorenko E 2003 Phys. Rev. Lett. 91 034101
|
[5] |
Wu X, Li J and Chen G 2008 Journal of the Franklin Institute 345 392
|
[6] |
Wang X Y and He Y J 2008 Acta Phys. Sin. 57 1485 (in Chinese)
|
[7] |
Zhang R X, Yang S P and Liu Y L 2010 Acta Phys. Sin. 59 1549 (in Chinese)
|
[8] |
Shao S Q, Gao X and Liu X W 2007 Acta Phys. Sin. 56 6815 (in Chinese)
|
[9] |
Zhang R X and Yang S P 2009 Chin. Phys. B 18 3295
|
[10] |
Chen X R, Liu C X and Wang F Q 2008 Chin. Phys. B 17 1664
|
[11] |
Zhou P, Cheng X F and Zhang N Y 2008 Chin. Phys. B 17 3252
|
[12] |
Zhang C F, Gao J F and Xu L 2007 Acta Phys. Sin. 56 5124 (in Chinese)
|
[13] |
Chen X R, Liu C X, Wang F Q and Li Y X 2008 Acta Phys. Sin. 57 1416 (in Chinese)
|
[14] |
Lu J J and Liu C X 2007 Chin. Phys. 16 1586
|
[15] |
Tavazoei M and Haeri M 2007 Phys. Lett. A 367 102
|
[16] |
Zgliczynski P 1997 Nonlinearity 10 243
|
[17] |
Li Q and Yang X 2006 Journal of Physics A: Mathematical and General 39 9139
|
[18] |
Galias Z and Zgliczy'nski P 1998 Physica D: Nonlinear Phenomena 115 165
|
[19] |
Yang X and Li Q 2006 Chaos, Solitons and Fractals 27 25
|
[20] |
Wiggins S 2003 Introduction to Applied Nonlinear Dynamical Systems and Chaos (New York: Springer Verlag)
|
[21] |
Yang X and Tang Y 2004 Chaos, Solitons and Fractals 19 841
|
[22] |
Sun K H, Ren J and Qiu S S 2008 Journ al of South China University of Technology (Natural Science Edition) 36 6 (in Chinese)
|
[23] |
Li Q 2008 A toolbox for finding horseshoes in 2D maps. In: http://www.mathworks.com/matlabcentral/fileexchange/ 14075
|
[24] |
Li Q and Yang X S 2009 International Journal of Bifurcation and Chaos 20 467
|
[25] |
Lohner R 1992 Computational Ordinary Differential Equations (London: Oxford University Press) p425
|
[26] |
Diethelm K, Ford N J and Freed A D 2002 Nonlinear Dynamics 29 3
|
[27] |
Eckmann J and Ruelle D 1985 Rev. Mod. Phys. 57 617 endfootnotesize
|
No Suggested Reading articles found! |
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
Altmetric
|
blogs
Facebook pages
Wikipedia page
Google+ users
|
Online attention
Altmetric calculates a score based on the online attention an article receives. Each coloured thread in the circle represents a different type of online attention. The number in the centre is the Altmetric score. Social media and mainstream news media are the main sources that calculate the score. Reference managers such as Mendeley are also tracked but do not contribute to the score. Older articles often score higher because they have had more time to get noticed. To account for this, Altmetric has included the context data for other articles of a similar age.
View more on Altmetrics
|
|
|