|
|
A new piecewise linear Chen system of fractional-order: Numerical approximation of stable attractors |
Marius-F. Dancaa b, M. A. Aziz-Alaouic, Michael Smalld |
a Department of Mathematics and Computer Science, Emanuel University of Oradea, 410597 Oradea, Romania; b Romanian Institute of Science and Technology, 400487 Cluj-Napoca, Romania; c Normandie University, France; ULH, LMAH, F-76600 Le Havre; FR CNRS 3335, ISCN, 25 rue Philippe Lebon 76600 Le Havre, France; d School of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia |
|
|
Abstract In this paper we present a new version of Chen's system: a piecewise linear (PWL) Chen system of fractional-order. Via a sigmoid-like function, the discontinuous system is transformed into a continuous system. By numerical simulations, we reveal chaotic behaviors and also multistability, i.e., the existence of small parameter windows where, for some fixed bifurcation parameter and depending on initial conditions, coexistence of stable attractors and chaotic attractors is possible. Moreover, we show that by using an algorithm to switch the bifurcation parameter, the stable attractors can be numerically approximated.
|
Received: 07 November 2014
Revised: 30 December 2014
Accepted manuscript online:
|
PACS:
|
05.45.Ac
|
(Low-dimensional chaos)
|
|
05.45.Gg
|
(Control of chaos, applications of chaos)
|
|
05.45.Pq
|
(Numerical simulations of chaotic systems)
|
|
Fund: Dedicated to Professor Chen Guan-Rong on the occasion of his 65th birthday. |
Corresponding Authors:
Marius-F. Danca, M. A. Aziz-Alaoui, Michael Small
E-mail: danca@rist.ro;aziz.alaoui@univ-lehavre.fr;michael.small@uwa.edu.au
|
About author: 05.45.Ac; 05.45.Gg; 05.45.Pq |
Cite this article:
Marius-F. Danca, M. A. Aziz-Alaoui, Michael Small A new piecewise linear Chen system of fractional-order: Numerical approximation of stable attractors 2015 Chin. Phys. B 24 060507
|
[1] |
Chen G and Ueta T 1999 Int. J. Bifur. Chaos 9 1465
|
[2] |
Aziz-Alaoui M A and Chen G 2002 Int. J. Bifur. Chaos 12 147
|
[3] |
Hu J B, Zhao L D and Xie Z G 2013 Chin. Phys. B 22 080506
|
[4] |
Liu J G 2013 Chin. Phys. B 22 060510
|
[5] |
Mainardi F 1996 Appl. Math. Lett. 9 23
|
[6] |
Li R H and Chen W S 2013 Chin. Phys. B 22 040503
|
[7] |
Kilbas A A, Srivastava H M and Trujillo J J 2006 Theory and Applications Fractional Differential Equations (Amsterdam: North-Holland Mathematics Studies, Vol. 204, North-Holland)
|
[8] |
Zhang R X, Yang Y, Yang S P and Wang X B 2009 Acta Phys. Sin. 58 6039 (in Chinese)
|
[9] |
Samko G, Kilbas A A and Marichev O I 1993 Fractional Integrals and Derivatives: Theory and Applications (Yverdon: Gordon and Breach)
|
[10] |
Ma T, Jiang W, Fu J, Chai Y, Chen L and Xue F 2012 Acta Phys. Sin. 61 160506 (in Chinese)
|
[11] |
Wang S P, Lao S K, Chen H K, Chen J H and Chen S Y 2013 Int. J. Bifurc. Chaos 23 1350030
|
[12] |
Petras I 2011 Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Hidelberg: Springer-Verlag)
|
[13] |
Arena P, Caponetto R, Fortuna L and Porto D 2000 Nonlinear Noninteger Order Circuits and Systems: An Introduction (Singapore: World Scientific)
|
[14] |
Wen S, Zeng Z, Huang T and Zhang Y 2014 IEEE Trans. Fuzzy Sys. 22 1704
|
[15] |
Wen S, Huang T, Zeng Z, Chen Y and Lie P 2014 Neural Networks 63 48
|
[16] |
Wen S P, Zeng Z G, Huang T W and Li C J 2013 International Journal of Robust and Nonlinear Control DOI: 10.1002/rnc.3112
|
[17] |
Li C and Chen G 2004 Chaos, Solitons & Fractals 22 549
|
[18] |
Caputo M 1967 Geophys. J. R. Astron. Soc. 13 529; reprinted in Fract. Calc. Appl. Anal. 2007 10 309
|
[19] |
Oldham K B and Spanier J 1974 The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order (New York: Academic Press)
|
[20] |
Podlubny I 1999 Fractional Differential Equations (San Diego: Academic Press)
|
[21] |
Heymans N and Podlubny I 2006 Rheol. Acta 45 765
|
[22] |
Diethelm K, Ford N J and Freed A D 2002 Nonlinear Dyn. 29 3
|
[23] |
Danca M F and Diethlem K 2011 Commun. Nonlinear Sci. Numer. Simul. 15 3745
|
[24] |
Danca M F 2013 Commun. Nonlinear Sci. Numer. Simul. 18 500
|
[25] |
Feckan M and Danca M F 2014 Mathematica Slovaca accepted
|
[26] |
Diethlem K 2010 The Analysis of Fractional Differential Equations (Berlin/Heidelberg: Springerr-Verlag)
|
[27] |
Odibat Z, Bertelle C, Aziz-Alaoui M A and Duchamp G 2010 Comput. Math. Appl. 59 1462
|
[28] |
Danca M F 2014 Int. J. Bifur. Chaos accepted
|
[29] |
Kaslik E and Sivasundaram S 2012 Nonlinear Anal-Real 13 1489
|
[30] |
Falconer K 1990 Fractal Geometry, Mathematical Foundations and Applications (Chichester: John Wiley and Sons)
|
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
|
|
|