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Parrondo's paradox for chaos control and anticontrol of fractional-order systems |
| Marius-F Danca1,2 and Wallace K S Tang3 |
1. Department of Mathematics and Computer Science, Avram Iancu University, 400380 Cluj-Napoca, Romania;
2. Romanian Institute for Science and Technology, 400487 Cluj-Napoca, Romania;
3. Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China |
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Abstract We present the generalized forms of Parrondo's paradox existing in fractional-order nonlinear systems. The generalization is implemented by applying a parameter switching (PS) algorithm to the corresponding initial value problems associated with the fractional-order nonlinear systems. The PS algorithm switches a system parameter within a specific set of N≥2 values when solving the system with some numerical integration method. It is proven that any attractor of the concerned system can be approximated numerically. By replacing the words “winning” and “loosing” in the classical Parrondo's paradox with “order” and “chaos”, respectively, the PS algorithm leads to the generalized Parrondo's paradox: chaos1+chaos2+…+chaosN=order and order1+order2+…+orderN=chaos. Finally, the concept is well demonstrated with the results based on the fractional-order Chen system.
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Received: 08 March 2015
Revised: 10 September 2015
Accepted manuscript online:
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PACS:
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05.45.Ac
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(Low-dimensional chaos)
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05.45.-a
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(Nonlinear dynamics and chaos)
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Corresponding Authors:
Marius-F Danca
E-mail: danca@rist.ro
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Cite this article:
Marius-F Danca, Wallace K S Tang Parrondo's paradox for chaos control and anticontrol of fractional-order systems 2016 Chin. Phys. B 25 010505
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| [1] |
Harmer G P and Abbott D 1999 Nature 402 864
|
| [2] |
Harmer G P and Abbott D 1999 Stat. Sci. 14 206
|
| [3] |
Raghuram I and Rajeev K 2003 Complexity 9 23
|
| [4] |
Lee Y, Allison A, Abbott D and Stanley H E 2003 Phys. Rev. Lett. 91 220601
|
| [5] |
Amengual P, Allison A Toral R and Abbott D 2004 Proc. R Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 460 2269
|
| [6] |
Blackwell D and Girshick M A 1954 Theory of Games and Statistical Decisions (New York: John Wiley & Sons)
|
| [7] |
Heath D, Kinderlehrer D and Kowalczyk M 2002 Discr. Cont. Dynam. Syst. B 2 153
|
| [8] |
Harmer G P and Abbott D 2002 Fluct. Noise Lett. 2 R71
|
| [9] |
Shu J J and Wang Q W 2014 Scientific Reports 4 4244
|
| [10] |
Danca M F 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 500
|
| [11] |
Mao Y, Tang W K S and Danca M F 2010 Appl. Math. Comput. 217 355
|
| [12] |
Danca M F, Aziz-Alaoui M A and Small M 2015 Chin. Phys. B 24 060507
|
| [13] |
Danca M F, Fečkan M and Romera M 2014 Int. J. Bifurcat. Chaos 24 1450008
|
| [14] |
Danca M F, Romera M and Pastor G 2009 Int. J. Bifurcat. Chaos 19 2123
|
| [15] |
Oustaloup A 1995 La Derivation Non Entiere: Theorie, Synthese et Applications (Paris: Hermes)
|
| [16] |
Podlubny I 1999 Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solutiona nd Some of Their Applications (San Diego: Academic Press)
|
| [17] |
Nakagava M and Sorimachi K 1992 IEICE Trans. Fundamentals E75-A(12) 1814
|
| [18] |
Cafagna D and Grassi G 2015 Chin. Phys. B 24 080502
|
| [19] |
Min F H, Shao S Y, Huang W D and Wang E R 2015 Chin. Phys. Lett. 32 030503
|
| [20] |
Zhang H, Chen D Y, Zhou K and Wang Y C 2015 Chin. Phys. B 24 030203
|
| [21] |
Xue W, Xu J K, Cang S J and Jia H Y 2014 Chin. Phys. B 23 060501
|
| [22] |
Zhoua P and Zhua W 2011 Nonlinear Anal. Real. 12 811
|
| [23] |
Oldham K B and Spanier J 1974 The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order (New York: Academic Press)
|
| [24] |
Diethelm K, Ford N J and Freed A D 2002 Nonlinear Dynam. 29 3
|
| [25] |
Charef A, Sun H H, Tsao Y Y and Onaral B 1992 IEEE Trans. Automatic Control 37 1465
|
| [26] |
Tavazoei M S and Haeri M 2010 Automatica 46 94
|
| [27] |
Caputo M 1969 Elasticity and Dissipation (Bologna: Zanichelli)
|
| [28] |
Hartley T T, Lorenzo C F, Trigeassou J C and Maamri N 2013 J. Comput. Nonlin. Dyn. ASME 8 041014
|
| [29] |
Diethelm K 2013 App. Anal. 93 2126
|
| [30] |
Diethelm K, Ford N J and Freed A D 2004 Numer. Algorithms 36 31
|
| [31] |
Danca M F, Romera M, Pastor G and Montoya F 2012 Nonlinear Dynam. 67 2317
|
| [32] |
Li C and Chen G 2004 Chaos, Solitons Fractal. 22 549
|
| [33] |
Garrappa R 2011 Predictor-corrector PECE Method for Fractional Differential Equations Matlab Central
|
| [34] |
Garrappa R 2010 Int. J. Comput. Math. 87 2281
|
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