Fractional-order systems without equilibria: The first example of hyperchaos and its application to synchronization

doi:10.1088/1674-1056/24/8/080502

Abstract A challenging topic in nonlinear dynamics concerns the study of fractional-order systems without equilibrium points. In particular, no paper has been published to date regarding the presence of hyperchaos in these systems. This paper aims to bridge the gap by introducing a new example of fractional-order hyperchaotic system without equilibrium points. The conducted analysis shows that hyperchaos exists in the proposed system when its order is as low as 3.84. Moreover, an interesting application of hyperchaotic synchronization to the considered fractional-order system is provided.

Keywords: fractional-order systems equilibrium points hyperchaotic systems synchronization

Received: 21 November 2014
Revised: 19 February 2015
Accepted manuscript online:

PACS:	05.45.Jn	(High-dimensional chaos)
	05.45.Xt	(Synchronization; coupled oscillators)

Corresponding Authors: Donato Cafagna, Giuseppe Grassi
E-mail: donato.cafagna@unisalento.it;giuseppe.grassi@unisalento.it

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Donato Cafagna, Giuseppe Grassi Fractional-order systems without equilibria: The first example of hyperchaos and its application to synchronization 2015 Chin. Phys. B 24 080502

[1]	Podlubny I 1999 Fractional differential equations (New York: Academic Press)
[2]	Arena P, Caponetto R, Fortuna L and Porto D 2000 Nonlinear non-integer order circuits and systems – An introduction (Singapore: World Scientific)
[3]	Cafagna D 2007 IEEE Industrial Electronics Magazine 1 35
[4]	Hilfer R (ed.) 2000 Applications of Fractional Calculus in Physics (Singapore: World Scientific)
[5]	Tenreiro Machado J A, Silva M F, Barbosa R S, Jesus I S, Reis C M, Marcos M G and Galhano A F 2010 Mathematical Problems in Engineering 2010 639801
[6]	Hartley T, Lorenzo C and Qammer H 1995 IEEE Trans. Circ. Sys. I 42 485
[7]	Li R H and Chen W S 2013 Chin. Phys. B 22 040503
[8]	Petras I 2008 Chaos, Solitons and Fractals 38 140
[9]	Cafagna D and Grassi G 2008 Int. J. Bifur. Chaos 18 615
[10]	Luo C and Wang X 2013 Nonlinear Dyn. 71 241
[11]	Lu J G and Chen G 2006 Chaos, Solitons and Fractals 27 685
[12]	Cafagna D and Grassi G 2008 Int. J. Bifur. Chaos 18 1845
[13]	Cafagna D and Grassi G 2009 Int. J. Bifur. Chaos 19 339
[14]	Deng W and Lu J 2006 Chaos 16 043120
[15]	Cafagna D and Grassi G 2009 Int. J. Bifur. Chaos 19 3329
[16]	Guckenheimer J and Holmes P 1983 Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (New York: Springer-Verlag)
[17]	Silva C P 1993 IEEE Trans. Circ. Sys. I 40 675
[18]	Li H, Liao X F and Luo M 2012 Nonlinear Dyn. 68 137
[19]	Zhou P and Huang K 2014 Commun. Nonlinear Sci. Numer. Simul. 19 2005
[20]	Cafagna D and Grassi G 2013 Mathematical Problems in Engineering 2013 380436
[21]	Wang Z, Cang S, Ochola E O and Sun Y 2012 Nonlinear Dyn. 69 531
[22]	Caponetto R and Fazzino S 2013 Commun. Nonlinear Sci. Numer. Simul. 18 22
[23]	Gorenflo R and Mainardi F 1997 Fractal and Fractional Calculus in Continuum Mechanics (Wien: Springer)
[24]	Caputo M 1967 Geophys. J. Roy. Astronom. Soc. 13 529
[25]	Diethelm K, Ford N J and Freed A D 2002 Nonlinear Dyn. 29 3
[26]	Diethelm K, Ford N J and Freed A D 2004 Numer. Algorithms 36 31
[27]	Yang J and Qi D L 2010 Chin. Phys. B 19 020508
[28]	Hu G S 2009 Chin. Phys. Lett. 26 120501
[29]	Yang F Y, Leng J L and Li Q D 2014 Acta Phys. Sin. 63 080502 (in Chinese)
[30]	Kuznetsov Y A 2004 Elements of Applied Bifurcation Theory (New York: Springer-Verlag)
[31]	Leonov G, Kuznetsov N, Kuznetsova O, Seldedzhi S and Vagaitsev V 2011 Trans. Sys. Control 6 54
[32]	Leonov G, Kuznetsov N and Vagaitsev V 2011 Phys. Lett. A 375 2230
[33]	Leonov G, Kuznetsov N and Vagaitsev V 2012 Physica D 241 1482
[34]	Pham V T, Volos C, Jafari S, Wei Z and Wang X 2014 Int. J. Bifur. Chaos 24 1450073
[35]	Wang X and Chen G 2012 Commun. Nonlinear Sci. Numer. Simul. 17 1264
[36]	Wei Z 2011 Phys. Lett. A 376 102
[37]	Jafari S, Sprott J and Golpayegani S 2013 Phys. Lett. A 377 699
[38]	Jafari S and Sprott J 2013 Chaos, Solitons and Fractals 57 79
[39]	Zhang R X and Yang S P 2010 Chin. Phys. B 19 020510
[40]	Wang S, Yu Y G, Wang H and Rahmani A 2014 Chin. Phys. B 23 040502
[41]	Wang L M, Tang Y G, Chai Y Q and Wu F 2014 Chin. Phys. B 23 100501
[42]	Feng C F and Wang Y H 2011 Chin. Phys. Lett. 28 120504
[43]	Lin L X and Peng X F 2014 Acta Phys. Sin. 63 080504 (in Chinese)
[44]	Zhong D Z, Deng T and Zheng G L 2014 Acta Phys. Sin. 63 070504 (in Chinese)
[45]	Rakkiyappan R, Sivasamy R and Lakshmanan S 2014 Chin. Phys. B 23 060504
[46]	Xue W, Xu J K, Cang S J and Jia H Y 2014 Chin. Phys. B 23 060501
[47]	Jia B 2014 Chin. Phys. B 23 050510
[48]	Wang J A, Nie R X and Sun Z Y 2014 Chin. Phys. B 23 050509
[49]	Grassi G and Mascolo S 1997 IEEE Trans. Circ. Sys. I 44 1011
[50]	Grassi G and Mascolo S 1999 IEEE Trans. Circ. Sys. II 46 478
[51]	Cafagna D and Grassi G 2011 Int. J. Bifur. Chaos 21 955
[52]	Cafagna D and Grassi G 2012 Nonlinear Dyn. 70 1185

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Fractional-order systems without equilibria: The first example of hyperchaos and its application to synchronization

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