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Shilnikov criteria believe that the emergence of chaos requires at least one unstable equilibrium, and an attractor is associated with the unstable equilibrium. However, some special chaotic systems have been proposed recently, each of which has one stable equilibrium, or no equilibrium at all, or has a linear equilibrium (infinite equilibrium). These special dynamical systems can present chaotic characteristics, and the attractors in these chaotic systems are called hidden attractors due to the fact that the attraction basins of chaotic systems do not intersect with small neighborhoods of any equilibrium points. Since they were first found and reported in 2011, the dynamical systems with hidden attractors have attracted much attention. Additionally, the fractional-order system, which can give a clearer physical meaning and a more accurate description of the physical phenomenon, has been broadly investigated in recent years. Motivated by these two considerations, in this paper, we propose a fractional-order chaotic system with hidden attractors, and the finite time synchronization of the fractional-order chaotic systems is also studied.Most of the researches mainly focus on dynamic analysis and control of integer-order chaotic systems with hidden attractors. In this paper, based on the Sprott E system, a fractional-order chaotic system is constructed by adding an appropriate constant term. The fractional-order chaotic system has only one stable equilibrium point, but it can generate various hidden attractors. Basic dynamical characteristics of the system are analyzed carefully through phase diagram, Poincare mapping and power spectrum, and the results show that the fractional-order system can present obvious chaotic characteristics. Based on bifurcation diagram of system order, it can be found that the fractional-order system can have period attractors, doubling period attractors, and chaotic attractors with various orders. Additionally, a finite time synchronization of the fractional-order chaotic system with hidden attractors is realized based on the finite time stable theorem, and the proposed controller is robust and can guarantee fast convergence. Finally, numerical simulation is carried out and the results verify the effectiveness of the proposed controller.The fractional-order chaotic system with hidden attractors has more complex and richer dynamic characteristics than integer-order chaotic systems, and chaotic range of parameters is more flexible, meanwhile the dynamics is more sensitive to system parameters. Therefore, the fractional-order chaotic system with hidden attractors can provide more key parameters and present better performance for practical applications, such as secure communication and image encryption, and it deserves to be further investigated.
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Keywords:
- hidden attractor /
- fractional order /
- chaos /
- finite-time synchronizing
[1] Lorenz E N 1963 J. Atmos. Sci. 20 130
[2] Rössler O E 1976 Phys. Lett. A 57 397
[3] Chen G R, Ueta T 1999 Int. J. Bifurcation Chaos 9 1465
[4] Liu C X, Liu T, Liu L, Liu K 2004 Chaos Solitions Fractals 22 1031
[5] L J H, Chen G R 2002 Int. J. Bifurcation Chaos 12 659
[6] Liu W B, Chen G R 2003 Int. J. Bifurcation Chaos 13 261
[7] Qi G Y, Chen G R, Du S Z, Chen Z Q, Yuan Z Z 2005 Physica A 352 295
[8] Bao B C, Liu Z, Xu J P 2009 J. Sys. Eng. Electron. 20 1179
[9] Shilnikov L P 1965 Sov. Math. Dokl. 6 163
[10] Leonov G A, Kuznetsov N V, Vagaitsev V I 2011 Phys. Lett. A 375 2230
[11] Molaie M, Jafari S, Sprott J C, Golpayegani S M R H 2013 Int. J. Bifurcation Chaos 23 1350188
[12] Wang X, Chen G R 2012 Commu. Nonlinear Sci. Numer. Simul. 17 1264
[13] Jafari S, Sprott J C, Golpayegani S M R H 2013 Phys. Lett. A 377 699
[14] Wei Z 2011 Phys. Lett. A 376 102
[15] Jafari S, Sprott J C 2013 Chaos Solitions Fractals 57 79
[16] Li Q D, Zeng H Z, Yang X S 2014 Nonlinear Dyn. 77 255
[17] Leonov G A, Kuznetsov N V, Mokaev T N 2015 Commu. Nonlinear Sci. Numer. Simul. 28 166
[18] Zhang Y A, Yu M Z, Wu H L 2016 Acta Electron. Sin. 44 607 (in Chinese) [张友安, 余名哲, 吴华丽 2016 电子学报 44 607]
[19] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[20] Li C G, Liao X F, Yu J B 2003 Phys. Rev. E 68 067203
[21] Jia H Y, Chen Z Q, Yuan Z Z 2010 Chin. Phys. B 19 020507
[22] Zhang L, Yan Y 2014 Nonlinear Dyn. 76 1761
[23] Wang D F, Zhang J Y, Wang X Y 2013 Chin. Phys. B 22 100504
[24] Zhao L D, Hu J B, Bao Z H, Zhang G A, Xu C, Zhang S B 2011 Acta Phys. Sin. 60 100507 (in Chinese) [赵灵冬, 胡建兵, 包志华, 章国安, 徐晨, 张士兵 2011 60 100507]
[25] Podlubny I 1999 Fractional Differential Equations (San Diego: Academic Press) p18
[26] Sprott J C 1994 Phys. Rev. E 50 647
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[1] Lorenz E N 1963 J. Atmos. Sci. 20 130
[2] Rössler O E 1976 Phys. Lett. A 57 397
[3] Chen G R, Ueta T 1999 Int. J. Bifurcation Chaos 9 1465
[4] Liu C X, Liu T, Liu L, Liu K 2004 Chaos Solitions Fractals 22 1031
[5] L J H, Chen G R 2002 Int. J. Bifurcation Chaos 12 659
[6] Liu W B, Chen G R 2003 Int. J. Bifurcation Chaos 13 261
[7] Qi G Y, Chen G R, Du S Z, Chen Z Q, Yuan Z Z 2005 Physica A 352 295
[8] Bao B C, Liu Z, Xu J P 2009 J. Sys. Eng. Electron. 20 1179
[9] Shilnikov L P 1965 Sov. Math. Dokl. 6 163
[10] Leonov G A, Kuznetsov N V, Vagaitsev V I 2011 Phys. Lett. A 375 2230
[11] Molaie M, Jafari S, Sprott J C, Golpayegani S M R H 2013 Int. J. Bifurcation Chaos 23 1350188
[12] Wang X, Chen G R 2012 Commu. Nonlinear Sci. Numer. Simul. 17 1264
[13] Jafari S, Sprott J C, Golpayegani S M R H 2013 Phys. Lett. A 377 699
[14] Wei Z 2011 Phys. Lett. A 376 102
[15] Jafari S, Sprott J C 2013 Chaos Solitions Fractals 57 79
[16] Li Q D, Zeng H Z, Yang X S 2014 Nonlinear Dyn. 77 255
[17] Leonov G A, Kuznetsov N V, Mokaev T N 2015 Commu. Nonlinear Sci. Numer. Simul. 28 166
[18] Zhang Y A, Yu M Z, Wu H L 2016 Acta Electron. Sin. 44 607 (in Chinese) [张友安, 余名哲, 吴华丽 2016 电子学报 44 607]
[19] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[20] Li C G, Liao X F, Yu J B 2003 Phys. Rev. E 68 067203
[21] Jia H Y, Chen Z Q, Yuan Z Z 2010 Chin. Phys. B 19 020507
[22] Zhang L, Yan Y 2014 Nonlinear Dyn. 76 1761
[23] Wang D F, Zhang J Y, Wang X Y 2013 Chin. Phys. B 22 100504
[24] Zhao L D, Hu J B, Bao Z H, Zhang G A, Xu C, Zhang S B 2011 Acta Phys. Sin. 60 100507 (in Chinese) [赵灵冬, 胡建兵, 包志华, 章国安, 徐晨, 张士兵 2011 60 100507]
[25] Podlubny I 1999 Fractional Differential Equations (San Diego: Academic Press) p18
[26] Sprott J C 1994 Phys. Rev. E 50 647
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