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Chin. Phys. B, 2020, Vol. 29(5): 050504    DOI: 10.1088/1674-1056/ab820d
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Hidden attractors in a new fractional-order discrete system: Chaos, complexity, entropy, and control

Adel Ouannas1,2, Amina Aicha Khennaoui3, Shaher Momani2,4, Viet-Thanh Pham5, Reyad El-Khazali6
1 Laboratory of Mathematics, Informatics and Systems(LAMIS), University of Laarbi Tebessi, Tebessa, 12002, Algeria;
2 College of Humanities and Sciences, Ajman University, Ajman, UAE;
3 Laboratory of Dynamical Systems and Control, University of Larbi Ben M'hidi, Oum El Bouaghi, Algeria;
4 Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan;
5 Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam;
6 ECCE Department, Khalifa University, Abu-Dhabi 127788, United Arab Emirates
Abstract  This paper studies the dynamics of a new fractional-order discrete system based on the Caputo-like difference operator. This is the first study to explore a three-dimensional fractional-order discrete chaotic system without equilibrium. Through phase portrait, bifurcation diagrams, and largest Lyapunov exponents, it is shown that the proposed fractional-order discrete system exhibits a range of different dynamical behaviors. Also, different tests are used to confirm the existence of chaos, such as 0-1 test and C0 complexity. In addition, the quantification of the level of chaos in the new fractional-order discrete system is measured by the approximate entropy technique. Furthermore, based on the fractional linearization method, a one-dimensional controller to stabilize the new system is proposed. Numerical results are presented to validate the findings of the paper.
Keywords:  discrete chaos      discrete fractional calculus      hidden attractor  
Received:  26 January 2020      Revised:  12 March 2020      Accepted manuscript online: 
PACS:  05.45.-a (Nonlinear dynamics and chaos)  
  05.45.Ac (Low-dimensional chaos)  
  05.45.Gg (Control of chaos, applications of chaos)  
  05.30.Pr (Fractional statistics systems)  
Fund: The author Adel Ouannas was supported by the Directorate General for Scientific Research and Technological Development of Algeria. The author Shaher Momani was supported by Ajman University in UAE.
Corresponding Authors:  Viet-Thanh Pham     E-mail:

Cite this article: 

Adel Ouannas, Amina Aicha Khennaoui, Shaher Momani, Viet-Thanh Pham, Reyad El-Khazali Hidden attractors in a new fractional-order discrete system: Chaos, complexity, entropy, and control 2020 Chin. Phys. B 29 050504

[1] Hénon M 1976 Comms. Math. Phys. 50 69
[2] Lozi R 1978 J. Phys. 39 9
[3] Hitzl D L and Zele F 1985 Phys. D Nonlinear Phenom. 14 305
[4] Baier G and Sahle S 1995 Phys. Rev. E 51 R2712
[5] Stefanski K 1998 Chaos Solitons Fract. 9 83
[6] Alamodi A A O, Sun K, Ai W, Chen C and Peng D 2019 Chin. Phys. B 28 020503
[7] Wang X Y, Zhang J J, Zhang F C and Cao G H 2019 Chin. Phys. B 28 040504
[8] Han F, Wang Z J, Fan H and Gong T 2015 Chin. Phys. Lett. 32 040502
[9] Ouannas A and Odibat Z 2015 Nonlinear Dyn. 81 765
[10] Ouannas A and Grassi G 2016 Chin. Phys. B 25 090503
[11] Ouannas A, Obidat Z, Alsaedi A and Ahmad B 2017 Appl. Math. Model. 45 636
[12] Jafari S, Sprott J C and Nazarimehr F 2015 Eur. Phys. J. Spec. Top. 224 1469
[13] Leonov G A and Kuznetsov N V 2013 Int. J. Bifurc. Chaos 23 1330002
[14] Danca M F, Kuznetsov N V and Chen G 2017 Nonlinear Dyn. 88 791
[15] Kuznetsov N V, Leonov G, Yuldashev M and Yuldashev R 2017 Commun. Nonlinear Sci. Numer. Simul. 51 39
[16] Jiang H, Liu Y, Wei Z and Zhang L 2016 Int. J. Bifurc. Chaos 26 1650206
[17] Wang C and Ding Q 2018 Entropy 20 322
[18] Jiang H, Liu Y, Wei Z and Zhang L 2016 Nonlinear Dyn. 85 2719
[19] Chen L, Wu R, He Y and Yin L 2015 Appl. Math. Comput. 257 274
[20] Chen L, Pan W, Wu R, Tenreiro Machado J A and Lopes A M 2016 Chaos 26 084303
[21] Chen L, Pan W, Wang K, Wu R, Machado J T and Lopes A M 2017 Chaos Solitons Fract. 105 244
[22] Atici F M and Eloe P W 2009 Electron. J. Qual. Theory Differ. Equ. Spec. Ed. I 3 1
[23] Anastassiou G A 2010 Math. Comput. Model 52 556
[24] Wu G C and Baleanu D 2015 Nonlinear Dyn. 80 1697
[25] Ouannas A, Khennaoui A A, Grassi G and Bendoukha S 2019 J. Comput. Appl. Math. 358 293
[26] Ouannas A, Khennaoui A A, Grassi G and Bendoukha S 2019 Int. J. Bifurc. Chaos 29 1950078
[27] Ouannas A, Khennaoui A A, Odibat Z, Pham V T and Grassi G 2019 Chaos Solitons Fract. 123 108
[28] Khennaoui A A, Ouannas A, Bendoukha S, Grassi G, Lozi R P and Pham V T 2019 Chaos Solitons Fract. 119 150
[29] Jouini L, Ouannas A, Khennaoui A A, Wang X, Grassi G and Pham V T 2019 Adv. Differ. Equ. 2019 122
[30] Khennaoui A A, Ouannas A, Bendoukha S, Wang X and Pham V T 2018 Entropy 20 530
[31] Cermak J, Gyori I and Nechvatal L 2015 Fract. Calc. Appl. Anal. 18 651
[32] Abdeljawad T 2011 Comput. Math. Appl. 62 1602
[33] Wu G C and Baleanu D 2015 Commun. Nonlinear. Sci. Numer. Simulat 22 95
[34] Gottwald G A and Melbourne I 2009 SIAM J. Appl. Dyn. Syst. 8 129
[35] He S, Sun K and Wang H 2016 Math. Method. Appl. Sci. 39 2965
[36] Pincus S M and Keefe D L 1992 Am. J. Physiol. Endocrinol. Metab. 262 E741
[37] Tang J and Liu X Q 2019 Acta Phys. Sin. 68 149801 (in Chinese)
[38] He J H 2020 A fractal variational theory for one-dimensional compressible flow in a microgravity space Fractals
[39] He C H, Shen Y, Ji F Y and He J H 2020 Fractals 28 2050011
[40] He J H 2019 J. Electroanal. Chem. 854 113565
[41] He J H and Ji F Y 2019 Therm. Sci. 23 2131
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