|
|
|
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.
|
Received: 26 January 2020
Revised: 12 March 2020
Published: 05 May 2020
|
|
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: phamvietthanh@tdtu.edu.vn
|
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
|
| No Suggested Reading articles found! |
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
| |
Shared |
|
|
|
|
| |
Discussed |
|
|
|
|
