Different linear control laws for fractional chaotic maps using Lyapunov functional

• Autorzy: Almatroud A. Othman , Ouannas Adel , Grassi Giuseppe , Batiha Iqbal M. , Gasri Ahlem , Al-sawalha M. Mossa

Identyfikatory

DOI

10.24425/acs.2021.139729

• Języki publikacji: EN

Abstrakty

EN

Dynamics and control of discrete chaotic systems of fractional-order have received considerable attention over the last few years. So far, nonlinear control laws have been mainly used for stabilizing at zero the chaotic dynamics of fractional maps. This article provides a further contribution to such research field by presenting simple linear control laws for stabilizing three fractional chaotic maps in regard to their dynamics. Specifically, a one-dimensional linear control law and a scalar control law are proposed for stabilizing at the origin the chaotic dynamics of the Zeraoulia-Sprott rational map and the Ikeda map, respectively. Additionally, a two-dimensional linear control law is developed to stabilize the chaotic fractional flow map. All the results have been achieved by exploiting new theorems based on the Lyapunov method as well as on the properties of the Caputo h-difference operator. The relevant simulation findings are implemented to confirm the validity of the established linear control scheme.

Słowa kluczowe

EN

discrete fractional calculus; chaotic map; linear control; Lyapunov method

• Wydawca: Polish Academy of Sciences, Committee of Automatic Control and Robotics
• Czasopismo: Archives of Control Sciences
• Rocznik: 2021
• Tom: Vol. 31, No. 4
• Strony: 765--780
• Opis fizyczny: Bibliogr. 29 poz., rys., wzory

Twórcy

autor

Almatroud A. Othman

• Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 81451, Saudi Arabia

autor

Ouannas Adel

• Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria

autor

Grassi Giuseppe

• Dipartimento Ingegneria Innovazione, Universita del Salento, 73100 Lecce, Italy

autor

Batiha Iqbal M.

• Department of Mathematics, Faculty of Science and Information Technology, Irbid National University, Irbid
• Jordan and Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE

autor

Gasri Ahlem

• Department of Mathematics, University of Larbi Tebessi, Tebessa 12002, Algeria

autor

Al-sawalha M. Mossa

• Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 81451, Saudi Arabia

Bibliografia

• [1] C. Goodrich and A.C. Peterson: Discrete Fractional Calculus. Springer: Berlin, Germany, 2015, ISBN 978-3-319-79809-7.
• [2] P. Ostalczyk: Discrete Fractional Calculus: Applications in Control and Image Processing. World Scientific, 2016.
• [3] K. Oprzedkiewicz and K. Dziedzic: Fractional discrete-continuous model of heat transfer process. Archives of Control Sciences, 31(2), (2021), 287- 306, DOI: 10.24425/acs.2021.137419.
• [4] T. Kaczorek and A. Ruszewski: Global stability of discrete-time nonlinear systems with descriptor standard and fractional positive linear parts and scalar feedbacks. Archives of Control Sciences, 30(4), (2020), 667-681, DOI: 10.24425/acs.2020.135846.
• [5] J.B. Diaz and T.J. Olser: Differences of fractional order. Mathematics of Computation, 28 (1974), 185-202, DOI: 10.1090/S0025-5718-1974-0346352-5.
• [6] F.M. Atici and P.W. Eloe: A transform method in discrete fractional calculus. International Journal of Difference Equations, 2 (2007), 165-176.
• [7] F.M. Atici and P.W. Eloe: Discrete fractional calculus with the nabla operator. Electronic Journal of Qualitative Theory of Differential Equations, Spec. Ed. I, 3 , (2009), 1-12.
• [8] G. Anastassiou: Principles of delta fractional calculus on time scales and inequalities. Mathematical and Computer Modelling, 52(3-4), (2010), 556- 566, DOI: 10.1016/j.mcm.2010.03.055.
• [9] T. Abdeljawad: On Riemann and Caputo fractional differences. Computers and Mathematics with Applications, 62(3), (2011), 1602-1611, DOI: 10.1016/j.camwa.2011.03.036.
• [10] M. Edelman, E.E.N. Macau, and M.A.F. Sanjun (Eds.): Chaotic, Fractional, and Complex Dynamics: New Insights and Perspectives. Springer International Publishing, 2018.
• [11] G.C. Wu, D. Baleanu, and S.D. Zeng: Discrete chaos in fractional sine and standard maps. Physics Letters A, 378(5-6), (2014), 484-487, DOI: 10.1016/j.physleta.2013.12.010.
• [12] G.C. Wu and D. Baleanu: Discrete fractional logistic map and its chaos. Nonlinear Dynamics, 75(1-2), (2014), 283-287, DOI: 10.1007/s11071-013-1065-7.
• [13] T. Hu: Discrete chaos in fractional Henon map. Applied Mathematics, 5(15), (2014), 2243-2248, DOI: 10.4236/am.2014.515218.
• [14] G.C. Wu and D. Baleanu: Discrete chaos in fractional delayed logistic maps. Nonlinear Dynamics, 80 (2015), 1697-1703, DOI: 10.1007/s11071-014-1250-3.
• [15] M.K. Shukla and B.B. Sharma: Investigation of chaos in fractional order generalized hyperchaotic Henon map. International Journal of Electronics and Communications, 78 (2017), 265-273, DOI: 10.1016/j.aeue.2017.05.009.
• [16] A. Ouannas, A.A. Khennaoui, S. Bendoukha, and G. Grassi: On the dynamics and control of a fractional form of the discrete double scroll. International Journal of Bifurcation and Chaos, 29(6), (2019), DOI: 10.1142/S0218127419500780.
• [17] L. Jouini, A. Ouannas, A.A. Khennaoui, X. Wang, G. Grassi, and V.T. Pham: The fractional form of a new three-dimensional generalized Henon map. Advances in Difference Equations, 122 (2019), DOI: 10.1186/s13662-019-2064-x.
• [18] F. Hadjabi, A. Ouannas,N. Shawagfeh, A.A. Khennaoui, and G. Grassi: On two-dimensional fractional chaotic maps with symmetries. Symmetry, 12(5), (2020), DOI: 10.3390/sym12050756.
• [19] D. Baleanu, G.C. Wu, Y.R. Bai, and F.L. Chen: Stability analysis of Caputo-like discrete fractional systems. Communications in Nonlinear Science and Numerical Simulation, 48 (2017), 520-530, DOI: 10.1016/j.cnsns.2017.01.002.
• [20] A-A. Khennaoui, A. Ouannas, S. Bendoukha, G. Grassi, X. Wang, V-T. Pham, and F.E. Alsaadi: Chaos, control, and synchronization in some fractional-order difference equations. Advances in Difference Equations, 412 (2019), DOI: 10.1186/s13662-019-2343-6.
• [21] A. Ouannas, A.A. Khennaoui, G. Grassi, and S. Bendoukha: On chaos in the fractional-order Grassi-Miller map and its control. Journal of Computational and Applied Mathematics, 358(2019), 293-305, DOI: 10.1016/j.cam.2019.03.031.
• [22] A. Ouannas, A.A. Khennaoui, S. Momani, G. Grassi and V.T. Pham: Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization. AIP Advances, 10 (2020), DOI: 10.1063/5.0004884.
• [23] A. Ouannas, A-A. Khennaoui, S. Momani, G. Grassi, V-T. Pham, R. El- Khazali, and D. Vo Hoang: A quadratic fractional map without equilibria: Bifurcation, 0-1 test, complexity, entropy, and control. Electronics, 9 (2020), DOI: 10.3390/electronics9050748.
• [24] A. Ouannas, A-A. Khennaoui, S. Bendoukha, Z.Wang, and V-T. Pham: The dynamics and control of the fractional forms of some rational chaotic maps. Journal of Systems Science and Complexity, 33 (2020), 584-603, DOI: 10.1007/s11424-020-8326-6.
• [25] A-A. Khennaoui, A. Ouannas, S. Bendoukha, G. Grassi, R.P. Lozi, and V-T. Pham: On fractional-order discrete-time systems: Chaos, stabilization and synchronization. Chaos, Solitons and Fractals, 119(C), (2019), 150-162, DOI: 10.1016/j.chaos.2018.12.019.
• [26] A. Ouannas, A-A. Khennaoui, Z. Odibat, V-T. Pham, and G. Grassi: On the dynamics, control and synchronization of fractional-order Ikeda map. Chaos, Solitons and Fractals, 123(C), (2015), 108-115, DOI: 10.1016/j.chaos.2019.04.002.
• [27] A. Ouannas, F. Mesdoui, S. Momani, I. Batiha, and G. Grassi: Synchronization of FitzHugh-Nagumo reaction-diffusion systems via one-dimensional linear control law. Archives of Control Sciences, 31(2), 2021, 333-345, DOI: 10.24425/acs.2021.137421.
• [28] Y. Li, C. Sun, H. Ling, A. Lu, and Y. Liu: Oligopolies price game in fractional order system. Chaos, Solitons and Fractals, 132(C), (2020), DOI: 10.1016/j.chaos.2019.109583.
• [29] D. Mozyrska and E. Girejko: Overview of fractional h-difference operators. In Advances in harmonic analysis and operator theory. Birkhäuser, Basel, 2013, 253-268.

Uwagi

1. This work has been supported by Scientific Research Deanship at University of Ha’il - Saudi Arabia though Project Number RG-20115.

2. Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).