The dynamic and discrete systems of variable fractional order in the sense of the Lozi structure map

Nadia M. G. Al-Saidi; Hayder Natiq; Dumitru Baleanu; Rabha W. Ibrahim; Nadia M. G. Al-Saidi; Hayder Natiq; Dumitru Baleanu; Rabha W. Ibrahim

doi:10.3934/math.2023035

AIMS Mathematics

2023, Volume 8, Issue 1: 733-751. doi: 10.3934/math.2023035

Previous Article Next Article

Research article

The dynamic and discrete systems of variable fractional order in the sense of the Lozi structure map

1.
Department of Applied Sciences, University of Technology, Baghdad 10066, Iraq
2.
Information Technology College, Imam Ja'afar Al-Sadiq University, Baghdad 10001, Iraq
3.
Department of Mathematics, Cankaya University, Balgat 06530, Ankara, Turkey
4.
Institute of Space Sciences, R76900 Magurele-Bucharest, Romania
5.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
6.
Mathematics Research Center, Department of Mathematics, Near East University, Near East Boulevard, Nicosia/Mersin 10, Turkey

Received: 15 August 2022 Revised: 08 September 2022 Accepted: 08 September 2022 Published: 12 October 2022
MSC : 26A33

The variable fractional Lozi map (VFLM) and the variable fractional flow map are two separate systems that we propose in this inquiry. We study several key dynamics of these maps. We also investigate the sufficient and necessary requirements for the stability and asymptotic stability of the variable fractional dynamic systems. As a result, we provide VFLM with the necessary criteria to produce stable and asymptotically stable zero solutions. Furthermore, we propose a combination of these maps in control rules intended to stabilize the system. In this analysis, we take the 1D- and 2D-controller laws as givens.
- fractional calculus,
- discrete system,
- variable fractional differential operators,
- Lozi system,
- fractional differential equation
Citation: Nadia M. G. Al-Saidi, Hayder Natiq, Dumitru Baleanu, Rabha W. Ibrahim. The dynamic and discrete systems of variable fractional order in the sense of the Lozi structure map[J]. AIMS Mathematics, 2023, 8(1): 733-751. doi: 10.3934/math.2023035

Related Papers:

Abstract

The variable fractional Lozi map (VFLM) and the variable fractional flow map are two separate systems that we propose in this inquiry. We study several key dynamics of these maps. We also investigate the sufficient and necessary requirements for the stability and asymptotic stability of the variable fractional dynamic systems. As a result, we provide VFLM with the necessary criteria to produce stable and asymptotically stable zero solutions. Furthermore, we propose a combination of these maps in control rules intended to stabilize the system. In this analysis, we take the 1D- and 2D-controller laws as givens.

References

[1]	T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602–1611. https://doi.org/10.1016/j.camwa.2011.03.036 doi: 10.1016/j.camwa.2011.03.036
[2]	N. M. G. Al-Saidi, S. S. Al-Bundi, N. J. Al-Jawari, A hybrid of fractal image coding and fractal dimension for an efficient retrieval method, Comput. Math. Appl., 37 (2018), 996–1011. https://doi.org/10.1007/s40314-016-0378-9 doi: 10.1007/s40314-016-0378-9
[3]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[4]	Z. P. Bazant, S. Baweja, Creep and shrinkage prediction model for analysis and design of concrete structures: Model B3, ACI Spec. Publ., 194 (2000), 1–84.
[5]	A. Beltempo, M. Zingales, O. S. Bursi, L. Deseri, A fractional order model for agingmaterials: An application to concrete, Int. J. Solids Struct., 138 (2018), 13–23. https://doi.org/10.1016/j.ijsolstr.2017.12.024 doi: 10.1016/j.ijsolstr.2017.12.024
[6]	A. Beltempo, O. S. Bursi, C. Cappello, D. Zonta, M. Zingales, A viscoelastic model for the long-term deflection of segmental prestressed box girders, Comput.-Aided Civ. Infrastruct. Eng., 33 (2018), 64–78. https://doi.org/10.1111/mice.12311 doi: 10.1111/mice.12311
[7]	A. Beltempo, A. Bonelli, O. S. Bursi, M. Zingales, A numerical integration approach for fractional-order viscoelastic analysis of hereditary-aging structures, Int. J. Numer. Methods Eng., 121 (2020), 1120–1146. https://doi.org/10.1002/nme.6259 doi: 10.1002/nme.6259
[8]	S. Bendoukha, Stabilization and synchronization of discrete-time fractional chaotic systems with non-identical dimensions, Acta Math. Appl. Sin. Engl. Ser., 37 (2021), 523–538. https://doi.org/10.1007/s10255-021-1029-5 doi: 10.1007/s10255-021-1029-5
[9]	J. Borcea, P. Brändén, T. M. Liggett, Negative dependence and the geometry of polynomials, J. Amer. Math. Soc., 22 (2009), 521–567. https://doi.org/10.1090/S0894-0347-08-00618-8 doi: 10.1090/S0894-0347-08-00618-8
[10]	J. Čermák, I. Győri, L. Nechvátal, On explicit stability conditions for a linear fractional difference system, FCAA, 18 (2015), 651–672. https://doi.org/10.1515/fca-2015-0040 doi: 10.1515/fca-2015-0040
[11]	N. Colinas-Armijo, M. Di Paola, A. Di Matteo, Fractional viscoelastic behaviour under stochastic temperature process, Probab. Eng. Mech., 54 (2018), 37–43. https://doi.org/10.1016/j.probengmech.2017.06.005 doi: 10.1016/j.probengmech.2017.06.005
[12]	A. K. Farhan, N. M. G. Al-Saidi, A. T. Maolood, F. Nazarimehr, I. Hussain, Entropy analysis and image encryption application based on a new chaotic system crossing a cylinder, Entropy, 21 (2019), 958. https://doi.org/10.3390/e21100958 doi: 10.3390/e21100958
[13]	X. Han, J. Mou, H. Jahanshahi, Y. Cao, F. Bu, A new set of hyperchaotic maps based on modulation and coupling, Eur. Phys. J. Plus, 137 (2022), 523. https://doi.org/10.1140/epjp/s13360-022-02734-3 doi: 10.1140/epjp/s13360-022-02734-3
[14]	R. W. Ibrahim, D. Baleanu, Global stability of local fractional Henon-Lozi map using fixed point theory, AIMS Math., 7 (2022), 11399–11416. https://doi.org/10.3934/math.2022636 doi: 10.3934/math.2022636
[15]	A. Khennaoui, A. Ouannas, S. Bendoukha, G. Grassi, R. P. Lozi, V. Pham, On fractional-order discrete-time systems: Chaos, stabilization and synchronization, Chaos Solitons Fract., 119 (2019), 150–162. https://doi.org/10.1016/j.chaos.2018.12.019 doi: 10.1016/j.chaos.2018.12.019
[16]	J. F. Li, H. Jahanshahi, S. Kacar, Y. M. Chu, J. F. Gomez-Aguilar, N. D. Alotaibi, et al., On the variable-order fractional memristor oscillator: Data security applications and synchronization using a type-2 fuzzy disturbance observer-based robust control, Chaos Solitons Fract., 145 (2021), 110681. https://doi.org/10.1016/j.chaos.2021.110681 doi: 10.1016/j.chaos.2021.110681
[17]	C. F. Lorenzo, T. T. Hartley Variable order and distributed order fractional operators, Nonlinear Dyn., 29 (2002), 57–98. https://doi.org/10.1023/A:1016586905654 doi: 10.1023/A:1016586905654
[18]	R. Lozi, Un attracteur étrange (?) du type attracteur de Hénon, J. Phys. Colloquia, 39 (1978), C5-9. https://doi.org/10.1051/jphyscol:1978505 doi: 10.1051/jphyscol:1978505
[19]	R. Lyons, A note on tail triviality for determinantal point processes, Electron. Commun. Probab., 23 (2018), 1–3. https://doi.org/10.1214/18-ECP175 doi: 10.1214/18-ECP175
[20]	S. Nemati, P. M. Lima, D. F. M. Torres, Numerical solution of variable-order fractional differential equations using Bernoulli polynomials, Fractal Fract., 5 (2021), 219. https://doi.org/10.3390/fractalfract5040219 doi: 10.3390/fractalfract5040219
[21]	A. Ouannas, A. Khennaoui, I. M. Batiha, V. Pham, Stabilization of different dimensional fractional chaotic maps, Fractional-Order Des., 2022,123–155. https://doi.org/10.1016/B978-0-32-390090-4.00010-X
[22]	B. Ross, S. G. Samko, Fractional integration operator of variable order in the Holder space $H\lambda(x)$, Int. J. Math. Math. Sci., 18 (1995), 713098. https://doi.org/10.1155/S0161171295001001 doi: 10.1155/S0161171295001001
[23]	S. H. Salih, N. M. G. Al-Saidi, 3D-Chaotic discrete system of vector borne diseases using environment factor with deep analysis, AIMS Math., 7 (2022), 3972–3987. https://doi.org/10.3934/math.2022219 doi: 10.3934/math.2022219
[24]	S. G. Samko, B. Ross, Integration and differentiation to a variable fractional order, Integr. Transf. Spec. F., 1 (1993), 277–300. https://doi.org/10.1080/10652469308819027 doi: 10.1080/10652469308819027
[25]	M. Uddin, I. U. Din, A numerical method for solving variable-order solute transport models, Compu. Appl. Math., 39 (2020), 320. https://doi.org/10.1007/s40314-020-01355-9 doi: 10.1007/s40314-020-01355-9
[26]	S. Umarov, S. Steinberg, Variable order differential equations with piecewise constant order-function and diffusion with changing modes, ZAA, 28 (2009), 431–450. https://doi.org/10.4171/ZAA/1392 doi: 10.4171/ZAA/1392
[27]	D. Valério, J. Sá da Costa, Variable-order fractional derivatives and their numerical approximations, Signal Process., 91 (2011), 470–483. https://doi.org/10.1016/j.sigpro.2010.04.006 doi: 10.1016/j.sigpro.2010.04.006
[28]	L. Wei, W. Li, Local discontinuous Galerkin approximations to variable-order time-fractional diffusion model based on the Caputo-Fabrizio fractional derivative, Math. Comput. Simul., 188 (2021), 280–290. https://doi.org/10.1016/j.matcom.2021.04.001 doi: 10.1016/j.matcom.2021.04.001

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)