On fractional discrete financial system: Bifurcation, chaos, and control

doi:10.1088/1674-1056/ad5d96

中国物理B ›› 2024, Vol. 33 ›› Issue (10): 100201-100201.doi: 10.1088/1674-1056/ad5d96

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On fractional discrete financial system: Bifurcation, chaos, and control

Louiza Diabi¹, Adel Ouannas², Amel Hioual^3,†, Shaher Momani^4,5, and Abderrahmane Abbes⁶

1 Laboratory of Dynamical Systems and Control, University of Larbi Ben M'hidi, Oum El Bouaghi 04000, Algeria;
2 Department of Mathematics and Computer Science, University of Larbi Ben M'hidi, Oum El Bouaghi 04000, Algeria;
3 Department of Mathematics and Computer Science, University of Larbi Ben M'hidi, Oum El Bouaghi 04000, Algeria;
4 Nonlinear Dynamics Research Center, Ajman University, Ajman 346, United Arab Emirates;
5 Department of Mathematics, The University of Jordan, Amman 11942, Jordan;
6 Laboratory of Mathematics, Dynamics and Modelization, University Badji Mokhtar, Annaba 23000, Algeria

收稿日期:2024-05-22 修回日期:2024-06-19 接受日期:2024-07-02 出版日期:2024-10-03 发布日期:2024-09-13
通讯作者: Amel Hioual E-mail:amel.hioual@univ-oeb.dz

On fractional discrete financial system: Bifurcation, chaos, and control

Louiza Diabi¹, Adel Ouannas², Amel Hioual^3,†, Shaher Momani^4,5, and Abderrahmane Abbes⁶

1 Laboratory of Dynamical Systems and Control, University of Larbi Ben M'hidi, Oum El Bouaghi 04000, Algeria;
2 Department of Mathematics and Computer Science, University of Larbi Ben M'hidi, Oum El Bouaghi 04000, Algeria;
3 Department of Mathematics and Computer Science, University of Larbi Ben M'hidi, Oum El Bouaghi 04000, Algeria;
4 Nonlinear Dynamics Research Center, Ajman University, Ajman 346, United Arab Emirates;
5 Department of Mathematics, The University of Jordan, Amman 11942, Jordan;
6 Laboratory of Mathematics, Dynamics and Modelization, University Badji Mokhtar, Annaba 23000, Algeria

Received:2024-05-22 Revised:2024-06-19 Accepted:2024-07-02 Online:2024-10-03 Published:2024-09-13
Contact: Amel Hioual E-mail:amel.hioual@univ-oeb.dz

摘要/Abstract

摘要： The dynamic analysis of financial systems is a developing field that combines mathematics and economics to understand and explain fluctuations in financial markets. This paper introduces a new three-dimensional (3D) fractional financial map and we dissect its nonlinear dynamics system under commensurate and incommensurate orders. As such, we evaluate when the equilibrium points are stable or unstable at various fractional orders. We use many numerical methods, phase plots in 2D and 3D projections, bifurcation diagrams and the maximum Lyapunov exponent. These techniques reveal that financial maps exhibit chaotic attractor behavior. This study is grounded on the Caputo-like discrete operator, which is specifically influenced by the variance of the commensurate and incommensurate orders. Furthermore, we confirm the presence and measure the complexity of chaos in financial maps by the 0-1 test and the approximate entropy algorithm. Additionally, we offer nonlinear-type controllers to stabilize the fractional financial map. The numerical results of this study are obtained using MATLAB.

关键词: financial model, stability, chaos, commensurate and incommensurate orders, complexity

Abstract: The dynamic analysis of financial systems is a developing field that combines mathematics and economics to understand and explain fluctuations in financial markets. This paper introduces a new three-dimensional (3D) fractional financial map and we dissect its nonlinear dynamics system under commensurate and incommensurate orders. As such, we evaluate when the equilibrium points are stable or unstable at various fractional orders. We use many numerical methods, phase plots in 2D and 3D projections, bifurcation diagrams and the maximum Lyapunov exponent. These techniques reveal that financial maps exhibit chaotic attractor behavior. This study is grounded on the Caputo-like discrete operator, which is specifically influenced by the variance of the commensurate and incommensurate orders. Furthermore, we confirm the presence and measure the complexity of chaos in financial maps by the 0-1 test and the approximate entropy algorithm. Additionally, we offer nonlinear-type controllers to stabilize the fractional financial map. The numerical results of this study are obtained using MATLAB.

Key words: financial model, stability, chaos, commensurate and incommensurate orders, complexity

中图分类号: (Control theory)

02.30.Yy

02.30.Oz (Bifurcation theory)

Louiza Diabi, Adel Ouannas, Amel Hioual, Shaher Momani, and Abderrahmane Abbes. On fractional discrete financial system: Bifurcation, chaos, and control[J]. 中国物理B, 2024, 33(10): 100201-100201.

Louiza Diabi, Adel Ouannas, Amel Hioual, Shaher Momani, and Abderrahmane Abbes. On fractional discrete financial system: Bifurcation, chaos, and control[J]. Chin. Phys. B, 2024, 33(10): 100201-100201.

参考文献

[1] Diouf M and Sene N 2020 Complexity 1 9845031
[2] Jahanshahi H, Yousefpour A, Wei Z, Alcaraz R and Bekiros S 2019 Chaos, Solitons. Fractals 126 66
[3] Hu Y and Hu G 2023 Mathematics 11 2994
[4] Akhmet M, Akhmetova Z and Fen M O 2014 Journal of Economic Behavior. Organization 106 95
[5] Desogus M and Venturi B 2023 Journal of Risk and Financial Management 16 171
[6] Zhou S S, Jahanshahi H, Din Q, Bekiros S, Alcaraz R, Alassafi M O and Chu Y M 2021 Chaos. Solitons. Fractals 142 110378
[7] Chen Q, Sabir Z, Raja M A Z, Gao W and Baskonus H M 2023 Alexandria Engineering Journal 65 761
[8] Wu Y, Wu W, Xing C, Zhou M and Li Z 2016 arXiv:1612.01627
[9] Liu J, Wang Z, Shu M, Zhang F, Leng S and Sun X 2019 Complexity 2019 7242791
[10] Tejado I, Valério D and Valério N 2014 Proceedings of the ICFDA’14th International Conference on Fractional Differentiation and Its Applications, June, 2014, The Portuguese case. pp. 1-6
[11] Alzaid S S, Kumar A, Kumar S and Alkahtani B S T 2023 Fractals 31 2340056
[12] Chu Y M, Bekiros S, Zambrano-Serrano E, Orozco-López O, Lahmiri S, Jahanshahi H and Aly A A 2021 Chaos, Solitons. Fractals 145 110776
[13] Pourrahimian E, Salhab D, Shehab L, Hamzeh F and AbouRizk S 2024 Application of Chaos Theory in Project Management, Construction Research Congress, 2024, March 25-27, 2024, Arlington, Virginia, USA, p. 601
[14] Batiha I M, Djenina N, Ouannas A, Oussaeif T E, Aoua L B and Momani S 2023 Control of chaos in incommensurate fractional order discrete system, 2023 International Conference on Fractional Differentiation and Its Applications (ICFDA), March 13-15, 2023, Paris, France, p. 1
[15] Ouannas A, Khennaoui A A, Batiha I M and Pham V T 2022 Stabilization of different dimensional fractional chaotic maps (San Diego: Academic Press) pp. 123-155
[16] Almatroud A O, Ouannas A, Grassi G, Batiha I M, Gasri A and AlSawalha M M 2021 Archives of Control Sciences 31 765
[17] Ouannas A, Batiha I M and Pham V T 2023 Fractional discrete chaos: theories, methods and applications (Vol. 3) (Singapore: World Scientific)
[18] Hioual, A, Oussaeif, T E, Ouannas, A, Grassi, G, Batiha, I M and Momani S 2022 Alexandria Engineering Journal 61 10359
[19] Hamadneh, T, Hioual, A, Alsayyed, O, Al-Khassawneh, Y A, AlHusban, A and Ouannas A 2023 Fractal and Fractional 7 616
[20] Khennaoui A A, Almatroud A O, Ouannas A, Al-sawalha M M, Grassi G and Pham V T 2021 Discrete. Continuous Dynamical Systems-Series B 26 4549
[21] Abbes A, Ouannas A and Shawagfeh N 2023 Chin. Phys. B 32 030203
[22] Wu J and Xia L 2024 Economics 1 4
[23] Khennaoui A A and Ouannas A 2021 International Conference on Recent Advances in Mathematics and Informatics (ICRAMI), September 10-12, 2021, Constantine, Algeria, p. 1
[24] Abbes A, Ouannas A, Shawagfeh N and Grassi G 2022 Results in Physics 39 105797
[25] Abualhomos M, Abbes A, Gharib G M, Shihadeh A, Al Soudi M S, Alsaraireh A A and Ouannas A 2023 Mathematics 11 4166
[26] Alsayyed O, Abbes A, Gharib G M, Abualhomos M, Al-Tarawneh H, Al Soudi M S and Ouannas A 2023 Fractal and Fractional 7 728
[27] Atici F M and Eloe P 2009 Electron. J. Qual. Theory Differ. Equ. 3 1
[28] Abdeljawad T 2011 Computers & Mathematics with Applications 62 1602
[29] Wu G C and Baleanu D 2014 Nonlinear Dyn. 75 283
[30] Čermák J, Györi I and Nechvátal L 2015 Fractional Calculus and Applied Analysis 18 651
[31] Shatnawi M T, Djenina N, Ouannas A, Batiha I M and Grassi G 2022 Alexandria Engineering Journal 61 1655
[32] Huang D and Li H 1993 Theory and method of the nonlinear economics (Chengdu: Publishing House of Sichuan University)
[33] Xin B, Chen T and Ma J 2010 Discrete Dynamics in Nature and Society 2010
[34] Wu G C and Baleanu D 2015 Communications in Nonlinear Science and Numerical Simulation 22 95
[35] Gottwald G A and Melbourne I 2016 Chaos detection and predictability pp. 221-247
[36] Pincus S M 1991 Proc. Natl. Acad. Sci. USA 88 2297
[37] Jakimowicz A 2020 Entropy 22 452

On fractional discrete financial system: Bifurcation, chaos, and control

On fractional discrete financial system: Bifurcation, chaos, and control

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