Chaotic behavior of a new fractional-order financial system and its predefined-time sliding mode control based on the RBF neural network

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Abstract

Fractional-order financial systems hold significant importance in practical situations. In this work, a novel fractional-order financial system considering the non-constant elasticity of demand was presented, and the system’s complex dynamics were studied. The results demonstrated that the system can exhibit diverse chaotic dynamics and periodic oscillations, which are influenced by different frac- tional orders and system parameters. Then, to stabilize the proposed chaotic system with uncertainties and achieve predefined-time synchronization of master-slave systems, an effective sliding mode con- trol strategy utilizing the RBF neural network was put forward. In real financial markets, uncertainties and perturbations occur suddenly, and excessive control input can lead to resource inefficiency. There- fore, unlike other papers that rely on conservative estimations using upper bounds, this paper used RBF neural network approximation to design a more flexible and robust controller while reducing the control input. Simulation findings reveal that this approach requires less control input than traditional methods without the RBF neural network and converges more rapidly than finite-time, fixed-time, and other predefined-time sliding mode control strategies with the RBF neural network, which validates the feasibility of this approach. Finally, the proposed chaotic system and control method were successfully applied to secure encryption, demonstrating their practical value.

Keywords

fractional-order financial system / sliding mode control / predefined-time synchronization / RBF neural network / secure encryption

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References

[1] I. Podlubny, Fractional Differential Equations , Academic Press, 1999.

[2] J. M. Munoz-Pacheco, E. Zambrano-Serrano, C. Volos, O. I. Tacha, I. N. Stouboulos, V. T. Pham, A fractional order chaotic system with a 3D grid of variable attractors, Chaos Solitons Fractals, 113 (2018), 69-78. https://doi.org/10.1016/j.chaos.2018.05.015

[3] P. Ostalczyk, Discrete Fractional Calculus: Applications in Control and Image Processing , World Scientific, 2016. https://doi.org/10.1142/9833

[4] L. C. de Barros, M. M. Lopes, F. S. Pedro, E. Esmi, J. P. C. dos Santos, D. E. Sánchez, The memory effect on fractional calculus: an application in the spread of COVID-19, Comput. Appl. Math., 40 (2021), 72. https://doi.org/10.1007/s40314-021-01456-z

[5] S. B. Chen, H. Jahanshahi, O. A. Abba, J. E. Solís-Pérez, S. Bekiros, J. F. Gómez-Aguilar, et al., The effect of market confidence on a financial system from the perspective of fractional calculus: Numerical investigation and circuit realization, Chaos Solitons Fractals, 140 (2020), 110223. https://doi.org/10.1016/j.chaos.2020.110223

[6] H. Jahanshahi, A. Yousefpour, J. M. Munoz-Pacheco, S. Kacar, V. T. Pham, F. E. Alsaadi, A new fractional-order hyperchaotic memristor oscillator: Dynamic analysis, robust adaptive syn- chronization, and its application to voice encryption, Appl. Math. Comput., 383 (2020), 125310. https://doi.org/10.1016/j.amc.2020.125310

[7] S. Vaidyanathan, C. Volos, Advances and Applications in Chaotic Systems , Springer, 2016. https://doi.org/10.1007/978-3-319-30279-9

[8] R. H. Strotz, J. C. McAnulty, J. B. Naines, Goodwin’s nonlinear theory of the business cycle: an electro-analog solution, Econometrica, J. Econometric Soc., 21 (1953), 390-411. https://doi.org/10.2307/1905446

[9] J. H. Ma, Y. S. Chen, Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (I), Appl. Math. Mech., 22 (2001), 1240-1251. https://doi.org/10.1007/bf02437847

[10] J. H. Ma, Y. S. Chen, Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (II), Appl. Math. Mech., 22 (2001), 1375-1382. https://doi.org/10.1007/bf02435540

[11] Q. Gao, J. H. Ma, Chaos and Hopf bifurcation of a finance system, Nonlinear Dyn., 58 (2009), 209-216. https://doi.org/10.1007/s11071-009-9472-5

[12] O. I. Tacha, C. K. Volos, I. N. Stouboulos, I. M. Kyprianidis, Analysis, adaptive control and circuit simulation of a novel finance system with dissaving, Arch. Control Sci., 26 (2016), 95-115. https://doi.org/10.1515/acsc-2016-0006

[13] S. Vaidyanathan, C. K. Volos, O. I. Tacha, I. M. Kyprianidis, I. N. Stouboulos, V. T. Pham, Analysis, control and circuit simulation of a novel 3-D finance chaotic system, in Advances and Applications in Chaotic Systems, Springer, (2016), 495-512. https://doi.org/10.1007/978-3-319-30279-921

[14] S. Vaidyanathan, A. Sambas, S. Kacar, U. Cavusoglu, A new finance chaotic system, its elec- tronic circuit realization, passivity based synchronization and an application to voice encryption, Nonlinear Eng., 8 (2019), 193-205. https://doi.org/10.1515/nleng-2018-0012

[15] T. Yang, Dynamical analysis on a finance system with nonconstant elasticity of demand, Int. J. Bifurcation Chaos, 30 (2020), 2050148. https://doi.org/10.1142/s0218127420501485

[16] B. Subartini, S. Vaidyanathan, A. Sambas, Multistability in the finance chaotic system, its bifur- cation analysis and global chaos synchronization via integral sliding mode control, IAENG Int. J. Appl. Math., 51 (2021), 995-1002.

[17] Y. F. Wei, C. R. Xie, X. Qing, Y. H. Xu, Control of a new financial risk contagion dynamic model based on finite-time disturbance, Entropy, 26 (2024), 999. https://doi.org/10.3390/e26120999

[18] S. R. Yan, A. Mohammadzadeh, E. Ghaderpour, Type-3 fuzzy logic and Lyapunov ap- proach for dynamic modeling and analysis of financial markets, Heliyon, 10 (2024), e33730. https://doi.org/10.1016/j.heliyon.2024.e33730

[19] M. D. Johansyah, S. Vaidyanathan, K. Benkouider, A. Sambas, K. M. Rao, K. Anjaneyulu, Ex- ploring chaos and stability in a 3-D finance model with absolute function nonlinearity, Mathe- matics, 13 (2025), 735. https://doi.org/10.3390/math13050735

[20] A. Phukan, H. K. Sarmah, Bifurcation analysis of a non linear 6D financial sys- tem with three time delay feedback, Chaos Solitons Fractals, 194 (2025), 116248. https://doi.org/10.1016/j.chaos.2025.116248

[21] C. G. Xu, C. Aouiti, M. X. Liao, P. L. Li, Z. X. Liu, Chaos control strategy for a fractional-order financial model, Adv. Differ. Equations, 2020 (2020), 573. https://doi.org/10.1186/s13662-020-02999-x

[22] Z. Yao, K. H. Sun, S. He, Dynamics of fractional-order chaotic rocard re- laxation econometric system, Int. J. Bifurcation Chaos, 32 (2022), 2250195. https://doi.org/10.1142/S0218127422501954

[23] T. Cui, P. J. Liu, Computational modeling of financial sistem via a new fractal-fractional mathe- matical model, Fractals, 31 (2023), 2340083. https://doi.org/10.1142/S0218348X23400832

[24] Q. N. Li, R. H. Li, D. M. Huang, Dynamic analysis of a new 4D fractional-order financial system and its finite-time fractional integral sliding mode control based on RBF neural network, Chaos Solitons Fractals, 177 (2023), 114156. https://doi.org/10.1016/j.chaos.2023.114156

[25] L. Diabi, A. Ouannas, A. Hioual, S. Momani, A. Abbes, On fractional discrete financial system: Bifurcation, chaos, and control, Chin. Phys. B, 33 (2024), 100201. https://doi.org/10.1088/1674-1056/ad5d96

[26] Z. Xu, K. H. Sun, H. H. Wang, Dynamics and function projection synchronization for the fractional-order financial risk system, Chaos Solitons Fractals, 188 (2024), 115599. https://doi.org/10.1016/j.chaos.2024.115599

[27] L. M. Pecora, T. L. Carroll, Synchronization in chaotic systems, Chaos Solitons Fractals, 64 (1990), 821-824. https://doi.org/10.1103/PhysRevLett.64.821

[28] H. Fu, Y. G. Gao, Synchronization of uncertain general fractional unified chaotic systems via finite-time adaptive sliding mode control, Chaos, 33 (2023), 043136. https://doi.org/10.1063/5.0130366

[29] H. Q. Wang, G. Chai, S. J. Kang, Adaptive fixed-time anti-synchronization and synchroniza- tion control for Liu-Chen-Liu chaotic systems with actuator faults, Int. J. Adapt. Control Signal Process., 38 (2024), 2158-2177. https://doi.org/10.1002/acs.3799

[30] L. P. Chen, M. Xue, A. M. Lopes, R. C. Wu, X. H. Zhang, Y. Q. Chen, New synchronization cri- terion of incommensurate fractional-order chaotic systems, IEEE Trans. Circuits Syst. II Express Briefs, 71 (2024), 455-459. https://doi.org/10.1109/TCSII.2023.3297174

[31] Y. L. Wang, H. Jahanshahi, S. Bekiros, F. Bezzina, Y. M. Chu, A. A. Aly, Deep recur- rent neural networks with finite-time terminal sliding mode control for a chaotic fractional- order financial system with market confidence, Chaos Solitons Fractals, 146 (2021), 110881. https://doi.org/10.1016/j.chaos.2021.110881

[32] S. H. Yuningsih, E. Rusyaman, S. Vaidyanathan, A. Sambas, Investigation of chaos behavior and integral sliding mode control on financial risk model, AIMS Math., 7 (2022), 18377-18392. https://doi.org/10.3934/math.20221012

[33] M. D. Johansyah, A. Sambas, S. Mobayen, B. Vaseghi, S. F. Al-Azzawi, Sukono, et al., Dynam- ical analysis and adaptive finite-time sliding mode control approach of the financial fractional- order chaotic system, Mathematics, 11 (2023), 100. https://doi.org/10.3390/math11010100

[34] X. M. Zhang, Q. L. Han, J. Wang, Admissible delay upper bounds for global asymptotic stability of neural networks with time-varying delays, IEEE Trans. Neural Networks Learn. Syst., 29 (2018), 5319-5329. https://doi.org/10.1109/TNNLS.2018.2797279

[35] L. M. Wang, T. D. Dong, M. F. Ge, Finite-time synchronization of memristor chaotic sys- tems and its application in image encryption, Appl. Math. Comput., 347 (2019), 293-305. https://doi.org/10.1016/j.amc.2018.11.017

[36] Y. C. Ding, H. Liu, A new fixed-time stability criterion for fractional-order systems, AIMS Math., 7 (2022), 6173-6181. https://doi.org/10.3934/math.2022343

[37] J. D. Sánchez-Torres, D. G. Gómez-Gutiérrez, E. López, A. G. Loukianov, A class of predefined-time stable dynamical systems, IMA J. Math. Control Inf., 35 (2018), i1-i29. https://doi.org/10.1093/imamci/dnx004

[38] C. A. Anguiano-Gijón, A. J. Muñoz-Vázquez, J. D. Sánchez-Torres, G. Romero-Galván, F. Martínez-Reyes, On predefined-time synchronisation of chaotic systems, Chaos Solitons Frac- tals, 122 (2019), 172-178. https://doi.org/10.1016/j.chaos.2019.03.015

[39] M. J. Zhang, H. Y. Zang, L. Y. Bai, A new predefined-Time sliding mode control scheme for synchronizing chaotic systems, Chaos Solitons Fractals, 164 (2022), 112745. https://doi.org/10.1016/j.chaos.2022.112745

[40] H. B. Xue, X. H. Liu, A novel fast terminal sliding mode with predefined-time synchronization, Chaos Solitons Fractals, 175 (2023), 114049. https://doi.org/10.1016/j.chaos.2023.114049

[41] W. Zheng, S. C. Qu, Q. Tang, X. N. Du, Predefined-time sliding mode control for synchro- nization of uncertain hyperchaotic system and application, J. Vib. Control, 31 (2025), 528-538. https://doi.org/10.1177/10775463241228649

[42] S. Sahoo, R. Nathasarma, B. K. Roy, Time-synchronized predefined-time synchronization between two non-identical chaotic systems, Chaos Solitons Fractals, 181 (2024), 114662. https://doi.org/10.1016/j.chaos.2024.114662

[43] C. Jia, X. H. Liu, X. Y. Li, J. Y. Xu, A new adaptive predefined time sliding mode con- trol for nonlinear systems, IEEE Trans. Circuits Syst. II Express Briefs, 71 (2024), 2094-2098. https://doi.org/10.1109/TCSII.2023.3332089

[44] M. Y. Wang, J. P. Li, Interpretable predictions of chaotic dynamical systems using dynamical system deep learning, Sci. Rep., 14 (2024), 3143. https://doi.org/10.1038/s41598-024-53169-y

[45] H. Jahanshahi, Z. H. Zhu, Review of machine learning in robotic grasping control in space application, Acta Astronaut., 220 (2024), 37-61. https://doi.org/10.1016/j.actaastro.2024.04.012

[46] R. Zahedifar, M. S. Baghshah, A. Taheri, LLM-controller: Dynamic robot con- trol adaptation using large language models, Rob. Auton. Syst., 186 (2025), 104913. https://doi.org/10.1016/j.robot.2024.104913

[47] J. L. Chen, X. S. Zhao, J. Y. Xiao, Novel predefined-time control for fractional-order systems and its application to chaotic synchronization, Math. Methods Appl. Sci., 47 (2024), 5427-5440. https://doi.org/10.1002/mma.9871

[48] M. Yang, Q. Zhang, K. Xu, M. Chen, Adaptive differentiator-based predefined-time control for nonlinear systems subject to pure-feedback form and unknown disturbance, Complexity, 2021 (2021), 7029058. https://doi.org/10.1155/2021/7029058

[49] H. Liu, H. Y. Tu, S. Huang, X. J. Zheng, Adaptive predefined-time sliding mode control for QUADROTOR formation with obstacle and inter-quadrotor avoidance, Sensors, 23 (2023), 2392. https://doi.org/10.3390/s23052392

[50] Y. Zhang, F. W. Meng, Y. Z. Tai, Neural network-based Predefined-time attitude control for rigid spacecraft with uncertainties and disturbances, J. Ind. Manage. Optim., 21 (2025), 1832-1845. https://doi.org/10.3934/jimo.2024151

[51] E. Ahmed, A. M. A. El-Sayed, H. A. A. El-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl., 325 (2007), 542-553. https://doi.org/10.1016/j.jmaa.2006.01.087

[52] C. P, Li, W. H. Deng, Remarks on fractional derivatives, Appl. Math. Comput., 187 (2007), 777-784. https://doi.org/10.1016/j.amc.2006.08.163

[53] K. Diethelm, The Analysis of Fractional Differential Equations , Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2

[54] N. X. Liu, J. T. Fei, Adaptive fractional sliding mode control of active power filter based on dual RBF neural networks, IEEE Access, 5 (2017), 27590-27598. https://doi.org/10.1109/access.2017.2774264

[55] K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the nu- merical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3-22. https://link.springer.com/article/10.1023/A:1016592219341

[56] G. C. Wu, Z. G. Deng, D. Baleanu, D. Q. Zeng, New variable-order fractional chaotic systems for fast image encryption, Chaos, 29 (2019), 083103. https://doi.org/10.1063/1.5096645

[57] L. L. Liu, Q. Zhang, X. P. Wei, A RGB image encryption algorithm based on DNA encoding and chaos map, Comput. Electr. Eng., 38 (2012), 1240-1248. https://doi.org/10.1016/j.compeleceng.2012.02.007

[58] Y. W. Yang, J. Gao, H. Imani, Design, analysis, circuit implementation, and synchronization of a new chaotic system with application to information encryption, AIP Adv., 13 (2023), 075116. https://doi.org/10.1063/5.0161382