This paper studies the effect of memory and delayed investment response in a Caputo fractional version of the Chen financial system. The model describes the interaction between the interest rate, investment demand, and price index through a three-dimensional fractional delay system. Two mechanisms are incorporated simultaneously: a Caputo derivative of order $ 0 < \rho < 1 $, which represents hereditary memory, and a discrete delay in the nonlinear investment feedback, which models the fact that investment decisions do not react instantaneously to previous market conditions.
The equilibrium points of the system are obtained explicitly. The local stability problem is then formulated through the characteristic equation of the linearized fractional delay system. For the equilibrium $ E_0 $, a corrected stability condition is derived in terms of the fractional stability criterion. For the nontrivial equilibria $ E_\pm $, where the delay enters the characteristic equation explicitly, a numerical crossing procedure is used to identify delay-dependent stability changes. This provides a concrete way to examine how the fractional order and the delay parameter influence the response near equilibrium.
The numerical dynamics are computed by a predictor-corrector method adapted to Caputo fractional delay equations. To strengthen the reliability of the simulations, the computed trajectories are verified by an independent residual diagnostic based on a quintic Caputo reconstruction. The numerical study is organized around the separate and combined effects of memory and delay. Variations of the fractional order show that memory can modify the amplitude, duration, and smoothing of transient excursions. Variations of the delay show that lagged investment feedback can shift and amplify the transient response. Additional comparisons between delayed and nondelayed dynamics, as well as between fractional and integer-order responses, clarify the distinct roles of these two mechanisms.
The results indicate that delayed investment response and fractional memory act in different directions in the organization of the financial dynamics. The delay tends to promote transient amplification and phase shifting, whereas fractional memory can moderate or postpone these effects. The study is therefore presented as a stability-oriented and residual-verified numerical analysis of a delayed fractional financial model. It does not claim a complete bifurcation or chaos classification; rather, it provides a reproducible framework and identifies Lyapunov-exponent computation, continuation analysis, and broader parameter exploration as natural directions for future work.
Citation: Alvaro H. Salas, Gilder Cieza Altamirano, Lorenzo Julio Martínez Hernández. Memory and delayed investment response in a Caputo fractional financial system: stability, transient dynamics, and residual-verified computation[J]. AIMS Mathematics, 2026, 11(6): 18553-18579. doi: 10.3934/math.2026754
This paper studies the effect of memory and delayed investment response in a Caputo fractional version of the Chen financial system. The model describes the interaction between the interest rate, investment demand, and price index through a three-dimensional fractional delay system. Two mechanisms are incorporated simultaneously: a Caputo derivative of order $ 0 < \rho < 1 $, which represents hereditary memory, and a discrete delay in the nonlinear investment feedback, which models the fact that investment decisions do not react instantaneously to previous market conditions.
The equilibrium points of the system are obtained explicitly. The local stability problem is then formulated through the characteristic equation of the linearized fractional delay system. For the equilibrium $ E_0 $, a corrected stability condition is derived in terms of the fractional stability criterion. For the nontrivial equilibria $ E_\pm $, where the delay enters the characteristic equation explicitly, a numerical crossing procedure is used to identify delay-dependent stability changes. This provides a concrete way to examine how the fractional order and the delay parameter influence the response near equilibrium.
The numerical dynamics are computed by a predictor-corrector method adapted to Caputo fractional delay equations. To strengthen the reliability of the simulations, the computed trajectories are verified by an independent residual diagnostic based on a quintic Caputo reconstruction. The numerical study is organized around the separate and combined effects of memory and delay. Variations of the fractional order show that memory can modify the amplitude, duration, and smoothing of transient excursions. Variations of the delay show that lagged investment feedback can shift and amplify the transient response. Additional comparisons between delayed and nondelayed dynamics, as well as between fractional and integer-order responses, clarify the distinct roles of these two mechanisms.
The results indicate that delayed investment response and fractional memory act in different directions in the organization of the financial dynamics. The delay tends to promote transient amplification and phase shifting, whereas fractional memory can moderate or postpone these effects. The study is therefore presented as a stability-oriented and residual-verified numerical analysis of a delayed fractional financial model. It does not claim a complete bifurcation or chaos classification; rather, it provides a reproducible framework and identifies Lyapunov-exponent computation, continuation analysis, and broader parameter exploration as natural directions for future work.
| [1] |
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 doi: 10.1007/BF02437847
|
| [2] | R. Shone, An introduction to economic dynamics, Cambridge: Cambridge University Press, 2012. https://doi.org/10.1017/CBO9781139164733 |
| [3] |
M. Jusup, P. Holme, K. Kanazawa, M. Takayasu, I. Romić, Z. Wang, et al., Social physics, Physics Reports, 948 (2022), 1–148. https://doi.org/10.1016/j.physrep.2021.10.005 doi: 10.1016/j.physrep.2021.10.005
|
| [4] |
V. E. Tarasov, On history of mathematical economics: application of fractional calculus, Mathematics, 7 (2019), 509. https://doi.org/10.3390/math7060509 doi: 10.3390/math7060509
|
| [5] |
W. C. Chen, Nonlinear dynamics and chaos in a fractional-order financial system, Chaos Soliton. Fract., 36 (2008), 1305–1314. https://doi.org/10.1016/j.chaos.2006.07.051 doi: 10.1016/j.chaos.2006.07.051
|
| [6] |
W. C. Chen, Dynamics and control of a financial system with time-delayed feedbacks, Chaos Soliton. Fract., 37 (2008), 1198–1207. https://doi.org/10.1016/j.chaos.2006.10.016 doi: 10.1016/j.chaos.2006.10.016
|
| [7] |
Q. Gao, J. Ma, Chaos and Hopf bifurcation of a finance system, Nonlinear Dyn., 58 (2009), 209–216. https://doi.org/10.1007/s11071-009-9472-5 doi: 10.1007/s11071-009-9472-5
|
| [8] |
Z. Wang, X. Huang, G. Shi, Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay, Comput. Math. Appl., 62 (2011), 1531–1539. https://doi.org/10.1016/j.camwa.2011.04.057 doi: 10.1016/j.camwa.2011.04.057
|
| [9] |
C. Huang, L. Cai, J. Cao, Linear control for synchronization of a fractional-order time-delayed chaotic financial system, Chaos Soliton. Fract., 113 (2018), 326–332. https://doi.org/10.1016/j.chaos.2018.05.022 doi: 10.1016/j.chaos.2018.05.022
|
| [10] |
S. Wang, S. He, A. Yousefpour, H. Jahanshahi, R. Repnik, M. Perc, Chaos and complexity in a fractional-order financial system with time delays, Chaos Soliton. Fract., 131 (2020), 109521. https://doi.org/10.1016/j.chaos.2019.109521 doi: 10.1016/j.chaos.2019.109521
|
| [11] |
K. Rajagopal, F. Parastesh, H. R. Abdolmohammadi, S. Jafari, M. Perc, E. Klemenčič, Effects of coupling on extremely multistable fractional-order systems, Chinese J. Phys., 87 (2024), 246–255. https://doi.org/10.1016/j.cjph.2023.12.011 doi: 10.1016/j.cjph.2023.12.011
|
| [12] | I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, San Diego: Academic Press, 1999. |
| [13] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006. |
| [14] | K. Diethelm, The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type, Berlin: Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2 |
| [15] |
C. Li, W. Deng, Remarks on fractional derivatives, Appl. Math. Comput., 187 (2007), 777–784. https://doi.org/10.1016/j.amc.2006.08.163 doi: 10.1016/j.amc.2006.08.163
|
| [16] |
K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3–22. https://doi.org/10.1023/A:1016592219341 doi: 10.1023/A:1016592219341
|
| [17] |
K. Diethelm, N. J. Ford, A. D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), 31–52. https://doi.org/10.1023/B:NUMA.0000027736.85078.be doi: 10.1023/B:NUMA.0000027736.85078.be
|
| [18] | S. Bhalekar, V. Daftardar-Gejji, A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order, J. Fract. Calc. Appl., 1 (2011), 1–9. |
Figures(11) / Tables(3)
Alvaro H. Salas, Gilder Cieza Altamirano, Lorenzo Julio Martínez Hernández. Memory and delayed investment response in a Caputo fractional financial system: stability, transient dynamics, and residual-verified computation[J]. AIMS Mathematics, 2026, 11(6): 18553-18579. doi: 10.3934/math.2026754
DownLoad: