Memory and delayed investment response in a Caputo fractional financial system: stability, transient dynamics, and residual-verified computation

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.