Memory‐Dependent Chaotic Dynamics and Stabilization of a Nonlinear Fractional‐Order Financial System With Optimal Control

This study presents an innovative nonlinear fractional‐order financial model that employs Caputo and Caputo–Fabrizio fractional derivatives to represent the dynamic interactions among interest rates, investment demand, price indices, and income/output. The model is formulated as a system of coupled nonlinear differential equations to encapsulate memory‐dependent dynamics and economic feedback mechanisms. Lyapunov functions, Carathéodory conditions, and Picard iteration techniques are used to prove that solutions exist analytically, are unique, and are stable everywhere. The classical and fractional forms of the system have been resolved numerically using techniques such as the Bernoulli wavelet approximation, Euler’s method, and the Nystrom scheme. Bifurcation diagrams, Lyapunov exponents, and the Kaplan–Yorke dimension have been used to study the dynamics and distinguish between stable and chaotic regimes. Optimal control mechanisms are developed by evaluating the effectiveness of fiscal and monetary policies, grounded in Pontryagin’s maximum principle. Fractional‐order dynamics yield more realistic macroeconomic trajectories than classical models, exhibiting reduced chaos, enhanced convergence, and increased stability; control mechanisms efficiently manage volatility and stabilize the system across all significant financial variables. The numerical simulations show that the interest rate lowers from 18 to 1.0 in 20 days, investment levels out at 1.0, inflation rises to 0.4, and production stays close to 2.3. Eigenvalue analysis demonstrates the fact that E 1 is unstable with λ 4 = 0.4143, but E 2 and E 3 are stable with λ 1 = −2.1682. The dynamic stability is verified, with the maximal Lyapunov exponent stabilizing at −0.23 and the Kaplan–Yorke dimension at 2.067.