A Variable-Order Fractional Financial Model with State-Dependent Memory

This study presents a novel state-dependent fractional-order financial model that advances the modeling of memory effects in dynamic economic systems. By allowing the memory order q (t) to adapt continuously as a function of the internal state variable—specifically, the investment demand y (t)—the model captures the essential nonlinear feedback between market sentiment, memory depth, and macroeconomic behavior. The fractional framework is based on the Caputo-Fabrizio operator, chosen for its smooth, non-singular kernel, which ensures physical consistency and superior numerical tractability. A hyperbolic tangent structure is employed to define the adaptive memory function, enabling bounded, smooth, and symmetric evolution of q (t) within a realistic economic range. This design reflects empirically observed financial phenomena: heightened responsiveness during periods of uncertainty (short memory) and persistent inertia during stable or optimistic regimes (long memory). Numerical simulations demonstrate the model’s ability to replicate key financial features—such as damped oscillations, delayed stabilization, and variable sensitivity—under changing market conditions. Furthermore, the model is validated against real data from the 2008 global financial crisis, illustrating its empirical relevance and practical forecasting potential. By integrating internal memory regulation into the core of financial system dynamics, this work contributes a flexible and realistic tool for modern economic analysis, with applications in policy modeling, risk evaluation, and adaptive control of financial systems.