Stability and bifurcation analysis of a discrete fractional financial risk system

In this paper we investigate a three-dimensional fractional-order financial risk system with Caputo time derivatives. We first establish local existence and uniqueness of solutions by reformulating the model as a Volterra integral equation and applying Schauder’s fixed point theorem together with a Weissinger-type argument. Then, following the discretization approach of El-Sayed and Salman, we derive an explicit discrete-time approximation. For the resulting three-dimensional map, we characterize the fixed points and analyze their local stability via Schur–Cohn/Jury-type criteria. Using determinant-based methods, we obtain sufficient conditions for Neimark–Sacker and period-doubling bifurcations in terms of the model parameters. Several numerical experiments illustrate the emergence of invariant closed curves, period-doubling cascades and chaotic attractors, and show how appropriate parameter choices can suppress large oscillations in the risk variables. The results provide a mathematically rigorous framework for understanding nonlinear and potentially chaotic behavior in fractional-order financial risk modeling.