Chaos and Bifurcation Analysis of a Discrete Time Financial Risk Model

Abstract

Financial risk dynamics often exhibit irregular fluctuations that are well described by nonlinear models with complex behavior. Motivated by the suitability of discrete-time representations for financial phenomena, this paper develops and studies a discrete-time financial risk evolution model obtained by applying the forward Euler scheme to the continuous three-dimensional risk system proposed in [Zhang et al., 2013]. The state variables represent occurrence, analysis and control value risks, and the resulting map captures the interaction-driven evolution of these risk components under positive parameters. We first derive the fixed points of the discrete system and investigate their local stability through the Jacobian matrix and its characteristic polynomial. Using a Schur-type criterion for cubic polynomials, we provide explicit stability conditions for the fixed points. We then establish analytical criteria for the onset of two fundamental bifurcations governing the transition from stable behavior to complex dynamics: Neimark–Sacker bifurcation and period-doubling bifurcation. In particular, we employ an explicit determinant-based method that bypasses direct eigenvalue calculations to verify the bifurcation conditions with respect to a chosen control parameter. Numerical simulations, including bifurcation diagrams, phase portraits and maximum Lyapunov exponents, are presented to validate the theoretical analysis and to illustrate the emergence of invariant closed curves, periodic orbits and chaotic-like regimes. The proposed discrete-time framework provides a rigorous basis for understanding how parameter variations can trigger instability and complex oscillations in financial risk, and it supports the development of practical monitoring and control strategies for mitigating adverse risk dynamics.