On chaos in fractional discrete financial risk model and its control approaches

Addressing financial risk models is essential to maintaining the sustainability and stability of economic setups in a world economy. The advantage of using fractional discrete-time difference equations to model financial risk management and economic processes is their capacity to detect random fluctuations in dynamic behaviors. This novel research introduces and analyzes the complex dynamics of a discrete-time financial risk model with fractional orders. The fundamental dynamics of this proposed model are examined, including the bifurcation analysis, chaotic attractor, and Lyapunov exponents. Furthermore, sequence complexity variations when fractional order and parameter of bifurcation risk (Formula presented.) varies are observed using the approximate entropy (ApEn) and 0-1 test. The findings highlight the sensitivity of the discrete-time financial risk model to fractional derivative orders, leading to the emergence of various rich, dynamic patterns such as chaos. Additionally, the suggested fractional-order discrete-time model is successfully stabilized and synchronized through the design of an active controller. Finally, MATLAB R2024b simulations was used to validate the research findings.