This study explores the intricate behavior of monetary systems through a fractional-order model based on the Caputo derivative, emphasizing the nonlinear interactions among key economic variables: interest rate ( \(I_r\) ), investment demand ( \(I_d\) ), and price exponent ( \(P_e\) ). Utilizing a system of Caputo fractional differential equations, we investigate how these variables evolve under different fractional orders. Through 3D visualizations, the model’s solution trajectories are analyzed to illustrate the system’s sensitivity to fractional-order variations and parameter shifts. The Caputo framework enables a nuanced examination of memory effects and long-term dependencies within economic systems, highlighting how small changes in fractional orders can significantly influence stability and dynamic patterns. Stability analysis reveals equilibrium points and their transition to chaos through bifurcation. Bifurcation analysis demonstrates that varying the fractional order q from 0.85 to 1 significantly influences system stability and the onset of chaos. Numerical simulations, including phase portraits and Lyapunov exponent computations, confirm the existence of chaotic attractors phenomena. Moreover, synchronization strategies are explored for potential applications in financial forecasting and risk mitigation. The research emphasizes how important fractional-order modeling is for understanding and controlling chaotic monetary systems. This study provides policymakers with sophisticated analytical methods for evaluating economic stability and informing policy choices. Through the integration of fractional calculus and chaos theory, the research enhances the modeling of intricate economic interactions and delivers greater insight into monetary system behavior and connects theoretical developments with real-world applications.
