Dynamical analysis of a time-delayed financial system with synchronization strategies

Financial systems reveal complex and nonlinear behavior in the presence of various macroeconomic drivers. In the present study, we investigate the nonlinear financial system behavior through the formulation of an ordinary differential equation model using interest rate, investment demand, and price index as the state variables. The model involves government debt and investment time delay as the most influential parameters of system dynamics. Firstly, we examined the elementary characteristics of the system's solutions, identified equilibria, and analyzed local stability. Our findings show that, within a certain range of government debt, the system maintains stability until it reaches a critical point, at which it becomes unstable due to a Hopf bifurcation. We illustrated how raising the investment delay can drive the system towards instability after passing a critical delay magnitude. To counteract the emergence of erratic behavior caused by unregulated debt and time delay, we introduced a synchronization control strategy. Numerical simulations are presented to validate the theoretical results, demonstrating the efficacy of the introduced control mechanism in stabilizing the financial system.