Stability and bifurcation analysis of a four-dimensional economic model

In this work a four dimensional financial system, depending on five parameters is analyzed. The system was obtained by adding a new feed-back control variable to a well-known 3D financial system, modelling the evolution of the interest rate, the investment demand and the inflation rate. The stability of economical relevant equilibria is established. All the Hopf singularities were analyzed and the existence of supercritical, subcritical, and degenerated Hopf bifurcations was proved. Corresponding to them we found stable limit cycles, saddle type limit cycles. In addition, for certain parameters strata, one of the equilibria becomes a center, an approximate two-dimensional center manifold is determined and isolated periodic solutions are emphasized numerically. A double-Hopf degenerated bifurcation is also found. The addition of the feed-back led to obtain new possibilities to stabilize the economic environment, either to a new stable equilibrium state or to stable periodic behavior.