Stability and Hopf bifurcation of a Goodwin model with four different delays

This paper investigates the stability and Hopf bifurcation of a Goodwin model with four different delays. Firstly, we present the existence and uniqueness of the positive equilibrium for the system. Then the sum of time delays is chosen as the bifurcation parameter. By analyzing the distribution of characteristic roots of the corresponding linearized system, we obtain the conditions for keeping the system to be stable. Moreover, it is illustrated that the Hopf bifurcation will occur when the delay passes through a critical value. Moreover, some specific formulas for determining the stability and direction of the Hopf bifurcation are obtained by using the normal form theory and the center manifold reduction. Finally, numerical simulation is given to verify the correctness of our theoretical analysis.