Stability and Analytical Approximation of Limit Cycles in Hopf Bifurcations of Four-dimensional Economic Models

Paper constructs approximate analytical expressions of periodic disequilibrium fluctuations business cycles occurring in connection with Hopf bifurcations in nonlinear problems of economic interactions described by four-dimensional continuous time dynamical systems. Two nonlinear macrodynamic models are employed as tests models. The region of equilibrium stability in parameter space is obtained in each case and a Hopf bifurcation curve is identified as a boundary of the region. Validity of the analytical approximations obtained for the cycles generated by the loss of equilibrium stability on this curve is confirmed by comparison to numerically determined cycles. Explicit analytical description of such limit cycles is of particular interest in the case of subcritical bifurcation, due to the difficulty of the numerical determination of the generated unstable cycles. Mathematics Subject Classification: 34A34, 34C07, 34C23, 37G10