Bifurcations in economic growth model with distributed time delay transformed to ODE

Document pages: 26 pages

Abstract: We consider the model of economic growth with time delayed investmentfunction. Assuming the investment is time distributed we can use the linearchain trick technique to transform delay differential equation system toequivalent system of ordinary differential system (ODE). The time delayparameter is a mean time delay of gamma distribution. We reduce the system withdistribution delay to both three and four-dimensional ODEs. We study the Hopfbifurcation in these systems with respect to two parameters: the time delayparameter and the rate of growth parameter. We derive the results from theanalytical as well as numerical investigations. From the former we obtain thesufficient criteria on the existence and stability of a limit cycle solutionthrough the Hopf bifurcation. In numerical studies with the Dana and Malgrangeinvestment function we found two Hopf bifurcations with respect to the rategrowth parameter and detect the existence of stable long-period cycles in theeconomy. We find that depending on the time delay and adjustment speedparameters the range of admissible values of the rate of growth parameterbreaks down into three intervals. First we have stable focus, then the limitcycle and again the stable solution with two Hopf bifurcations. Such behaviourappears for some middle interval of admissible range of values of the rate ofgrowth parameter.