Bifurkasi Hopf Pada Model Siklus Bisnis Investment Saving-Liquidity Money dengan Waktu Tunda

Abstract

The economy is not always good, meaning it experiences ups and downs. This situation requires forecasting. One of which is the Investment Saving-Liquidity Money business cycle model, which was first introduced by Kalecki in 1935 with the assumption that profits will be saved to be used as initial investment capital, thus causing a delay in the investment process (delay time). The addition of time delay in the system of differential equations causes a change in stability at the equilibrium point so that bifurcation occurs. Another modification given to the model is the use of time delays. The model analyzed in this study is a non-linear IS-LM business cycle model with time delay. Delay time is required for the gross product, capital, and interest rate to be used as an investment. This model was observed in two cases. First Case is a model without time delay. The second case is a model with a time delay. In this case, the critical value of delay is obtained. Hopf bifurcation occurs when the value of delay time is equal to the critical value of delay and satisfies the transversality condition. Observations on the model simulation are carried out by varying the delay time value. When the Hopf bifurcation occurs, the graph on the solution plane shows a constant oscillatory movement. If the value of delay time given is less than the critical value of delay, the controlled system solution goes to a balanced condition. Then when the delay time value is greater than the critical value of delay, the system solution continues to fluctuate, causing an unstable system condition.