Periodic evolution of nonlinear economic cycle systems under exogenous harmonic and stochastic drivers

The study is conducted for analyzing the periodic behavior of nonlinear economic cycle systems with an exogenous harmonic driver in random environment. The case is taken in the form of a harmonic cosine function acting on income for the exogenous driver and Gaussian white noise for the random environment. It theoretically can model the periodic adjustment of an authority in a random market, which is more practical in reality. The probability density function (PDF) of the systems is used to describe the behaviors. Two different nonlinear economic cycle models are further studied to evaluate the reaction of economic cycle systems with different cases. One model is about a van der Pol-polynomial economic cycle model. The other model is about a van der Pol-type economic cycle model. To obtain the PDF evolution of the two economic cycle models, a path integration technique is adopted. The analysis reveals that the PDFs of the nonlinear models exhibit stable periodic behaviors after several cycles. Furthermore, the van der Pol-polynomial model has a unimodal PDF, with the PDFs of income and income growth rate exhibiting similar evolutionary patterns. By contrast, the van der Pol-type economic cycle model performs a repetitive alternation between unimodal and bimodal patterns in the PDF distributions. The shape of these distributions is significantly affected by different magnitudes of driver frequencies. Therefore, the authority should focus on the effect of the adjustment frequency on the random market besides the adjustment amplitude.