Probabilistic solution of a nonlinear economic cycle model simultaneously subjected to harmonic and random drivers

Nonlinear dynamical systems are widely applied in the study of nonlinear economic systems. However, the stochastic response of these nonlinear economic systems is less investigated when they are simultaneously subjected to harmonic and random drivers. This paper is concerned with this limitation. To describe the economic fluctuations, a forced van der Pol oscillator model is presented, and a harmonic cosine function and Gaussian white noise are used as exogenous drivers. The evolution of the probability density function (PDF) for the van der Pol-type economic cycle model develops using a path integral method. Monte Carlo simulation is used to validate the obtained PDFs. The numerical analysis investigates the effect of nonlinearity degree, driver frequency, and randomness on the evolution of the PDF. The findings show that the PDF evolves periodically as it fluctuates over time. Typically, the periodic evolution of the PDF results in the formation of unimodal or bimodal PDF distributions. The shape of the PDF distribution is significantly influenced by the examined factors.