Abstract: The evolution of a business cycle has  nonlinear and stochastic characteristics, and it is improtant to find its evolution law. Firstly, a nonlinear business cycle dynamic model with first-order and third-order terms of the difference between preincome and prior income to simulate a generic nonlinear impact of incomes on the investment is established in this paper. Two independent Gaussian white noise random functions are used to simulate the interference of uncertain factors and random income, respectively. Then a path integral method, based on short-term Gaussian transition probability density and Gauss-Legendre integration scheme, is applied to solve the probability density function of income and income change rate. Finally, the effects of income random disturbance and supplementary saving rate on the evolution of the nonlinear business cycle are studied. The results show that the fluctuation of the income in the random interference is significant in the early stage, and it tends to be stable in the later period. The increase of income random interference significantly increases the randomness of income, making income more difficult to be predicted and controlled. In addition, the enhancement of supplementary saving rate decreases the probability of high income.

Key words: stochastic nonlinear dynamics, business cycle, probability density, Gaussian white noise, random disturbance