Numerical scheme and dynamic analysis for variable-order fractional van der Pol model of nonlinear economic cycle

Considering the fact that the memory in economic series changes with dynamic economic environment, this paper is devoted to the proposal of a variable-order fractional van der Pol model (VOFVDPM), where the order of the derivative is replaced by a time-dependent function. A numeric scheme for this model is designed by the Adams-Bashforth-Moulton method. The dynamic behaviors of the VOFVDPM with linear and periodic variable-order functions are investigated through numerical experiment. Some dynamic characteristics of the VOFVDPM that do not exist in a fractional order van der Pol model are discovered in the numerical simulation, such as existing limit point when the linear order functions have the same ranges and opposite slopes.