Asymptotic limit cycle of fractional van der Pol oscillator by homotopy analysis method and memory-free principle

Abstract

This paper presents an analytical approach to analyze the asymptotic behaviors of the steady state responses of a fractional van der Pol oscillator. First, an equivalent equation is deduced by replacing the Caputo-type fractional derivative with an improper integral based on a memory-free principle. The homotopy analysis method is then employed to directly solve the asymptotic limit cycle of the equivalent system. The presented approach can provide asymptotic limit cycles very accurately compared with some direct simulation results. This method can keep high accuracy and efficiency for both weakly and strongly nonlinear problems with any given fractional order. In addition, it is capable of tracking unstable limit cycles which cannot be generated by some time-marching numerical techniques. According to the obtained results, we analyze the influence of the fractional order and the nonlinear coefficient on limit cycle amplitude and frequency.