Analysis on limit cycle of fractional-order van der Pol oscillator

In this paper the approximately analytical solution of van der Pol (VDP) oscillator with two kinds of fractional-order derivatives is obtained based on averaging method. Two equivalent system parameters, i.e. equivalent damping coefficient and equivalent stiffness coefficient, are defined, which could characterize the effects of the fractional parameters on the limit cycle in fractional-order VDP oscillator. The same points and differences between the traditional integer-order and fractional-order VDP oscillator are analyzed and summarized in detail. The differences are focused on the convergence speed and frequency characteristic of the limit cycle in VDP oscillator. The comparison between the analytical and numerical solution verifies the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the fractional parameters on the convergence speed and frequency characteristic of the limit cycle in fractional-order VDP oscillator are illustrated based on some typical system parameters.