Primary resonance of fractional-order van der Pol oscillator

In this paper the primary resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied analytically and numerically. At first the approximately analytical solution is obtained by the averaging method, and it is found that the fractional-order derivative could affect the dynamical properties of VDP oscillator, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. Moreover, the amplitude-frequency equation for steady-state solution is established, and the corresponding stability condition is also presented based on Lyapunov theory. Then, the comparisons of several different amplitude-frequency curves by the approximately analytical solution and the numerical integration are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the two fractional parameters, i.e., the fractional coefficient and the fractional order, on the amplitude-frequency curves are investigated for some typical excitation amplitudes, which are different from the traditional integer-order VDP oscillator.