Super-harmonic resonance of fractional-order van der Pol oscillator

Abstract

The dynamical characteristics of super-harmonic resonance of van der Pol oscillator with fractional-order derivative are studied. First the approximate analytical solution are obtained by the averaging method, and the definitions of equivalent linear damping and equivalent linear stiffness for super-harmonic resonance are established. Effects of the fractional-order parameters on the dynamical characteristics of the system are also studied through the equivalent linear damping and equivalent linear stiffness. Moreover, the amplitude-frequency equation and the stability condition for the steady-state solution are analytically presented, and the definitions of equivalent nonlinear damping coefficient and nonlinear stability parameter are also established. Finally, the comparisons of the fractional-order and the traditional integer-order van der Pol oscillators are carried out by numerical simulation. The effects of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude-frequency curves, and the system stability are also analyzed.