Simultaneously primary and super-harmonic resonance of a van der Pol oscillator with fractional-order derivative

Abstract

The dynamic characteristics and stability of a fractional-order van der Pol oscillator under simultaneously primary and super-harmonic resonance are analyzed by the multiscale method. At first, the first-order approximate analytical solution of the system is obtained, and the analytical results are verified by numerical simulation. In addition, the concepts of equivalent linear damping and equivalent linear stiffness of simultaneously primary and super-harmonic resonance are proposed to enhance the comprehension of fractional parameters roles. Based on Lyapunov's stability method, qualitative analysis is performed on the stability region and the type of equilibrium points of the steady-state solution. The influences of fractional order and coefficient on system stability, equivalent damping, and equivalent stiffness are discussed. It is found that the change of fractional order and coefficient will cause a deviation of the vibration curve and affect the multiple solutions, resonant frequency, and stability range of the system, which is useful to design or control the similar dynamic system.