Primary and super-harmonic simultaneous resonance of van der Pol oscillator

Abstract

In this paper the stability and the dynamic behavior of primary and super-harmonic simultaneous resonance of van der Pol (VDP) oscillator are researched. At first, the first-order approximate analytical solution is obtained by the method of multiple scales, and the correctness and satisfactory precision of the approximate solution are verified by comparing with the numerical solution. Moreover, the amplitude–frequency and phase-frequency equations of steady-state solution is derived from the approximate analytical solution, and the stability condition is discussed based on Lyapunov theory. Then the stability region and the types of equilibrium points are qualitatively analyzed and the result shows that the amplitude of the super-harmonic excitation is the major contributor to the stability and the types of equilibrium points. Finally, the periodic motion and chaos of VDP oscillator are analyzed numerically. It is found the VDP oscillator with small primary excitation amplitude and large super-harmonic excitation amplitude will generate complex dynamic phenomena including periodic and chaotic motions, which are certified by bifurcation diagram, largest Lyapunov exponent and power spectrum. These results are helpful to the analysis and design of the similar systems.