The pitchfork bifurcation and vibrational resonance in a quintic oscillator

Abstract

The pitchfork bifurcation and vibrational resonance are investigated in this paper. Based on the method of separating slow motion from fast motion, the equivalent equation to the slow motion is obtained. Then, the pitchfork bifurcation is studied. The results show that the amplitude of the high-frequency signal can induce the subcritical pitchfork bifurcation, while both the frequency of the high-frequency signal and the value of the fractional-order can induce supercritical pitchfork bifurcation. The pattern of the vibrational resonance depends on the pitchfork bifurcation. The vibrational resonance presents double-resonance pattern when the pitchfork bifurcation occurs. Or else, the vibrational resonance presents single-resonance pattern. The analytical predications are in good agreement with the numerical calculation results, which verifies the validity of the theoretical results.