Effects of Different Fast Periodic Excitations on the Pitchfork Bifurcation and Vibrational Resonance

We investigate the effects on the pitchfork bifurcation and the vibrational resonance of an overdamped bistable system subjected to both a slow harmonic excitation and a fast periodic excitation with different waveforms. We use numerical simulations along with theoretical explanations to analyze some interesting phenomena. The bifurcation configuration depends closely on the form of the fast excitation. As a result, we have found that the key factor to influence the bifurcation configuration is the symmetry property of the fast excitation. Further, due to the relationship of the vibrational resonance with the pitchfork bifurcation, the vibrational resonance also depends closely on the form of the fast periodic excitation. Moreover, for anharmonic fast excitations, if it is asymmetric, the vibrational resonance usually depends closely on the initial conditions.