# Impact of parametric excitation on vibrational resonance in a symmetric triple-well potential system

### Abstract

We analyze the impact of parametric excitation on vibrational resonance (VR) in a symmetric triple-well potential system driven by a biharmonic external force with two widely different frequencies $\omega$ and $\Omega$ ($\Omega \gg \omega$). Our analysis is limited to the following parametric choices: (i) $\omega_0^2, \gamma > 0$, $\beta < 0$, and $\beta^{2} = \frac{16}{3}\omega_{0}^{2}\gamma$ (type-1 triple-well), (ii) $\omega_0^2, \gamma > 0$, $\beta < 0$, and $4\omega_{0}^{2}\gamma < \beta^{2} < \frac{16}{3} \omega_{0}^{2} \gamma$ (type-2 triple-well), and (iii) $\omega_0^2, \gamma > 0$, $\beta < 0$, and $\beta^{2} > \frac{16}{3}\omega_{0}^{2}\gamma$ (type-3 triple-well). Using an approximate method involving direct separation of timescales and flow equations, we derive the equation of slow motion and obtain the response amplitude. Theoretical results are validated through numerical simulations. In addition to uncovering the existence of biharmonic force-induced VR, our findings demonstrate that parameters of parametric excitation exert a significant influence on VR, enabling either suppression or modulation of resonance peaks and, consequently, control over resonances. Furthermore, we explore the effects of biharmonic force and parametric excitation on various attractors of the triple-well system, presenting appropriate bifurcation diagrams, phase portraits, and time series plots.