Nonlinear mechanical resonators display rich and complex dynamics and are important in many areas of fundamental and applied sciences. Here, we present a general strategy to generate mechanical nonlinearities for trapped particles by transverse driving in a funnel-shaped potential. Employing a trapped ion platform, we study the nonlinear oscillation, bifurcation, and hysteresis of a single calcium ion and demonstrate a 20-fold enhancement of the signal from a zeptonewton-magnitude harmonic force through the effect of vibrational resonance. Our results represent a first step in combining the rich nonlinear dynamics with the precision control over mechanical motions offered by atomic physics and open up possibilities for exploiting nonlinear mechanical phenomena in the quantum regime.