The equations of motion for a bead on a smooth, freely rotating rod with unit length are
$$ \left\{\begin{aligned} 0&= \omega_0^2\sin\theta - x\dot{\theta}^2+\ddot{x}\\ 0&=3\omega_0^2\cos\theta (1+2\mu x)+ 12\mu x \dot{x}\dot{\theta}+2\ddot{\theta}(1+3\mu x^2) \end{aligned}\right. $$
Is it possible to tell without solving the equations what parameters $\{\omega_0,\mu\}$ and initial conditions $\{x(0), \dot{x}(0),\theta(0),\dot{\theta}(0)\}$ would lead to a bounded $x(t)?$
I tried to solve the equations numerically. With some $\{\omega_0,\mu\}$, $x(t)$ would oscillate for a while. But it would always end up diverging with a long enough time scale.
Linear divergence at $\omega_0=10^{-3},\mu=10,\theta'(0)=-2,x(0)=x'(0)=\theta(0)=0$
Technically the solution is meaningless for $x(t)>1$ since the length of the rod is $1$.