Balancing act 2

We have a pendulum made of a rigid, massless rod of length $$l$$ and a small ball of mass $$m$$ attached to the end of the rod. The period of small amplitude oscillations is $$T_0= 2 \pi \sqrt{\frac lg}$$.

The pivot point (axis of rotation) of the pendulum is not fixed, but it is attached to a mechanism that shakes the pivot point in the vertical directions with amplitude $$a\ll l$$ and period $$T\ll T_0$$. (For $$a=0$$ we get back the traditional pendulum.)

We set up the rod such that it is in the vertical, unstable position with the ball right above the pivot point. When $$a=0,$$ this position is unstable; for any small error in positioning the ball, the pendulum will turn over.

Is it possible that if we switch on the shaking $$a\ne 0,$$ the position becomes stable and the ball will remain over the pivot point indefinitely?

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