Exponential stability

[1] A discrete-time input-to-output LTI system is exponentially stable if and only if the poles of its transfer function lie strictly within the unit circle centered on the origin of the complex plane.

An exponentially stable LTI system is one that will not "blow up" (i.e., give an unbounded output) when given a finite input or non-zero initial condition.

Now imagine giving the ball a push, which is an approximation to a Dirac delta impulse.

Drawing the horizontal position of the marble over time would give a gradually diminishing sinusoid rather like the blue curve in the image above.

A step input in this case requires supporting the marble away from the bottom of the ladle, so that it cannot roll back.

It will stay in the same position and will not, as would be the case if the system were only marginally stable or entirely unstable, continue to move away from the bottom of the ladle under this constant force equal to its weight.