The dripping faucet is a classic example of chaos in everyday life.

What the OP describes is possibly the first step in a complex bifurcation route to chaos, where

As the flow speeds up, the dripping pattern changes abruptly at certain thresholds, their mathematics predicts. For slow speeds, there is regular, metronome-like dripping. Then the sequence splits, or 'bifurcates': instead of repeating every drip, it repeats every other drip. For instance, the researchers saw this bifurcation experimentally too, as an alternation between two drops of different size.

The system has been studied since at least the 1980s and its details are pretty complicated, because fluid dynamics plays such a decisive role in it. The specific shape of the drop (e-print), as well as its vibrations (-eprint) are also important.

Also, the system is not only chaotic, it also displays hysteresis, i.e.,

in some ranges of flow speed, the behaviour differs according to whether the speed is increasing or decreasing -- whether the 'tap' is being turned on or off. The tap, in other words, remembers its own 'history'.

The dripping faucet is a classic example of chaos in everyday life.

What the OP describes is possibly the first step in a complex bifurcation route to chaos, where

As the flow speeds up, the dripping pattern changes abruptly at certain thresholds, their mathematics predicts. For slow speeds, there is regular, metronome-like dripping. Then the sequence splits, or 'bifurcates': instead of repeating every drip, it repeats every other drip. For instance, the researchers saw this bifurcation experimentally too, as an alternation between two drops of different size.

The system has been studied since at least the 1980s and there's quite some literature on it so, yes, there are quantitative models which can predict its behavior - though I'm not aware of any for the OP's specific configuration of an overflowing container.

Its details are pretty complicated, because fluid dynamics plays such a decisive role in it. For instance, the specific shape of the drop (e-print), as well as its vibrations (e-print) are also important.

Also, the system is not only chaotic, it also displays hysteresis, i.e.,

in some ranges of flow speed, the behaviour differs according to whether the speed is increasing or decreasing -- whether the 'tap' is being turned on or off. The tap, in other words, remembers its own 'history'.