Here is the toy in question (Drinking bird - Wikipedia)
Given all the physical characteristics like toy dimensions, room temperature, fluid characteristics etc. what is (theta) as a function of time ? I am asking for the mathematical formulation (I have a fairly good idea of what theta will depend on). Assume theta is 0 when the toy is vertical.
Bonus question : what’s the time interval between each dip of the beak ?
FYI - I am looking for the equations of motion. For example look at the equations of motion for an inverted pendulum here : Inverted pendulum - Wikipedia
I may be wrong, but it sounds like someone needs ‘help with their homework’. 
This is NOT a simple mechanics problem. You’ll need to model evaporative cooling, heat flow, thermal expansion, and fluid mechanics in addition to your inverted pendulum problem. Good luck with that!
Yeah - but most of the phenomenon have been modeled 50 years or more back. So there’s no models ?
Of course there are models. And they all interact in a complicated fashion. You could model it, with accurate models of all the effects.
It would be far easier for you to buy one, set it up, video tape it, and then plot theta as a function of time. It will be a very complicated, non-periodic function. ETA: You could probably find videos on the web. Give that a try and see what you come up with.
This Article seem to contain the answer to your questions:
J. Guemez et al. , Experiments with the drinking bird,Am. J. Phys., Vol. 71, No. 12, pp 1257-1263
Thank you Anaglyph. Superb!! that’s what I was looking for.
There will be non-periodic transients initially, but they’ll die out (probably exponentially) leaving a purely periodic motion in the long run.