So when i give the pet fresh water in a stainless steel bowl that i place on a mat according to the attached picture, from $t=0$ the bowl is at rest, the water normally oscillates in the bowl like a pendulum following the action, but after one or two oscillations, the water seems to gain in momentum and starts to overflow from the bowl. why?
(bowl is a body of revolution)
A guess. Perhaps there are multiple modes present. It takes a while for them to have an interference maximum at the edge of the bowl.
If the mat is thick and soft enough to be compliant, then it will couple with the moving mass of the dish and water and allow the dish to tip back and forth, causing the spillage.
This can be directly tested by setting the bowl down on the bare floor instead and seeing if it spills.
Also, try this: carefully place a full water bowl down on top of the mat after first allowing all the sloshing to die down. then press down on the rim of the bowl to tip it slightly and release. Note the sloshing. press briefly down again, timing the press to add to the slosh, and release. With a little care you should be able to reproduce this effect manually, furnishing evidence that the bowl is tipping along with the sloshing water because of the mat's springiness.
Be sure to report back here to share your results!
The two horizontal sloshing modes could have slightly different frequencies (e.g. if the rotational symmetry is not perfect). If you then start with the combination of both, but out of phase, which is a circular sloshing pattern, then sometime later you will have both in the same phase which is diagonal sloshing, with $\sqrt 2$ times the original amplitude if no energy has been lost.
Of course the initial movement does not have to be exactly circular, which would probably be noticed immediately. But if there only is some circular component in it, this can still lead to increasing amplitude, just not by the full factor $\sqrt 2$.
And the symmetry breaking could come from the mat below it, which might have different compliance in orthogonal directions caused by the way it is manufactured.
The reason may be simple, because your container not only has angular movement, initially T-2 tilts forward, then T-1 tilts backward, but also horizontal movement from right to left during this process. This way, water definitely has momentum. So at T-0, in fact, your container is tilted backwards, so water will not overflow, but when the container is level, it may overflow. You can use a transparent container and conduct the experiment using a camera. This will enable more accurate observation.
If you check the tide chart for your local area you will find that the bowl overflows only during the incoming tide and is directly proportionate to what stage the tide is at when you set the bowl down. As the mean high tide approaches, the swirling water in the bowl will continue to rise as well. An outgoing tide will have the opposite effect, negatively impacting the water level and causing each swirl around the bowl to be at a lower level than the previous one. Additionally, a waxing moon phase increases water level even further regardless of tidal stage. Conversely a waning moon will negatively affect the swirling water level in the bowl during all tidal stages.
