Dynamical system example of an object that falls over when its COM falls outside its base of support? I am trying to understand in a rigorous way why an object falls over when its COM falls outside its base of support.
The following picture illustrates in crude terms why the center-of-mass causes an object to topple.

From what I can tell, an essential aspect of this seems to be the ease with which a given physical system acquires potential energy in such a way that it will change states. For instance, consider this example with a refrigerator

Notice, in Case A, the normal force and the force of gravity offset each other and consequently, the refrigerator remains at rest. In Case B, the refrigerator does have potential energy but it will likely fall back into its original position. By contrast, in case C, the refrigerator is tilted enough it will likely fall over and lay on its side and will not return to its original position.
These examples can also be seen in the following YouTube Video:

Source: https://www.youtube.com/watch?v=4rG9u478X1Q
Now, my issue is, I want an example that's a bit more rigorous. In particular, I would like an example of a system like the cars above but that is describable using dynamical systems terminology.
One simple example of a dynamical system I am familiar with is a pendulum. The phase space diagram for a dampened pendulum is as follows:

Can someone provide me with an example of a system like the refrigerator or the cars above that is describable using dynamical systems terminology?
 A: 
I am trying to understand in a rigorous way why an object falls over when its COM falls outside its base of support.

Please have in mind that you should actually consider the center of gravity (COG) - there is a subtle difference to the center of mass (COM), although for all practical purposes they can be considered equal. The center of gravity is a point where the (entire) pull exerted by the gravitational force on the object is concentrated. For objects of relatively small height, COG equals COM because acceleration due to the gravitational force $g$ does not change much. For typical skyscrapers the difference between COG and COM is in the range of centimeters.

Can someone provide me with an example of a system like the refrigerator or the cars above that is describable using dynamical systems terminology?

I am not sure what do you mean by "dynamical system terminology", but let me try to explain. It all comes down to torque.
There are two conditions for object to be in equilibrium: (i) net force on the object has to be zero, and (ii) object must not have tendency to rotate, i.e. for a nonrotating object to remain nonrotating the net external torque around any point on the object must be zero. Since gravitational force always acts downwards, this effectively means that the center of gravity must be directly above the support area; otherwise, it will produce net torque about closest support point and as a consequence the object will start rotating.
For the refrigerator example, net force in all three cases is obviously zero. But in cases (b) and (c), since COG is not directly above the support, there will be some torque about the support point and the refrigerator will rotate about that point in the direction determined by the torque.
A: 
I am trying to understand in a rigorous way why an object falls over
when its COM falls outside its base of support.

The short answer is the net torque acting about the contact point with the supporting surface causes the object to either topple or right itself. It is as simple and rigorous as that.
For the car on the right of Fig 2, the torque causes it to right itself. See the Fig below showing the torque causing the car to rotate clockwise about the point of contact of the wheel with the road, so that the car rights itself.
For the car on the left, although the tilt angle is the same as the car on the right, the higher location of the COG results in the line of action of the force of gravity acting to the left of the contact point rather than the right, resulting in a counter clockwise torque causing the car to topple.
The same applies to the examples of the bus and refrigerator.
Hope this helps.

