Stability as a state of equilibrium in a phase-space
Ashby[1] gives some examples of "stability". In the first figure 
below, the cone, the sphere, and the cube are considered as being in 
a state of equilbrium, but in terms of "stability", the cone 
is "unstable", the sphere is "neutral", and the cube is "stable".

In the central figure, the card can be called stable along one
dimension while along another dimension it is "unstable" in terms
of its simultaneous states of equilibrium.

 

In the lower figure, there is a dualism. While it can be said that
the sphere is unstable because it has rolled away from the ridge,
it can also be said that it is stable because it is rolling towards
the trough. If friction is considerably large, momentum may be 
neglected, and this half-empty/half-full dualism may be 'resolved' 
by considering only the surface, in which case as Ashby writes:
  "...the system is composed of a single variable 'the distance of 
   the ball laterally' is absolute and has a definite, permanent field...".

The point 'A' becomes a convergent position of behaviour
and the point 'B' becomes a divergent position of behaviour and
the property of stability is seen as not belonging to the sphere
but to the surface of behaviour (which Ashby refers to as a field
which he defines as: "the phase-space containing all the lines 
of behaviour found by releasing the system from all possible 
initial states" ).
    

[1] W. Ross Ashby, "Design for a Brain",1952 John Wiley and Sons, Chap. 4


 
Commentary:

Stability in Space-time

Spatial Extrema
   An infinite line, is inherently stable and has no "center of gravity".
   Every point on the infinite line is "stable". And infinite line
   is a random access memory.
  
   A point in space, is inherently "neutral" with regards to the 
   concept of "stability". You cannot "balance" a point; the concept
   of "balance" for an infinitesimal point has no meaning.

Temporal Extrema
   An infinite duration of time, is inherently static and all points
   upon it may be arbitrarily labelled "the present". An infinite
   duration of time is a random access history.

   An infinitesimal point in time, is "neutral" with regards to the 
   concept of "the present". The infinitesimal point in time has no 
   "past" or "future" from to distinguish a "present" from. 

Space-Time
   Perspective geometry on the infinite line may determine a superposition
   of mass values from their distribution on a infinite line which gives a 
   local illusion of a "center of gravity". In the same manner, a perspective
   on the infinite time line may determine a local illusion of "the present"
   via a superposition of local time distributions. This is a "dynamic" or
   unstable model of space-time.

   A more "absolute" determination of a "center of gravity" and "the present"
   may be derived from treating the measurement as a filter rather than
   as a reflection, as above. In such a case, an interferometer may be 
   employed to survey a local space-time of finite bounds as a "static" 
   or stable model of space-time. 

 Rotational Stability

   TBC