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

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