Special Shapes and Their Stability

1 Singular points

Definition: Those points in the n-dimensional phase space in which each of the differential equations describing the motion of the point is zero are called singular points. Where a point is stationary it is steady, that is, a singular point corresponds to a steady state or equilibrium.

Experiment: There is no separate experiment here, but the peak of the hilltop and the lowest point in the bottom of the pit to be seen at the stable and unstable singular points are of this kind.

Explanation: In case the ball is not moved it will stay at this point. At other points however it will roll away without being moved, so these points are different from the others. They are singular.

2 Singular formations

No literature reference can be given for this point, as this notion was introduced by me in a paper of mine written in 1963 but never published by my institute. (In 2010 the institute acknowledged officially that the paper remained unpublished through their fault.) For this reason, it is published here as taken from that paper.

Definition: A singular formation (generalisation of the singular point) is the geometric place in the n-dimensional phase space of the points which make a closed formation and in which the value of each of the equations in the equation system consisting of n-differential equations describing motion in the phase space is zero. (A zero-dimensional case of this is a singular point.)

Experiment: There is no separate experiment here. Same applies for this as for the above.

Explanation: Neither the hilltop, nor the bottom of the pit is a point, but both are an area with length and width. (It will be ignored here that they themselves also have small hillocks and pits, as the fact that the ball is stationary here is not affected by these.) So, these are singular formations in a three-dimensional phase space, and are approximately plains.

3 Stable singular points

Definition:

Stability according to Liapounoff: A state of equilibrium is stable whenever given any region “e” containing it there is another “d(e)” in “e” such that any motion starting in the region “d” remains in the region “e”.

Asymptotic stability: Namely, for all initial conditions, the system, after a sufficiently long period of time, will return as close as one wishes to the state of equilibrium. Such a stability, for which the initial deviations do not increase but on the contrary dampen, will be called asymptotic stability.

It is also written down in the unpublished paper referred to above that all that applies for the stability of singular points applies for the stability or instability of singular formations as well.

Experiment: Placed in the bottom of the pit, the ball remained stationary. When let go on the sidewall of the pit, it rolled down to the bottom.

Explanation: Wherever the ball was let go on the sidewall of the pit, it would always roll down to the bottom. So, this singular point is stable, or, to use another term, is an attractor.

4 Unstable singular points

Definition: Those singular points for which the condition of stability does not apply are called unstable.

Experiment: Placed on the hilltop, the ball remained stationary. When put anywhere else on the hill, it rolled down the hill.

Explanation: There were not any points on the hill from which the ball, when let go, rolled up to the hilltop. So this is an unstable singular point, or a repellent, to use another term. (How easy to understand everything so far, is it not? After all, everybody knows that an object let go on a slope will roll downhill and not uphill. And it is easy to learn this is what it is called.)

5 Stable depending on coordinate values

For the same reason as above, no literature references can be given here either. Similarly, it is now taken from the unpublished paper.

Definition: With the exception of certain coordinate values of the n-dimensional phase space, it is stable in a part of the space around the singular formation.

Experiment: Part of the sidewall of the pit was formed in such a way that the ball was not able to go through it. So, when the ball started its journey form a point higher than that area and rolled towards it, it stopped there and did not make it to the bottom of the pit.

Explanation: The ball rolls to the bottom of the pit, that is, to the singular point, from every point from which it does not get to the area in question. Regarding these points therefore the bottom is a stable singular point. Regarding those points from which it gets to that area, however, the bottom is an unstable singular point, as the ball stops at that area and does not make it to the singular point.

6 Global stability

Definition: In case a singular point’s stable domain is of infinite dimension regarding all the coordinates in an n-dimensional phase space, it is called globally stable.

Experiment: No experiment could have been carried out, as it would have required the building an infinitely big hill and pit.

Explanation: It is easy to see that nothing demonstrated earlier is different in any respect just because everything is extremely huge here.

7 Remark

Global stability can be related to the coordinate-values-dependent stability or instability of the singular formation (and of the singular point of course) examined. Thus, a singular point can be locally stable but globally stable only depending on the coordinate values. (This applies for the experiment in point 5. For, when the ball started its journey from a point lower than the hurdle, it reached the bottom of the pit from every direction, thus in this case it is stable. However, if the ball starts from a higher point, the phenomenon described in point 5 will take place. Therefore, if the sidewall of the pit were infinitely high, the ball would not make it to the bottom of the pit from those directions, just as it did not in the experiment in point 5.

Literature:
– Andronov-Haikin-Vitt: Oscillation Theory, 1959, Moskow

Dr. Endre Simonyi