Indices of the Stability of Dynamic Systems
In the previous pieces of this series of articles I added new concepts to already existing groups of concepts. In this writing and in the ones to follow, however, I will introduce a hitherto non-existent group of concepts and will expound on the components of this group of concepts that have also been introduced by me.
A novelty in my 1963 study referred to in the first piece of the series was the attempt to make it possible to compare dynamic systems from the point of view of their stability. (That study however failed to get published through fault of the institution I was working for at the time-a fault acknowledged by them since-and so the profession never got to know about it. I already mentioned this in the first piece.) To the best of my knowledge, that which is contained in the study in question was not only something new at the time but remains so today.
The examination of dynamic systems practically is restricted to giving qualitative features. It establishes whether
– a system is stable or not,
– and perhaps defines the boundaries of the stable range.
The above is all that is looked at.
The methods applied so far have not answered the following questions: – How stable a system is;
– Of more than one system, which is more stable.
The evaluation of a dynamic system and the comparison of two or more dynamic systems on the basis of their quantitative features must, similarly to any other comparison, be based on the application of indices.
For this reason, I have introduced a few indices for the characterization of the size of stability. It is these this writing is about.
1 Indices of singular formations
1.1 Size of the stable range
1.1.1 Absolute size (SA)
Definition: The absolute size of the stable range is the n-dimensional ‘volume’, that is, the space it contains, of the stable range in the n-dimensional phase space.
Note: To compare dynamic systems, the topology of the systems to be compared must be given. Topology here includes that of a singular formation as well, and also its location within the range. It has no significance however where the systems to be compared are located in a system of coordinates, so they can be moved freely within the given system of coordinates.
Systems of the same topology are comparable, as are those of different topologies where one of the systems contains the other, and the contained system can be obtained by cutting it out from the one containing it.
Thus, the two systems shown in Figure 1 and 2 are comparable, but the ones in Figure 3 and 4 are not. (In the figures, the points marked “S” are singular points.)
Thesis: In case SA1 and SA2 meet a precondition for comparability and SA1 > SA2, then the first is the more stable of the second – to the same extent as it is bigger, but I have not dealt with the meaning of this.
One thing coming from the thesis: systems difficult to define can be examined if
– a system is found which contains the examined system and the ‘volume’ of which can be defined;
– also, a system is found which is contained in the examined system and the ‘volume’ of which can also be defined.
In such a case these two systems give, from the point of view of stability, the lower and upper limits of the examined system. If the two limits are close to each other, the definition will be sufficiently precise.
Shown in Figure 5 is the making of such a limit. The section of circle in green is the examined system, the black circle shows the lower limit, while the circle comprised of the green and red sections shows the upper limit. The singular point of each of these is the origin.
1.1.2 Relative size (SR12)
Definition: the relative size of the stable range is the quotient of the n-dimensional ‘volume’ of the stable range of system 1 and the n-dimensional ‘volume’ of the stable range of system 2 in the n-dimensional phase space.
The formula is the following: SR12 = SA1 / SA2
The preconditions mentioned in point 1.1.1 also holds true here.
Its applicability is the same as that of the absolute size.
1.1.3 Their applicability as indices of stability
Defining the ‘volume’ of n-dimensional figures belongs to another branch of mathematics and therefore it is not my task to define ‘volume’. These indices can be applied for those figures in the case of which the ‘volume’ can be defined.
2 Reserve stability vector
The methods used to examine stability do not say anything about whether
– it is easy or difficult, and to what extent, to dislocate a point from within the stable range;
– to what extent it is difficult to take a point from the non-stable range into the stable range.
All this however would be highly important information.
Let us take a simple example:
Figure 6 shows a pit, at the bottom of which is the examined point.
– In Figure 6a, the rim of the pit is near,
– in Figure 6b, it is far.
In Figure 6a, the way to the rim of the pit is longer than in Figure 6b.
Also important is the proportion of this distance to the distance between the rim and the deepest point, which is the singular point. It is this that shows which fraction of this side of the pit the point is in.
Figure 7 shows the rim of the pit.
In Figure 7a, the rim rises above the bottom only slightly, while in Figure 7b it rises above the bottom markedly.
In the case of Figure 7a, it is much easier to get from the pit onto the rim than in that of Figure 7b.
Here it is also important to know which fraction of the difference in level between the very bottom and the rim of the pit the point is in. It is this that shows what proportion of the ‘potential energy’ is still required to get onto the rim. (Inverted commas are used here because although in the present example potential energy is indeed involved, generally speaking it is the dependent variable of the n-dimensional phase space that is involved.)
However, each piece of information is important.
2.2 Reserve stability vector
Definition: A reserve absolute-stability vector (S) is a vector which points from the examined point towards that point of the boundary of the stability range which the examination concerns, and the absolute value of which is the index of the distance.
Experiment: No experiment has been conducted. Getting the signed distance between the point from which the ball would roll off and the point from which it would roll from the rim to this point would serve as an experiment.
Relative stability reserve (rs) is the quotient of the absolute value of the reserve stability vector and the absolute value of the radius vector belonging to that point of the stability range which the examination concerns, and which point was referred to above, in connection with the reserve stability vector.
Experiment: No experiment has been conducted here either. The only difference from the above is that the distance calculated there has to be divided by the distance between the bottom of the pit and the point of the rim mentioned there.
S = rps – rp
rs = ABS(S)/ABS(rps)
where rps is the radius vector of the point from which it gets from the boundary of stability to the examined point, rp is the radius vector of the examined point.
2.3 Stability function vector
2.3.1 Definition: Absolute-stability function vector (Sr): The component of the absolute-stability reserve vector located in the direction of dependent variable in the n-dimensional phase space.
The formula is the following: Sn = Sy
Experiment: No experiment has been conducted here either. It differs from what is written in connection with the absolute-stability vector that instead of distances, heights measured from a given point are involved here.
Relative-stability function (rsn): The absolute value of the component of the relative-stability reserve located in the direction of the dependent variable.
The formula is the following: rsn = rsy
Experiment: The difference between this and the previous one is the same as in the case of the relative-stability reserve.
With the application of these indices more reliable physical systems can be planned, as it makes it possible to establish to what extent the planned system will be sensitive to interferences, what degree of interference it will be able to cope with.
First of all, the location and shape of the boundary of the stable range (in experiments, this is the rim) has to be defined. (In literature, it is called separatrice. It is indeed a separator, as it separates the stable range of the singular formation and the limit cycle from the instable range.) No method applicable for all cases exists.
The method based on the preconditions of stability worked out by me is in many cases suitable for this purpose too, if the satisfaction of the preconditions of stability (distance and directions from the singular formation and the limit cycle) are examined.
All this is made simpler by the fact that it is not necessary to find the entire separatrice; it is enough to find its point expounded on below.
3.3 A property of reserve functions
Are the two reserve functions way functions or state functions?
In the case of ‘potential energy’, it seems easy to answer this question. For, as is well known, real potential energy is a state function, and here the reserve depends solely on the value of the dependent variable of the n-dimensional phase space. The question is, on the value taken at which point?
The answer seems even easier in the second case, as here a way is involved in the first place. Which way it is, is not important from this point of view. Whichever way it is, the correct answer is way function.
If in the experiments a point is chosen in the sidewall of the pit and the ball is pushed there in the direction of the bottom of the pit, it will roll down into the bottom in the shortest way. In case the push done with the same amount of strength is sidewise, the ball will get into the bottom moving sideways and downwards (doing perhaps one or two circles meanwhile.) This way will surely be longer than the shortest one. However, from the same point it will get to the same level. The ‘potential energy’ of the starting point and the endpoint will therefore be the same.
Thus everything is known so far,
It is not known, however, where the ball would have got from to the point of departure if it had been sent off from the rim.
In case the rim, that is, the boundary of stability is of the same height everywhere, the ‘potential energy’ is the same everywhere. Thus from the point of view of ‘potential energy’ in a system like this it does not matter which point of the rim the ball would depart from. This is not the case however if the different points of the rim are at different ‘heights’. In this case the ‘potential energy’ is also dependent on the way the ball travels, that is, it is a way function.
Therefore, to calculate the proportions it is necessary to define the point of the rim from which the ball would have rolled into the bottom of the pit through the examined point. This means that the trajectory (the image of the course of change in the n-dimensional phase space) which starts from the rim, goes through the examined point and reaches the bottom of the pit (the singular formation and the limit cycle) has to be found.
For this to be done, it is necessary to travel along the trajectory back in time until either the boundary of the stable range or the negative infinite is reached, and forward, until the singular formation and the limit cycle is reached.
In the possession of the figures relating to the singular formation and the limit cycle, this system of differential equations can be done by moving from point to point on a computer and examining point by point the satisfaction or non-satisfaction of the preconditions of the trajectory and stability. (The method does not differ from the one developed in 1963, but the enormous development in information technology since then makes a much more widespread application possible.)
Dr. Simonyi Endre