Correlation between qualitative changes and time and space
In the first piece of the series the correlation between qualitative chamges and stability reserve was defined. Let us now have a look at the relationship between these on the one hand, and between qualitative changes and time and space on the other.
It will be demonstrated in a group of three pictures.
Picture one shows an island with a hill on its right-hand side. Picture 2 is of the same island, but only with the hill emerging from the water. In picture 3 the whole island is under water.
1 Patterns in time
Change in the island’s position in time is described by the following equation: dh/dt = c(t), with h being the height in relation to water level, t being time and c(t), some variable dependent on time.
The stability reserve is the value of “h”. Thus, a qualitative change takes place only if “h” changes sign. (In our case, the islamd “is born” or “ceases to exist”, that is, it emerges above water level or submerges under it.)
1.1 Let us say c = const.
Then h = h0 + ct with h0 being the level of the island at the point in time of t = 0
1.1.1 Let us say
h0 < 0, which is the case shown in picture 3.
126.96.36.199 Let us say
c 0, that is, the island is emerging. As a consequence, first the island emerges from the water only partially ( picture 2), then fully (picture 1) from the water.
That is, a qualitative change takes place.
1.1.1 Let us say
h0 > 0, that is, at the beginning there is an island (picture 1).
188.8.131.52 Let us say
c 0, that is, the island is emerging.
Thus the island gets higher and higher, but nothing else happens. That is, the quantitative change does not lead to a qualitative change.
1.1 A more general case
Let us say c = c0 – c1t
In this case h = h0 + c0t – 0.5c1t2
Here we look only at the case where h0 < 0, hm > 0, with hm being the maximum value of “h”. In another form, this is c0 > SQR(-2h0c1), with SQR being the taking of the square root.
In the case examined, first the island does not emerge from under the water (picture 3), then it does more amd more (picture 2 then picture 1). So, a qualitative change has taken place.
As opposed to the previous cases, however, the process does not end here but reverses, and the island gradually submerges. Thus, another qualitative change occurs.
This is what happens to islands that come into being in the wake of volcanic activity, then,, following the cecassion of that activity, are gradually destroyed. (A good example is Hawai, where following a volcanic eruption a new sland builds up gradually, and when the esrth turns away and the volcano “moves on”, the new island is gradually destroyed ny the ocean.)
1.1 An even more general case
Now let us say c = – ac0SIN(c0t + b), with “a”, „”b”, „”c0” being positive constants, and SIN being the so-called main value of the sinus function.
In this case h = aCOS(c0t + b), h0 = aCOS(b)
Thus at the beginning the island can be under the water (picture 3) if COS(b) 0.
With the passing of time, the value of c0t + b increases, and when the case becomes c0t + b = 0.5(2k + 1)PI the COS – function will change sign. (Here, „k” – 0, 1, 2…; and PI is the well-known Ludolf number.)
So, the process reverses, and the island, up to now emerging more and more, starts to submerge, and vice versa. This going on, the process will reverse again.
Thus, depemdent on he values of the comstants, the followimg can be the cases:
1.3.1 The island always remains under the water, only the depth of the water above it changes. So, there is no qualitative change. 1.3.2 The island is always above the water, only its height changes. So, there is no qualitative change here either 1.3.3 Finally, there is the case of the island now emerging from, now submerging under the water, which pattern keeps repeating. Thus here a qualitative change keeps repeating now with a positive, now with a negative direction.
I mentioned above that this is a case even more general than the previous one. This holds true not only for the form of the equation, but also for the frequency of its occurance. For this is what happens in the case of many islamds as a result of the cyclic changes of low tide and high tide.
With the phenomenon of low tide-high tide, it was not the island itself that got higher or lower but the level of the water surrounding it. As the value of “h” was given as the the height relative to water level, it did not matter if it was the island or the water level that moved up and down.
1 Pattern in space
If in the case last examined above the whole of the island and the hill area are examined separately, there can be differences.
It can be the case that no qualitative change takes place regarding the hill because it is always above water, while the other parts of the island are already under water, or only the hill emerges from the water, but the other parts of rhe island do not.
To sum up the effects of the two variables, there are systems the features of which change in time and / or space, and there are ones whose features do not. ( Here, “system” also includes its environment.) A change in features changes the the value of the stability reserve, and this, in case of a change of sign, leads to a qualitative change.