When the dynamic system can vary continuously, small dis-
Turbances are, in practice, usually acting on it incessantly. Elec-
Tronic systems are disturbed by thermal agitation, mechanical
Systems by vibration, and biological systems by a host of minor
Disturbances. For this reason the only states of equilibrium that
Can, in practice, persist are those that are stable in the sense of the
Previous section. States of unstable equilibrium are of small prac-
Tical importance in the continuous system (though they may be of
Importance in the system that can change only by a discrete jump).
The concept of unstable equilibrium is, however, of some theo-
Retical importance. For if we are working with the theory of some
Mechanism, the algebraic manipulations (S.5/3) will give us all
The states of equilibrium— stable, neutral, and unstable— and a
Good deal of elimination may be necessary if this set is to be
Reduced to the set states that have a real chance of persistence.
Ex.:Make up a transformation with two states of equilibrium, a and b, and two
Disturbances, D and E, so that a is stable to D but not to E, and b is stable to
E but not to D.
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A N I N T R O D UC T I O N T O C Y B E R NE T I C S
STA BI LIT Y
The subject soon becomes somewhat mathematical; here it is suf-
Ficient to notice that these questions are always capable of being
Answered, at least in principle, by the process of actually tracing
The changes through the states D(a), TD(a), T2D(a), etc. (Com-
Pare S.3/9). The sole objection to this simple, fundamental and
Reliable method is that it is apt to become exceedingly laborious
In the complicated cases. It is, however, capable of giving an
Answer in cases to which the more specialised methods are inap-
Plicable. In biological material, the methods described in this
Chapter are likely to prove more useful than the more specialised;
For the latter often are applicable only when the system is contin-
Uous and linear, whereas the methods of this chapter are applica-
Ble always.
A specially simple and well known case occurs when the sys-
Tem consists of parts between which there is feedback, and when
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This has the very simple form of a single loop. A simple test for
Stability (from a state of equilibrium assumed) is to consider the
Sequence of changes that follow a small displacement, as it travels
Round the loop. If the displacement ultimately arrives back at its
Place of origin with size and sign so that, when added algebra-
Ically to the initial displacement, the initial displacement is dimin-
Ished, i.e. is (commonly) stable. The feedback, in this case, is said
To be “negative” (for it causes an eventual subtraction from the
Initial displacement).
The test is simple and convenient, and can often be carried out
Mentally; but in the presence of any complications it is unreliable
If carried out in the simple form described above. The next section
Gives an example of one way in which the rule may break down if
Applied crudely.
Ex. 7: A bus service starts with its buses equally spaced along the route. If a bus
Is delayed, extra passengers collect at the stopping points, so it has to take
Up, and set down, more passengers than usual. The bus that follows has fewer
Passengers to handle and is delayed less than usual. Are irregularities of
Spacing self-correcting or self-aggravating?
Ex. 8: What would happen if an increase of carbon dioxide in the blood made the
Respiratory centre less active?
Ex. 9: Is the system x'= 1/2y, y' = 1/2 x stable around (0,0)?
Positive feedback. The system described in the last exercise
Deserves closer attention.
From (10,10) it goes to (5,5)
,, (10,12) ,, ,, ,, (6,5);
So an increase in y (from 10 to 12) leads to an increase in x (from
To 6). (Compare S.4/13.) Similarly,
From (10,10) it goes to (5,5)
,, (12,10) ,, ,, ,, (5,6)
So an increase in x (from 10 to 12) leads to an increase in y (from
To 6). Each variable is thus having a positive effect on the other
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And if the system were discussed in plain words these facts might
Be used to “prove” that it is unstable, for a vicious circle seems to
Be acting.
The system’s behaviour, by converging back to (0,0), declares
Indisputably that the system is stable around this state of equilib-
Rium. It shows clearly that arguments based on some short cut, e.g.
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