Way of representing the effects of time, the arbitrary and discrete
Tabular form of S.2/3 looking somewhat improper at first sight. He
Should notice, however, that the algebraic way is a restricted way,
Applicable only when the phenomena show the special property of
Continuity (S.7/20). The tabular form, on the other hand, can be
Used always; for the tabular form includes the algebraic. This is of
Some importance to the biologist, who often has to deal with phe-
Nomena that will not fit naturally into the algebraic form. When
This happens, he should remember that the tabular form can always
Provide the generality, and the rigour, that he needs. The rest of
This book will illustrate in many ways how naturally and easily the
Tabular form can be used to represent biological systems.
36
Of Ex. 3/6/7. If we take axes x and y, we can represent each partic-
Ular vector, such as (8,4), by the point whose x-co-ordinate is 8
And whose y- co-ordinate is 4. The state of the system is thus rep-
Resented initially by the point P in Fig. 3/10/l (I).
The transformation changes the vector to (6,6), and thus changes
the system’s state to P'. The movement is, of course, none other than
The change drawn in the kinematic graph of S.2/17, now drawn in a
Plane with rectangular axes which contain numerical scales. This
Two- dimensional space, in which the operands and transforms can
Be represented by points, is called the phase-space of the system.
(The “button and string” freedom of S.2/17 is no longer possible.)
37
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
TH E D ET ERM IN A TE MA C HI N E
In II of the same figure are shown enough arrows to specify
Generally what happens when any point is transformed. Here the
Arrows show the other changes that would have occurred had
Other states been taken as the operands. It is easy to see, and to
Prove geometrically, that all the arrows in this case are given by
one rule: with any given point as operand, run the arrow at 45 ° up
And to the left (or down and to the right) till it meets the diagonal
represented by the line y = x.
Ex.: Sketch the phase-spaces, with detail merely sufficient to show the main fea-
Tures, of some of the systems in S.3/4 and 6.
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What is a “system”? In S.3/1 it was stated that every real
Determinate machine or dynamic system corresponds to a closed,
Single-valued transformation; and the intervening sections have
Illustrated the thesis with many examples. It does not, however,
Follow that the correspondence is always obvious; on the contrary,
Any attempt to apply the thesis generally will soon encounter cer-
Tain difficulties, which must now be considered.
Suppose we have before us a particular real dynamic system—
A swinging pendulum, or a growing culture of bacteria, or an auto-
Matic pilot, or a native village, or a heart-lung preparation— and
We want to discover the corresponding transformation, starting
,from the beginning and working from first principles. Suppose it
Is actually a simple pendulum, 40 cm long. We provide a suitable
recorder, draw the pendulum through 30 ° to one side, let it go, and
Record its position every quarter-second. We find the successive
deviations to be 30 ° (initially), 10 °, and – 24° (on the other side).
So our first estimate of the transformation, under the given condi-
Tions, is
30 ° 10 °
10 ° – 24°
Next, as good scientists, we check that transition from 10 °: we
draw the pendulum aside to 10 °, let it go, and find that, a quar-
ter-second later, it is at +3 °! Evidently the change from 10 ° is not
Single-valued— the system is contradicting itself. What are we to
Do now?
Our difficulty is typical in scientific investigation and is funda-
Mental: we want the transformation to be single-valued but it will
Not come so. We cannot give up the demand for singleness, for to
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