Not have to be numerical to be well defined, is given by certain
Forms of reflex animal behaviour. Thus the male and female
Threespined stickleback form, with certain parts of their environ-
Ment, a determinate dynamic system. Tinbergen (in his Study of
Instinct) describes the system’s successive states as follows: “Each
Reaction of either male or female is released by the preceding reac-
Tion of the partner. Each arrow (in the diagram below) represents a
Causal relation that by means of dummy tests has actually been
Proved to exist. The male’s first reaction, the zigzag dance, is
Dependent on a visual stimulus from the female, in which the sign
Stimuli “swollen abdomen” and the special movements play a part.
The female reacts to the red colour of the male and to his zigzag
Dance by swimming right towards him. This movement induces
The male to turn round and to swim rapidly to the nest. This, in turn,
Entices the female to follow him, thereby stimulating the male to
Point its head into the entrance. His behaviour now releases the
Female’s next reaction: she enters the nest.... This again releases
The quivering reaction in the male which induces spawning. The
Presence of fresh eggs in the nest makes the male fertilise them.”
Tinbergen summarises the succession of states as follows:
26
Appears
Courts
Follows
Female
Enters nest
Spawns
Leads
MaleShows nest entrance
Trembles
Fertilises
Zigzag dance
He thus describes a typical trajectory.
Further examples are hardly necessary, for the various branches
Of science to which cybernetics is applied will provide an abun-
Dance, and each reader should supply examples to suit his own
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Speciality.
By relating machine and transformation we enter the discipline
That relates the behaviours of real physical systems to the proper-
Ties of symbolic expressions, written with pen on paper. The
Whole subject of “mathematical physics” is a part of this disci-
Pline. The methods used in this book are however somewhat
Broader in scope for mathematical physics tends to treat chiefly
Systems that are continuous and linear (S.3/7). The restriction
Makes its methods hardly applicable to biological subjects, for in
Biology the systems arc almost always non- linear, often
Non-continuous, and in many cases not even metrical, i.e. express-
Ible in number, The exercises below (S.3/4) are arranged as a
Sequence, to show the gradation from the very general methods
Used in this book to those commonly used in mathematical phys-
Ics. The exercises are also important as illustrations of the corre-
Spondences between transformations and real systems.
To summarise: Every machine or dynamic system has many
Distinguishable states. If it is a determinate machine, fixing its cir-
Cumstances and the state it is at will determine, i.e. make unique
The state it next moves to. These transitions of state correspond to
Those of a transformation on operands, each state corresponding to
A particular operand. Each state that the machine next moves to
Corresponds to that operand’s transform. The successive powers
Of the transformation correspond, in the machine, to allowing
Double, treble, etc., the unit time-interval to elapse before record-
Ing the next state. And since a determinate machine cannot go to
Two states at once, the corresponding transformation must be sin-
Gle-valued.
27
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
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Ex.: Name two states that are related as operand and transform, with
Time as the operator, taking the dynamic system from:
A) Cooking, (b) Lighting a fire; (c) The petrol engine; (d) Embryo-
Logical development; (e) Meteorology; (f) Endocrinology; (g) Econom-
Ics; (h) Animal behaviour; (i) Cosmology. (Meticulous accuracy is not
Required.)
Machine and transformation. The parallelism between
Machine and transformation is shown most obviously when we
Compare the machine’s behaviour, as state succeeds state, with the
Kinematic graph (S.2/17), as the arrows lead from element to ele-
Ment. If a particular machine and a particular graph show full cor-
Respondence it will be found that:
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