Bilised definition”. Since the ideas behind the word are of great
Practical importance, we shall examine the subject with some care,
Distinguishing the various types that occur.
Today’s terminology is unsatisfactory and confused; I shall not
Attempt to establish a better. Rather I shall focus attention on the
Actual facts to which the various words apply, so that the reader will
Tend to think of the facts rather than the words. So far as the words
Used are concerned, I shall try only to do no violence to established
Usages, and to be consistent within the book. Each word used will
Be carefully defined, and the defined meaning will be adhered to.
Invariant. Through all the meanings runs the basic idea of an
“invariant”: that although the system is passing through a series of
Changes, there is some aspect that is unchanging; so some state-
Ment can be made that, in spite of the incessant changing, is true
Unchangingly. Thus, if we take a cube that is resting on one face
And tilt it by 5 degrees and let it go, a whole series of changes of
position follow. A statement such as “its tilt is 1 °” may be true at
One moment but it is false at the next. On the other hand, the state-
ment “its tilt does not exceed 6 °” remains true permanently. This
Truth is invariant for the system. Next consider a cone stood on its
point and released, like the cube, from a tilt of 5 °. The statement
“its tilt does not exceed 6 °” is soon falsified, and (if we exclude
Reference to other subjects) so are the statements with wider lim-
Its. This inability to put a bound to the system’s states along some
Trajectory corresponds to “instability”.
These are the basic ideas. To make them incapable of ambiguity
We must go back to first principles.
State of equilibrium. The simplest case occurs when a state
And a transformation are so related that the transformation does
73
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
not cause the state to change. Algebraically it occurs when T(x) =
X. Thus if T is
T: ↓
A b c d e f g h
D b h a e f b e
then since T(b) = b, the state b is a state of equilibrium under T.
So also are e and f.
If the states are defined by vectors, then, for a vector to be
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Unchanged, each component must be unchanged (by S.3/5). Thus
If the state is a vector (x, y), and the transformation is
2U: x'' = x x –yy++32 y = +
then, at a state of equilibrium (x', y') must equal (x, y), and values
For x and y must satisfy the equations
Ex. 4: Find all the states of equilibrium of the transformation:
dx/dt = e- sin x, dy/dt = x2.
Ex. 5: If x' = 2x– y + j, y' = x + y + k, find values for j and k that will give a state
Of equilibrium at (1,1). (Hint: First modify the equations to represent the
State of equilibrium.)
Ex. 6: If T(b) = b, must T2(b), T3(b), etc., all also equal b?
Ex. 7: Can an absolute system have more states of equilibrium than it has basins ?
Ex. 8: What is the characteristic appearance of the kinematic graph of a transfor-
Mation whose states are all equilibrial ?
Ex. 9: (Continued.) What special name was such a transformation given in an
Earlier chapter ?
Ex. 10: If the transformation is changed (the set of operands remaining the same)
Are the states of equilibrium changed?
Ex. 11: If a machine’s input is changed, do its states of equilibrium change?
Hint: See Ex.5.)
x = 2x – y + 2
y =x + y + 3
x–y =– 2
x =– 3
I.e.
S/4. Cycle. Related to the states of equilibrium is the cycle, a
Sequence of states such that repeated application of the transforma-
Tion takes the representative point repeatedly round the sequence.
Thus if T is
a b c d e f g hT: ↓ c h b h a c c g
Then, from a, T generates the trajectory
A c b h g c b h g c b ...
And the representative point repeatedly traverses the cycle
c → b
↑↓
g ← h
Ex. 1: Write down a transformation that contains two distinct cycles and three
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States of equilibrium.
Ex. 2: (Continued.) Draw its kinematic graph.
Ex. 3: Can a state of equilibrium occur in a cycle ?
Ex. 4: Can an absolute system have more cycles than it has basins?
Ex. 5: Can one basin contain two cycles ?
*Ex. 6: Has the system dx/dt = y, dy/dt = – x a cycle?
*Ex. 7: If the transformation has a finite number of states and is closed and sin-
Gle-valued, can a trajectory end in any way other than at a state of equilib-
Rium or in a cycle?
So this system has only one state of equilibrium, at ( – 3, – 1). Had
The equations not been linear there might have been more.
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