Objects: Tokens for (Eigen-)Behaviors - Understanding Understanding: Essays on Cybernetics and Cognition - page 268

Appendix A

Examples:
A1. Consider the operator (linear transform) Op 1 :
Op 1 = “divide by two and add one”
and apply it (recursively) to x 0 ,x 1 , etc., (whose domains are the real
numbers).
Choose an initial x 0 , say x 0 = 4.

4
2

() =+=+=

xOp = 4 1213 ;

1 1

xOp
xOp
xOp
xOp
xOp
x

= () =
= ( ) =
= ( ) =
= ( ) =
= ( ) =
= () =
=

3 2 500
2 500

. ;

2 1

. 2 250 . ;

3 1

2 250 . 2 125 . ;

4 1

2 125 . 2 063 . ;

5 1

2 063 . 2 031

2 001

. ;

6 1

px . ;

11 1 10

x px

( ) = 2 000

.

• 1 •

Choose another initial value; say x 0 = 1

xOp
xOp
xOp
xOpx
x px
xOpx

= () =
= ( ) =
= ( ) =
= () =
= () =
= () =

1 1 500
1 500

. ;

1 1

. 1 750 . ;

2 1

1 750 . 1 875

1 996
1 999
2 000

. ;

3 1

. ;

8 1 7

. ;

10 1 9

.

• 1 •

And indeed:

1
2

◊+= 212

Op 1 (2) = 2
i.e., “2” is the (only eigenvalue of Op 1 .
A2. Consider the operator Op 2 :
Op 2 = exp(cos ).
There are three eigenvalues, two of which imply each other (“bi-stability”),
and the third one being instable:
Op 2 (2.4452. . .) = 0.4643. . .

stable

Op 2 (0.4643. . .) = 2.4452. . .
Op 2 (1.3029. . .) = 1.3092. . .

instable