142

12

Systems and Networks

forbidding. Hence, one must turn to statistical properties of the system. Equation

(12.6) can be written compactly as

ModifyingAbove bold g With dot equals upper A bold g˙g = Ag

(12.9)

where bold gg is the vector left parenthesis g 1 comma g 2 comma ellipsis right parenthesis(g1, g2, . . .), ModifyingAbove bold g With dot˙g its time differential, and upper AA the matrix of the

coefficients a 11 comma a 12a11, a12, etc. connecting the elements of the vector. The binary connec-

tivityupper C 2C2 ofupper AA is defined as the proportion of nonzero coefficients. ⁵ In order to decide

whether the system is stable or unstable, we merely need to ascertain that none of

the roots of the characteristic equation are positive, for which the Routh–Hurwitz

criterion can be used without actually having to solve the equation. Gardner and

Ashby determined the dependence of the probability of stability on upper C 2C2 by distribut-

ing nonzero coefficients at random in the matrix upper AA for various values of the number

of variables nn. They found a sharp transition between stability and instability: for

upper C less than 0.13C < 0.13, a system will almost certainly be stable, and for upper C greater than 0.13C > 0.13, almost cer-

tainly unstable. For very smallnn the transition became rather gradual, viz. forn equals 7n = 7

the probability of stability is 0.5 at upper C 2 almost equals 0.3C2 ≈0.3, and for n equals 4n = 4, at upper C 2 almost equals 0.7C2 ≈0.7.

Problem. Evaluate Berlinski (1978)’s criticism of systems theory.

12.1.1

Automata

We can generalize the Markov chains from Sect. 11.2 by writing Eq. (12.9) in discrete

form:

bold g prime equals upper A bold g commag^, = Ag,

(12.10)

i.e., the transformationupper AA is applied at discrete intervals andbold g primeg^, denotes the values of

gg at the epoch following the starting one. The value of g Subscript igi now depends not only on

its previous value but also on the previous values of some or all of the other n minus 1n −1

components. Generalizations to the higher order coefficients are obvious but difficult

to write down; we should bear in mind that the application of this approach to the

living cell is likely to require perhaps third- or fourth-order coefficients, but that the

corresponding matrices will be extremely sparse.

The analysis of such systems usually proceeds by restricting the values of the

gg to integers, and preferably to just zero or one (Boolean automata). Consider an

automaton with just three components, each of which has an output connected to the

other two. Equation (12.10) becomes

Start 3 By 1 Matrix 1st Row g 1 2nd Row g 2 3rd Row g 3 EndMatrix prime equals Start 3 By 3 Matrix 1st Row 1st Column 0 2nd Column 1 3rd Column 1 2nd Row 1st Column 1 2nd Column 0 3rd Column 1 3rd Row 1st Column 0 2nd Column 0 3rd Column 1 EndMatrix normal ♢ Start 3 By 1 Matrix 1st Row g 1 2nd Row g 2 3rd Row g 3 EndMatrix

⎛

⎝

g1

g2

g3

⎞

⎠

,

=

⎛

⎝

0 1 1

1 0 1

0 0 1

⎞

⎠♦

⎛

⎝

g1

g2

g3

⎞

⎠

(12.11)

5 The ternary connectivity takes into account connexions between three elements, i.e., contains

coefficients likea 123a123, etc.