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12

Systems and Networks

Another corollary is that the concept of feedback, which is usually clear enough to

apply to two-component systems, is practically useless in more complex systems. ²

In this chapter, we shall first consider the approach of general systems theory,

largely pioneered by Bertalanffy. This allows some insight into the behaviour of

very simple systems with not more than two components, but thereafter statistical

approaches have to be used. ³ This is successful for very large systems, in which

statistical regularities can be perceived; the most difficult cases are those of interme-

diate size. Some properties of networks per se will then be examined, followed by a

brief look at synergetics (systems with a diffusion term), and the final section deals

with complex evolving systems.

Problem. Consider various familiar objects, and ascertain using the above criteria

whether they are systems.

12.1

General Systems Theory

Consider a system containingnn interacting elementsupper G 1 comma upper G 2 comma ellipsis comma upper G Subscript n BaselineG1, G2, . . . , Gn. Let the values

of these elements beg 1 comma g 2 comma ellipsis comma g Subscript n Baselineg1, g2, . . . , gn. For example, if theupper GG denote species of animals,

then g 1g1 could be the number of individual animals of species upper G 1G1. The temporal

evolution of the system is then described by

StartLayout 1st Row 1st Column StartFraction normal d g 1 Over normal d t EndFraction 2nd Column equals 3rd Column script upper G 1 left parenthesis g 1 comma g 2 comma ellipsis comma g Subscript n Baseline right parenthesis 2nd Row 1st Column StartFraction normal d g 2 Over normal d t EndFraction 2nd Column equals 3rd Column script upper G 2 left parenthesis g 1 comma g 2 comma ellipsis comma g Subscript n Baseline right parenthesis 3rd Row 1st Column Blank 2nd Column vertical ellipsis 3rd Column Blank 4th Row 1st Column StartFraction normal d g Subscript n Baseline Over normal d t EndFraction 2nd Column equals 3rd Column script upper G Subscript n Baseline left parenthesis g 1 comma g 2 comma ellipsis comma g Subscript n Baseline right parenthesis EndLayout^dg1

dt ^{= G1(g1, g2, . . . , gn)}

dg2

dt ^{= G2(g1, g2, . . . , gn)}

...

(12.1)

dgn

dt ^{= Gn(g1, g2, . . . , gn)}

where the functionsscript upper GG include terms proportional tog 1 comma g 1 squared comma g 1 cubed comma ellipsis comma g 1 g 2 comma g 1 g 2 g 3g1, g²

1^{, g3}

1^{, . . . , g1g2, g1g2g3, etc.}

In practice, many of the coefficients of these terms will be close or equal to zero.

If we only consider one variable,

StartFraction normal d g 1 Over normal d t EndFraction equals script upper G 1 left parenthesis g 1 right parenthesis period^dg1

dt ^{= G1(g1) .}

(12.2)

Expanding gives

StartFraction normal d g 1 Over normal d t EndFraction equals r g 1 minus StartFraction r Over upper K EndFraction g 1 squared plus midline horizontal ellipsis^dg1

dt ^{= rg1 −r}

K ^g2

1 ^{+ · · ·}

(12.3)

2 Even in two component systems its nature can be elusive. For example, as Ashby (1956) has

pointed out, are we to speak of feedback between the position and momentum of a pendulum?

Their interrelation certainly fulfils all the formal criteria for the existence of feedback.

3 Robinson (1998) has proved that all possible chaotic dynamics can be approximated in only three

dimensions.