Epochal trajectories

Steve.Keen at unsw.edu.au Steve.Keen at unsw.edu.au
Mon Mar 6 23:35:54 MST 1995


Howie Chodos put a tangent at the end of a recent post:

|Several other people have pointed to the various social regressions that
|have occurred, so I won't pursue that any further. However, Ron Press raised
|some interesting issues regarding complex systems. I have no understanding
|of the math involved, but I think that it is interesting to note that
|complexity theory would also seem amenable to forms of determinism.

Complexity theory has in fact arisen from the realisation that *completely*
deterministic models--i.e., models which have no chance or random
components whatsoever--can generate results that appear "chaotic"
or random. That is, the outcomes look like the result of a chance
process like rolling a dice, but are in fact determined by a process
that has no chance component whatsoever.

As well, a system which consists of a large (greater than 2!) number
of components with positive and negative feedback effects on each other
can also generate behaviour which, seen from some perspectives, is almost,
but not quite, periodic. So there is structure as well as apparent
randomness.

In some systems, parameter values have to be within certain ranges to
generate this "randomness out of order". The simplest example is the
idea of the logistic curve, which--when used as a model of population
growth--says that this period's population is a function of last
period's population:

x(k+1) = a * x(k) *(1 - b * x(k))

All this equation reflects is the propositions that:
(a) in the absence of any constraints on resources, etc., a given
population will increase exponentially; and
(b) constraints do exist, so that any environment can be said
to have a maximum carrying capacity.

The parameters a and b are constants, which respectively reflect
the rate of unconstrained population growth, and the carrying
capacity. If population at time k is less than the carrying
capacity, population will increase; if it is greater than
the carrying capacity, it will decrease.

Note that there is no chance element at all in the above:
given a known population at time k, and pre-set values for a
and b, the population at time k+1 is predicted exactly (this is
in a model of course).

You might think that such a system would eventually converge
to the carrying capacity, and for low values of a and b, it
does. But for higher values, you start to see, first of all,
periodic behaviour--values for k=values for k+2; values for
k+1=values for k+3. Then there are more complex periods, and
finally, no pattern at all--randomness from a deterministic
system. You could even try this out on a spreadsheet--it's
a common exercise in introductory courses in dynamics these
days. Try small values of b (say .001) and start with low
values of a (but greater than 1), moving up to values of
about 5. Allow about 50 iterations. For values of a below
3, you will eventually converge to one number (the carrying
capacity). For values above 3, first of all you will
get two numbers alternating; then 4; then 8; then really
large cycles; then no apparent pattern from one iteration to
the next. It looks random, and yet it is completely
deterministic.

That is an example of the processes behind the quote you give:

|In short, physics is
|beginning to discover ways in which very complex systems nevertheless
|exhibit remarkable order. No reflective biologist can view these
|developments without wondering whether the origins of order in nonliving
|sytems augurs new insights for the origins of order in the living ones as |well.

In fact, the logistic curve IS a model from population biology. The
extent to which living systems seem to occur at points right at the
border of transition between periodicity and randomness has led to
the feeling that perhaps life is a process which occurs at the
"edge of chaos"--your next quote. Self-organisation is the term
now applied, and it implies that order evolves to process and
simultaneously dissipate an external energy input--which has
inspired some researchers to talk of a third law of
thermodynamics which tempers the oft-cited second law of
increasing entropy, leading instead to increasing complexity
in open systems.

Cheers,
Steve Keen


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