Re: Chaos theory versus complexity theory

[P8475423@xxxxxxxxxxxxxxxxxxxxxxxx]

I basically agree with Chris B's summary of chaos, and chaos v complexity
(I think the latter is more a question of the emphasis of the researchers
than any substantive difference, BTW).

And no, I don't think I can change my "call sign" -:( ; but
steve.keen@xxxxxxxxxxx still works as an address

|I am totally confident that P8475423 (Steve, can't you do anything about
|your inhuman name? ) will have read this post. He has referred to Goodwin's
|interesting predator-prey model for the economy. I have got Goowin's book
|but am not confident with the maths, and get destracted by this marxism
|list. I am hoping Steve can say whether this predator-prey model is the
|same as the population equation in chaos theory.

I wouldn't recommend reading Goodwin's book if the maths worries you, not
because of the maths per se but because Goodwin, as imaginative as he is
a practitioner, is generally a pretty poor explainer of maths. He tends to
leave variables undefined, and makes great (but logically correct) leaps
without warning.

A *FAR* better source is another Aussie (unfortunately deceased): JM Blatt's
_Dynamic Economic Systems_, ME Sharpe, 1983 (I think) which has a superb
explanation of Goodwin's model.

The predator-prey model is not the same as the population equation. To explain
the latter to any non-afficionados still reading, the population equation
encapsulates two details about population growth:

* that if resources were unconstrained a population can expand exponentially

* that the resources are ultimately limited, so that in a given environment,
there will be a maximum carrying capacity.

In mathematical terms, it says that the percentage rate of change of a
population will be a constant (the exponential bit) minus some number times
the current population (with the number representing the carrying capacity
of the environment):

1/x dx/dt = a - bx

Re-organise and you get

dx/dt = ax - bx^2

which is known as the population equation. If you express it in discrete
time form, what you get is

N(t+1) =  a * N(t) (1 - b * N(t))

If you start with low values of b/a, you get a stable equilibrium outcome;
the population converges to the carrying capacity. But with higher values
of b/a, the system can "flip" from one side of the carrying capacity to the
other.

The idea is pretty simple to express in verbal terms: if you have a rapid
growth rate and you're very near the carrying capacity, you'll overshoot
it with the next round of births, and the excess population will lead to
starvation-caused deaths, reducing you below the capacity; and then next
time you overshoot it again.

For a range of b/a values, the system has a "two cycle" -- it flip-flops
between the under-capacity level and the over-capacity. Then it goes into
a three stage cycle, then 4 stage,... and then, apparent randomness--
deterministic chaos.

Continuing:

|I am hoping he will explain the term "limit-cycle", which he has suggested
|the capitalist economy looks like to him, Marx, and intuitively to me too.
|But I am not quite sure of this terminology. It sounds like the fish and
|the lake. What is a limit cycle?

It is not the fish and the lake but the fish and the sharks (initially,
literally: it was used to explain why the population of sharks rose and
fish fell during WWI, when the lack of fishing led people to expect an
increase in fish numbers).

Fish grow exponentially depending on the current number of fish, and
they decline at some rate times the likekihood of a fish "bumping into"
a shark:

df/dt = a * f - b * f * s

On the other hand, the sharks die at an exponential rate in the absence
of fish, but increase at an exponential rate depending upon the
likelihood of finding fish:

ds/dt = - c * s + d * f * s

Solve this and you get a fixed limit cycle: start from a small number
of fish, and the shark population falls. The fish population rises,
leading eventually to an abundance of fish which are easy pickings
for the few sharks. The sharks then breed, reducing fish numbers,
and the cycle repeats (exactly).

Goodwin drew a very valid analogy between this process and the wage/
employment struggle in capitalism, as described in Ch 25 of Capital.

A limit cycle in general is any cyclical pattern towards which a
system converges. Goodwin's one above is rigid: the same pattern
is repeated over time. A more general one will converge from the
inside out if you start near the equilibrium point (which is
inside the limit cycle and unstable, obviously); ir will converge
from the outside in if you start far away.

A chaotic one will (as my Minsky model with government does) cycle
inside and outside as it converges.

Strange attractors in nonlinear models result when there are two
or more unstable equilibria, and their regions of attaction and
repulsion overlap. Imagine drawing a box in the sand, and then
making two "Mexican sombrero" patterns in the sand, so that the
rims of the sombreros just overlap. The you hit the sand with a
small, "nice" French atomic bomb, turning the sand into glass
without disturbing the sombreros ( -:)). Now roll a ball within
the box. The ball will have to fall into the rims of the
sombreros, but the path it takes depends on where you pushed it
from, and how hard you pushed it. At one level, it might
spin around one sombrero once, then move to the other.
A slight difference, and it might spin around one sombrero 5
times, then shoot across.

That ain't strictly chaos, but I hope it's a useful mental
picture of the process.

What does this have to do with capitalism? Lots, if the
underlying processes are nonlinear, which I assert they
are. This kind of picture could help explain why, for example,
we might move from 20 years of high growth and low inflation,
suddenly to low growth and high inflation, and then suddenly
again, to low growth and low inflation...

Cheers,
Steve K


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