"Lo ! thy dread empire Chaos !
is restored:
Light dies before thy uncreating word;
Thy hand, great Anarch ! lets the curtain fall,
And universal darkness buries all."
Alexander Pope, The Dunciad,
1728-1743, bk. IV
The terms
"linear" and "nonlinear" are often made synonymous
mathematically with simple and difficult. Linear is a property of straight
lines, of simple proportions, of predictability and good behaviour. Nonlinear
on the other hand applies to systems that do unpredictable things, that cannot
be solved exactly and need to be approximated, the one-offs that don't fit the
expected pattern.
So now that we understand
these terms, what things are nonlinear exactly ?
Answer - almost everything !
Nearly all of the
mathematics and science with which we are familiar is presented in a linear
form - that is, the output of the equation varies in direct proportion to the
input y = f(x). This seems very natural to us, and we often take for granted
that the world works like this - a ball hit twice as hard will go twice as
fast; a hole twice as big will leak twice as much water. If this were not true
then we would have great difficulty in predicting anything wouldn't we ?
It comes as a surprise to
many people that the natural state of affairs in the world is non-linear.
Linearity is merely an approximation. Let us demonstrate this by a simple
thought experiment. One of the most frequent demonstrations of Newtonian linear
dynamics is the billiard ball model of atoms. Conservation of momentum and
energy allow us to predict exactly the trajectories of atoms moving in straight
lines. A neat trick on the college blackboard, but does it work in real life ?
Unfortunately the answer is a qualified no.
Let us take the air itself.
We will assume that it is at a constant temperature and pressure and comprises
atoms that collide elastically (no energy lost as heat). For how long could we
predict the trajectories of the molecules (given an ideal computer) ? The
answer is almost no time at all, a few collisions only, a tiny fraction of a
second. The system is nonlinear, as are all real systems. This also
means that they are irreversible, contrary to conventional assumptions.
Why ? Well, the billiard
ball model, like most models, is a simplified treatment. It ignores gravity for
instance. Atoms are subject to it in the same way as are planets. It can be
shown that the gravitational attraction of a single electron, at the edge of
the known universe (10 billion light years away), is sufficient to deflect an
oxygen molecule in the air on earth by enough to miss a predicted destination
molecule within about 50 collisions, around one hundred millionth of a second !
This phenomena is known as
sensitivity to initial conditions, or the Butterfly Effect. It arises because
the errors that accumulate from each collision do not simply add (as linear
analyses assume), but increase exponentially and this geometric progression
rapidly diverges any initial state to one that is unpredictably far from the
estimate. This behaviour is responsible for what we call Chaos, a term that has
a technical meaning but is roughly equivalent to the common notion of future
randomness, except that the states in which the system may be found are often restricted and known in total - we just don't know which
one it will be in at any future time. The system is unstable, a small change
leads to a massive reaction...
Each molecule is acted upon
by every object in the universe - an infinite sum of terms is thus necessary
for an exact model. If we take the totality of the interactions between
molecules (non-stationary - hence irreversibility, the return path would
experience different forces) we can see that the normal linear treatment
(discarding most terms) is woefully inadequate, so why does it ever work ?
Well, we have four factors at play here.
Firstly, we do not
generally require to predict the trajectories of the individual components of
gasses (and similar multi-agent systems), instead we adopt a statistical
treatment - we smooth out and average the uncertainties (sum-over in the
parlance). We can do this because even a small system contains vast numbers of
particles. If we are only interested in overall effects, then the minor short
term individual deviations will, on average, cancel out.
Secondly for larger one-off
objects we do not need to predict over long time periods. Thirdly, usually in
these cases we need take no account of any collisions with systems of the same
order of size. Planetary predictions cover a short cosmic time scale relative
to the frequency of collision with similar sized asteroids, and the
gravitational effects of distant planets or collisions with smaller objects
(meteors) are assumed negligible over these short periods - these terms can
be neglected, in some circumstances...
But what about longer
periods ? It is true to say that in that case our predictions are known to
break down - the Earth's orbit is chaotic over a mere 5 million years, all
systems of 3 or more bodies may exhibit such chaotic dynamics. We cannot predict
forward or backwards for very long in astronomical terms, so extrapolations to
the beginning or end of time would be little more than guesswork - apart,
perhaps, for our final factor.
That fourth reason is most
interesting, and concerns the constraints
on the system. Most familiar systems do not have the freedom to move that we
attribute to ideal gasses (on which traditional dynamics are based). A planet
is held together by gravity, a chemical substance by molecular bonds. Structure
implies the presence of forces acting to prevent the system disintegrating -
the system does not have available the maximum degrees of freedom.
If a molecule in a solid is
attracted away from its position, then the forces from neighbouring molecules
act to restore that position - there is a negative feedback effect, the displacement is no longer linear.
Any time that we have a system that has limits, minimum or maximum settings,
then we have a nonlinear system. Populations of animals are limited by food,
space or predators; trignometrical functions are limited, elastic stretching is
limited, pendulum swings are limited. All aspects of our perceived reality are,
at root, nonlinear - none are completely unrestrained. The formula we now need
is x = f(x), the output is a function of itself. Positive feedback, hence we
have potential instability.
Let us compare the two
formulae, y = f(x) and x = f(x).
Take first the equation y = x * n. Suppose n = 2, then plotting for x = 1, 2, 3
we have y = 2, 4, 6 - a straight line, of slope n and linearity. Now for x = x
* n. Again for n = 2, starting with x = 1 we have x = 2, 4, 8 - an exponential
progression towards infinity, if n = 1 we have 1, 1, 1 - stagnation and for n =
0.5 we have 0.5, 0.25, 0.125 a progression towards zero. Three very different
behaviours from the same formula, dependant upon the constant. If we assume
that it is one then anything even slightly more will take us eventually to
infinity, any less to zero - our 'Butterfly Effect', sensitivity to initial
conditions appears even for such a trivial formula.
In real life we will
usually have many coupled variables, so in practice x,y,z,... = f(x,y,z,...) -
a matrix of inter-dependencies. The 'constants' in nature (say
'reproduction rate') are usually nothing of the sort, merely our assumptions.
The behaviours we then predict are highly dependent upon what we have chosen to
assume initially - our axioms !
Is it possible to control
these nonlinear systems ? For us to do this we need to understand under what
circumstances they become chaotic. To some extent we have already tried to do
this with our linear approximations, effectively we have only operated our
predictions in regions where the system behaves almost linearly. This means
restricting the parameters of the system to areas that do not possess the
sensitivity to initial conditions or studying only simplified aspects of
systems.
Unfortunately, when we deal
with complex systems of the type of interest to researchers in our fields, that
is rarely an option. Strong feedback (positive as well as negative) and many
interactions mean that chaotic behaviour is potentially always just around the
corner, disastrously if unsuspected (say in an aircraft control system). We
need to find a way of controlling chaos, to understand what may happen long
term. Luckily we have some tools at our disposal.
Chaotic systems are not
totally random, they usually have well defined limits (trajectories in phase space),
so to a first approximation we can determine limits for the system. If we are
happy to use statistical analysis then this may suffice, we can (like in
quantum mechanics) assign a probability to the system being in a particular
state (actually within a small region of phase space).
We have seen that small perturbations
can send chaotic systems into very different states, so a further technique
relates to this. We can arrange to perturb the system in a controlled way to
keep it in the region of behaviour that we need. A similar idea is to add
additional constraints to change the global dynamics until all the states
available to the system fall within permitted boundaries.
If we plot the behaviour of
many nonlinear formulae against changes in parameters we see an interesting
phenomenon. At certain values of a control parameter the system changes phase -
it moves from a regular behaviour to a chaotic one. The first stage in this is
what is known as a Bifurcation, the system splits from one state to two
possible states. As the parameter changes further, additional splits take place
at intervals dependent upon a constant, the Feigenbaum number (4.669...),
giving more complex limit cycles progressively until the system enters a
chaotic state. We can therefore determine the parameter ranges that lead to the
behaviour we desire, and try to constrain the system accordingly.
What about systems whose
formulae are unknown ? Here we have a problem. If we can only sample a few
states of a system we cannot with any certainty determine its limits. It is
here that computer simulations come to the fore, allowing us to
investigate the general behaviour of nonlinear systems and to try to understand
and categorise them. If we are able to assign to natural systems an appropriate
general behaviour, then we may be well on the way to controlling that most
difficult and unpredictable of chaotic human problems - our social interactions.