Non-Linear Dynamics, Self-Organization and Cellular Automata Models - Environmental Modelling: Finding Simplicity in Complexity - page 46

branches of science, which permitted scientists to under-
take previously infeasible numerical analyses, and to
become less reliant on linear approaches. Alongside this
increase in computational capability was a developing
mathematical focus on nonlinear systems (i.e. systems
whose components are related by curvilinear functions:
see e.g. Jordan and Smith, 2007). These two threads led to
what became subsequently known as nonlinear dynamics,
which itself has two main strands: deterministic chaos (see
e.g. Kellert, 1993), and fractals (see e.g. Mandelbrot, 1982).
Deterministic chaos was first discovered by a number of
workers such as the atmospheric physicist Edward Lorentz
(Ruelle, 2001). However, the mathematical roots of chaos
are a good deal older: notably the work of mathematician
Henri Poincare around the beginning of the twentieth
century (e.g. Jones, 1991). Deterministic chaos took some
time to enter the scientific mainstream. It began in the
1970s, with for example the work of biologist Robert May
on chaotic population dynamics (May, 1976).
The tongue-in-cheek question. 3 'Does the flap of a
butterfly's wings in Brazil set off a tornado in Texas?'
summarizes one major attribute of deterministic chaos.
In effect, the question asks if a large effect has a tiny
cause. It can: such nonlinear 'extreme sensitivity to ini-
tial conditions' is a hallmark of chaotic systems. Small
uncertainties in measurement or specification of initial
conditions become exponentially larger, and the eventual
state of the system cannot be predicted. Thus, for example,
the chaotic component of the Earth's atmosphere means
that weather forecasts rapidly diminish in reliability as one
moves more than a few days into the future. It is important
to note though that whereas weather (i.e. the particular set
of meteorological conditions on a specific day, at a specific
location) cannot be predicted, climate (here, the range of
meteorological conditions of a number of replicates of
the meteorological conditions simulated for that day and
location) can be predicted. This notion is at the heart of
'ensemble forecasting' techniques, which are carried out
using atmospheric models (Washington, 2000).
Deterministic chaos is also often associated with fractal
(i.e. self-similar, scale-independent) patterns. Following
seminal work by Benoit Mandelbrot (1975; see also
Andrle, 1996), fractal patterns were acknowledged to
be present in a wide variety of natural situations (but
see e.g. Evans and McClean, 1995). This linkage between
fractals and systems exhibiting deterministic chaos is

suggestive of some deeper connection (cf. Cohen and
Stewart, 1994).
But for environmental modellers, perhaps the most
interesting insight from chaotic systems is that they do
not have to be complicated to produce complex results.
Lorentz's atmospheric model comprised only three non-
linear equations, and May's population-dynamics models
were even simpler. In all such models, the results at the end
of one iteration (in the case of a time-series model - see
Chapter 3 - at the end of one 'timestep') are fed back into
the model and used to calculate results for the next iter-
ation. This procedure produces a feedback loop. Some
values of the model's parameters will cause it eventu-
ally 4 to settle down to a static equilibrium output value;
with other values, the model's output will eventually settle
down to cycle forever between a finite number of endpoint
values; but for others, the model will switch unpredictably
between output values in an apparently random way.
Thus in such chaotic systems, complex patterns can be
the results of simple underlying relationships.
Both positive and negative implications follow from
the discovery of such systems. For those of a deterministic
cast of mind this is sobering because it represents the final
death rattle of the notion of a predictable, 'clockwork',
universe, even at the macroscale. 5 But there is also a
strongly positive philosophical implication: complexity
does not have to be the result of complexity!
Nonetheless, while this early work on deterministic
chaos was intriguing and suggestive, it was not immedi-
ately 'useful' for most environmental modellers.

While the output from chaotic functions is complex, it
includes little in the way of immediately recognizable
structure: at first glance it more resembles random
noise. It is therefore qualitatively very different from
the complex but highly structured patterns that we
observe in many environmental systems.

•

When analysing real-world measurements, which
plausibly possess a chaotic component, it has proved to

•

4 The word 'eventually' is important here. The repetitive calcula-
tions that are often necessary are ideally suited to a computer, but
not to a human. This is one reason why deterministic chaos had
to wait for the widespread use of computers for its discovery.
5 'But what we've realized now is that unpredictability is very
common, it's not just some special case. It's very common for
dynamical systems to exhibit extreme unpredictability, in the sense
that you can have perfectly definite equations, but the solutions
can be unpredictable to a degree that makes it quite unreasonable
to use the formal causality built into the equations as the basis for
any intelligent philosophy of prediction' (Berry, 1988: 49).

3 Originally the title of a 1970s lecture by Lorentz. There are now
many variants.