The "Butterfly Effect" is the propensity of a system to be sensitive to initial conditions.Such systems over time become unpredictable,this idea gave rise to the notion of a butterfly flapping it's wings in one area of the world,causing a tornado or some such weather event to occur in another remote area of the world.
Comparing this effect to the domino effect,is slightly
misleading.There is dependence on the initial sensitivity,but whereas a simple
linear row of dominoes would cause one event to initiate another similar
one,the butterfly effect amplifies the condition upon
each iteration.
The butterfly effect has been most commonly associated with the Weather
system as this is where the
discovery of
"nonlinear"
phenomenon began when
Edward Lorenz found anomalies in computer
models of the weather.But
Henri Poincaré had already
made inroads into this area. Mapping the results in
"phase space" produced a twolobe map called the
Lorenz Attractor. The word
attractor meaning that events tended
to be attracted towards the two lobes,and events outside of the lobes are
such things like snow in the desert.
The attractor acts like an egg whisk,teasing apart parameters that
may initially be close together,this is why the weather is so hard to predict.
Super computers run several models of the weather in
parallel to discover whether they stay close together or diverge away from
each other.Models that stay similar in nature give an indication that the
weather is relatively predictable,and are used to indicate the confidence
level that Meteorologists have in a prediction.
It is not just the weather though that is subject to such
phenomena.Any
"Newtonian Classical" system where one system
is in competition with another,such as the "Chaotic Pendulum" which plays
magnetism off against gravity will exhibit "sensitivity to initial conditions".
Animal populations may also be subject to the same
phenomena.Work done by Robert May,suggests that
predatorprey systems have complex dynamics making them prone to "boom" and
"bust",due to the difference equations that model them.Such
a system even with two variables such as Rabbits and
Foxes can create a system that is much
more complex than would be thought to be the case.Lack of Foxes means
that the Rabbit population can increase,but increasing numbers of Rabbits
means Foxes have more food and are likely to survive and reproduce,which
in turn decreases the number of Rabbits.It is possible for such systems to
find a steady state or equilibrium,and even though species can become
extinct,there is a tendency for
populations to be robust,but they
can vary dramatically under certain circumstances. Real populations of
course,have more than two variables making them ever more complex.But as
can be seen from the diagram, such systems are not as simple as might be
thought.
The chemical world is also not free from such intrusions of
nonlinearity.In certain cases chemical
feedback produces effects as that in the
BelousovZhabotinsky reaction, creating
concentric rings, which are produced by a chemical change, whose decision
to change from one state to another cannot be predicted.The BZ chemical
system is currently being trialled as a means to achieve artificially intelligent
states in robots.
Phase space portraits of liquid flow show that they too are subject to the
same kind of nonlinearity that is inherent in other physical systems.It
may be apparent when turning on a tap that sporadic drips become "laminar"
as the flow increases.What might not be apparent is the nature of the change
from semirandom to continuous.It may seem rather at odds with intuition
that such natural systems have inherent behaviour that is not random,or indeed
that is not capable of being predicted.It may also seem that "not random"
means "predictable".
Natural systems can present a tangled mix of determinism and
randomness,or "order" and
"chaos".In
such cases as water moving from drips to continuous flow, pictures called
"Bifurcation diagrams" demonstrate the nature of movement from order into
chaos.This bifurcation is based on Robert May's work,but one of the intriguing
things about bifurcations is that the same pattern occurs no matter what
system is iterated.In fact
Mitchell
Feigenbaum discovered that there was a "constant of doubling"
hidden in amongst all these systems.
Electronic apparatus is also not free from such effects,and it is perhaps
ironic,that we think of electronic apparatus as as being the epitome
of predictable determinism and ruthless clockwork efficiency.Indeed the powerful
computers used to predict weather,would seem ineffectual if they were not
ruthless automatons.But such effects occur only in certain circumstances
where there is "sensitivity to initial conditions".Amplifiers for
instance,produce a howl when feedback occurs as they go into a stable
state of oscillation.Logic gates as used in computers
have to select a "0" or a "1",and this relies on choosing between two
states whose boundary is indeterminate,and it is when a computer confuses
a "0" for a "1" or vice versa that mistakes occur.


Phase space portrait of regular flow in a TaylorCouette system 
Phase space portrait of chaotic flow as the attractor becomes "strange" 

Phase space portraits of a system of coupled pendulums 
Many of the shapes that describe nonlinear systems are fractal,a set of shapes that are selfsimilar on smaller and smaller scales with no limit to the size of the scale. Fractals were discovered by Benoit Mandelbrot at IBM.
A picture of a real fern and an affine transformation performed by "Winfract" (Inset1).Inset 2 is a real image of a conifer,and bears an uncanny resemblance to a diffusion limited aggregate. 
Fractals have been seen as describing naturally occurring phenomena
such as the cragginess of mountains or the shapes of certain plant forms,
such as ferns,which can be modelled by affine transformations.
Whether in fact Nature is fractal,or whether it just describes it better than the simple geometry of Euclid depends on the philosophical view taken of mathematics as a whole. Some people think mathematics is just a tool or a creation of man,and therefore Nature is only described or mapped by mathematics.
Others think that the description is real at least in the sense that the
similarity is not superficial,that in fact natural objects that look fractal,or
which fractals look like,are similar in appearance because at some fundamental
level the natural objects are obeying some form of rule system that bears
a similarity to the sort of rules which govern fractals.
Whichever way you look at it,one thing no one can say is that mathematics is irrelevant to Nature. From butterflies to plants,from the weather to chemistry,mathematics is modelling or displaying attributes of Nature,and helping us to understand what we see.
A new conception was being made....that
whatever fundamental units the world is put together from,they are more
delicate,more fugitive,more startling than we catch in the Butterfly Net
of our senses.
Jacob Bronowski "The Ascent of Man"
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