have you ever wondered why when we look
around us in the natural world we see
very few straight lines but the straight
line is ubiquitous in the systems we
engineer from buildings to circuit
boards it appears to be the default
position one way to understand this is
that the systems we engineer are based
upon a scientific and mathematical
understanding of the world which has
inevitably started by describing the
simplest and most orderly systems that
is to say those that are composed of
linear forms and relations from Euclid
to Newton and on science has been
focused upon the orderly systems of
perfect squares triangles and linear
relations have caused an effect that can
be encoded in beautifully compact
equations
thus we described real world as a kind
of approximation to these perfect linear
forms but let's take a look at some of
the basic principles underlining the
theory of linear systems linear is
basically a fancy word for line and a
line is often understood as the shortest
or most direct path from one point to
another to say there is a linear
relationship between two things is to
say that there's a direct relationship
of cause and effect between them let's
take an example of this say I'm playing
baseball and I hit a ball with a bat if
we make a model of this system we see
that their energy inputted by my swing
of the bat will be directly proportional
to the output of the ball's momentum as
it travels off in the opposite direction
this is a simplified model but it
illustrates the direct or linear
relationship between cause and effect my
swinging of the bat and the balls motion
in response there are of course many
examples of linear systems particularly
within physics but we can capture the
underlining logic of linear systems in
general with a model that has only two
simple rules called the superposition
principles which states firstly that the
output to the system will always be
directly proportional to the input so if
there's a linear relationship between
the amount of fuel I put in my car and
how far it will go well then if I put
twice as much fuel in the car it will go
twice as far and secondly that if we add
two or more systems together
then the output to this combined system
will simply be the sum of the two
outputs of the original systems so say
we have two tractor factories each
producing a million tractors a year well
if we merge them then we will get a
factory that will produce 2 million
tractors a year linear systems are
deterministic meaning that if we know
their present state we can fully
determine their past and future States
this can be seen by plotting a linear
system as a graph where it will always
be depicted as a straight line
although linear systems modeling has
proven highly successful in many areas
and is often a very good approximation
the reality is that we live in a world
with eco systems economies societies and
physical systems that are not governed
by the superposition principles and thus
are what we call nonlinear an example of
non-linearity might be listening to two
of your favorite pieces of music at the
same time because there is a
relationship of interference between
them the results of the experience will
not be a simple equation of adding the
enjoyment from listening to each
independently this illustrates how
non-linearity arises whenever there is
some relationship between elements
within the system but can be either
synergistic making the output of the
system greater than some of its parts or
one of interference making the output
less than the sum of its individual
components
to illustrate this further let's take an
example of four workers producing
clothing in isolation
each seamstress can sew a given amount
of clothes within a day now if we put
them together we might get one of three
results firstly they might not interact
with each other very much meaning the
whole would simply be the sum of its
individual parts but equally likely they
might form some cooperative relationship
which lets them each specialize in a
particular function making them more
efficient as a whole and thus the output
of the system would be greater than the
sum of its parts
due to these synergistic relations or
inversely they might start all talking
with each other getting little work done
and thus the total output would be less
than the sum of the individual outputs
due to these relations of interference
non-linearity can also arise from
feedback loops whereby the same process
is iterated with the output fed back as
the input to the next cycle a classical
example of this is compound interest
where at the end of each period the
balance plus interest is fed back into
the formula to compute the next cycle of
interest accumulation iterative
functions are an important concept
within nonlinear science and have been
used to create a whole new type of
geometry called fractal geometry where
by iterating a simple function generates
irregular organic looking patterns that
can model many of the geometric forms we
see in nature from the structure of
seashells to the rugged formation of
mountains
in these nonlinear systems superposition
fails meaning one can operate the system
down into smaller subproblems and then
add their solutions we must consider a
nonlinear problem in total it is this
need to approach nonlinear systems as a
whole that is giving rise to new more
holistic approaches to science that are
developing under the canopy of the term
complex system science non-linearity in
all the shapes and forms is at the heart
of many of the 21st century challenges
to science as we try to extend the
scientific framework beyond its
dependency upon linear systems theory to
finding new ways to embrace the complex
world we live in on its own irregular
and imperfect terms